LINEAR TOPOLOGICAL SPACES OF CONTINUOUS ...linear topological spaces of continuous bounded vector-valued functions endowed with the uniform, strict, compact-open and weighted topologies

LINEAR TOPOLOGICAL SPACES OF

CONTINUOUS VECTOR-VALUED FUNCTIONS

Liaqat Ali Khan

PAAcademic Publications

Linear Topological Spaces of

Continuous Vector-Valued Functions

Authors: Liaqat Ali KhanAddress: Department of Mathematics

Faculty of ScienceKing Abdulaziz University, Jeddah

DOI: 10.12732/acadpubl.201301

Pages: 350

Typesetting system: LATEX

c©Academic Publications, Ltd., 2013

All rights reserved. This work may not be translated or copied in whole or in

part without the written permission of the publisher (Academic Publications,

Ltd., http://www.acadpubl.eu), except for scientific or scholarly analysis.

c©Academic Publications, Ltd., 2013http://www.acadpubl.eu

Contents

Introduction vi

Preface vii

Acknowledgements xiii

Chapter 1. The Strict and Weighted Topologies 11. Strict and related Topologies on Cb(X,E) 22. Weighted Topology on CVb(X,E) 203. Notes and Comments 24

Chapter 2. Completeness in Function Spaces 271. Completeness in the uniform topology 282. Completeness in the Strict Topology 333. Completion of (Cb(X,E), β) 384. Completeness in the Weighted Topology 415. Notes and Comments 43

Chapter 3. The Arzela-Ascoli type Theorems 451. Arzela-Ascoli Theorem for k and β Topologies 462. Arzela-Ascoli Theorem for Weighted Topology 533. Notes and Comments 61

Chapter 4. The Stone-Weierstrass type Theorems 631. Approximation in the Strict Topology 642. Approximation in the Weighted Topology 723. Weierstrass Polynomial Approximation 774. Maximal Submodules in (Cb(X,E), β) 795. The Approximation Property 826. Notes and Comments 87

Chapter 5. Maximal Ideal Spaces 891. Maximal Ideals in (Cb(X,A), β) 902. Maximal Ideal Space of (Cb(X,A), β) 943. Notes and Comments 99

iii

iv CONTENTS

Chapter 6. Separability and Trans-separability 1011. Separability of Function Spaces 1022. Trans-separability of Function Spaces 1073. Notes and Comments 114

Chapter 7. Weak Approximation in Function Spaces 1151. Weak Approximation in (Cb(X,E), β) 1162. Weak Approximation in CVo(X,E) 1203. Notes and Comments 126

Chapter 8. The Riesz Representation type Theorems 1271. Vector-valued Measures and Integration 1282. Integral Representation Theorems 1353. Notes and Comments 140

Chapter 9. Weighted Composition Operators 1411. Multiplication Operators on CVo(X,E) 1422. Weighted Composition Operators on CVo(X,E) 1523. Compact Weighted Composition Operators 1594. Notes and Comments 166

Chapter 10. The General Strict Topology 1671. Strict Topology on Topological Modules 1682. Essentiality of Modules of Vector-valued Functions 1773. Topological Modules of Homomorphisms 1804. Notes and Comments 187

Chapter 11. Mean Value Theorem and Almost Periodicity 1891. The Mean Value Theorem in TVSs 1902. Almost Periodic Functions with Values in a TVS 1943. Notes and Comments 203

Chapter 12. Non-Archimedean Function Spaces 2051. Compact-open Topology on C(X,E) 2062. Strict Topology on Cb(X,E) 2143. Maximal Ideals in NA Function Algebras 2224. Notes and Comments 227

Appendix A. Topology and Functional Analysis 2291. Topological Spaces 2302. Topological Vector Spaces 2433. Spaces of Continuous Linear Mappings 2614. Shrinkable Neighborhoods in a TVS 268

CONTENTS v

5. Non-Archimedean Functional Analysis 2726. Topological Algebras and Modules 2757. Measure Theory 2968. Uniform Spaces and Topological Groups 3009. Notes and Comments 310

Appendix. Bibliography 313

Appendix B. List of Symbols 329

Appendix C. Index 333

vi CONTENTS

Introduction

This monograph is devoted to the study of linear topological spacesof continuous bounded vector-valued functions endowed with the uni-form, strict, compact-open and weighted topologies. In the past fourdecades, several major results on vector-valued function spaces have beenextended to the non-locally setting. These include generalized versions ofsome classical results such as the Stone-Weierstrass theorem, the Arzela-Ascoli theorem, and the Riesz representation theorem. These also in-clude maximal ideal spaces of function algebras, separability and trans-separability, weak approximation, composition operators, general stricttopology on topological modules, the mean value theorem and almostperiodicity for vector-valued functions, and non-Archimedean functionspaces. Our main objective is to present recent developments in theseareas. Several examples and counter-examples are included in the text.Background material on Topology and Functional Analysis is includedin the Appendix. Also, an up-to-date bibliography is included to assistresearch in further studies.

AMS Subject Classification. [2000] Primary 46E10, 46E40; Sec-ondary 46A10, 28A25, 46H25, 47B33

Preface

The purpose of this monograph is to present an up-to-date study oflinear topological spaces of continuous bounded vector-valued functionsendowed with the uniform, strict, compact-open and weighted topologies.In the past four decades, several new results have had been obtainedin this direction. The main topics include generalized versions of someclassical results such as the Stone-Weierstrass theorem, the Arzela-Ascolitheorem, and the Riesz representation theorem. These also include thestudy of maximal ideal spaces of function algebras, separability and trans-separability, weak approximation, composition operators, general stricttopology on topological modules, the mean value theorem and almostperiodicity for vector-valued functions, and non-Archimedean functionspaces. Our main objective is to present recent developments in the non-locally convex setting, that is, in topological vector spaces, not necessarilylocally convex. So it covers both the cases: (a) locally convex spaces, (b)topological vector spaces which are not locally convex. We mention that”non-convexity” has also been the main theme in the monographs byL. Waelbroeck (1971), N. Adasch, B. Ernst and D. Keim (1978), N.J.Kalton, N.T. Peck and J.W. Roberts (1984), S. Rolewicz (1985) and A.Bayoumi (2003); see also the papers by W. Robertson (1958), V. Klee(1960a, 1960b) and J. Kakol (1985, 1987, 1990, 1992).

Our exposition relies mostly on the original papers (published during1972-2012) with some additional clarifications. We have tried to makethe text easily readable and as self-contained as possible. The only pre-requisites for reading the book are topology, functional analysis and mea-sure theory of the undergraduate level (e.g. the material given in H.L.Royden’s book: Real Analysis). For this purpose, we have included back-ground material on Topology and Functional Analysis (such as topolog-ical spaces, topological vector spaces, non-Archimedean functional anal-ysis, topological algebras and measure theory) in Appendix A. Also, anup-to-date bibliography is included to assist research in further studies.The monograph is intended basically a monograph for researchers work-ing in vector-valued function spaces. However, those working in this and

vii

viii PREFACE

related fields may conveniently choose some topics for a one or two se-mester course work. In fact, the author has been using portions of thetext in his lectures to M.Sc./M.Phil. students as a one semester coursebefore initiating their research in a relevant field of function spaces.

To introduce the material in its historical perspective, consider atopological space X and a topological vector space E over K (= R orC). Let C(X,E) (resp. Cb(X,E)) be the vector space of all continuous(resp. continuous and bounded) E-valued functions on X ; if E = K,these spaces are simply denoted by C(X) and Cb(X). If X is compact,then C(X,E) = Cb(X,E) and C(X) = Cb(X); in this case the uni-form topology (i.e. the sup norm topology) u is the appropriate one tostudy on C(X). After the appearance of M.H. Stone’s paper of 1937, thespace (C(X), u) in the case of compact X has been an intensively studiedmathematical object. Its interest arises in part from its rich structure:under the uniform topology u, C(X) is a Banach space; under pointwisemultiplication, it is an algebra; under the natural ordering, C(X,R) isa lattice (see, e.g., Kakutani (1941), Kaplansky (1947b), Hewitt (1948),Myers (1950), M. and S. Krein (1940), etc.). If X is not compact, theuniform topology u on C(X) is not well-defined since C(X) may containsome unbounded functions. However, in this case, the compact-opentopology k is the most useful topology to be considered on C(X) and onC(X,E) with E even a topological space. It has been studied by severalauthors over the past sixty years and significant contributions have beenmade in this field, among others, by Fox (1945), Arens (1946), Myers(1946), Hewitt (1948), Warner (1958), Wheeler (1976), McCoy (1980);see also the monographs by Semadeni (1971), Schmets (1983), McCoyand Ntantu (1988), Arkhangel’skii (1992) and Tkachuk (2011). If X isagain non-compact and we consider the spaces Cb(X) and Cb(X,E), thenboth u and k topologies are well-defined and we have k 6 u.

In 1958, R.C. Buck introduced the notion of strict topology β onCb(X,E) in the case of X locally compact and E a locally convex space.The problems discussed in the Buck’s paper (1958) are:

(1) Relationship between the β, k and u topologies on Cb(X,E) (e.g.k 6 β 6 u with k = u iff X is compact; k = β on u-bounded sets);

(2) Completeness of Cb(X,E), β);(3) Stone-Weierstrass theorems for (Cb(X), β) and Cb(X,En), β);(4) Characterization of maximal β-closed ideals in Cb(X);(5) Identification of the β-dual of Cb(X) with the space M(Bo(X))

of regular Boreal measures on X , via the integral representations;(6) The open problem whether or not (Cb(X), β) a Mackey space.

PREFACE ix

After the appearance of the Buck’s paper, a large number of pa-pers have appeared in the literature concerned with extending Buck’sresults to more general cases or studying further properties of β, andalso with obtaining some variants of β. In particular Todd (1965) andWells (1965)], independently, established the Stone-Weierstrass theoremfor (Cb(X), β). Todd (1965) also characterized the maximal β-closedCb(X)-submodules of Cb(X,E) while Wells (1965) identified the β-dualof Cb(X,E) with a certain space M(X,E∗) of E∗-valued measures on X.Conway (1967) and LeCam (1957), independently, proved that, if X islocally compact and paracompact, (Cb(X,E), β) is a Mackey space butit is not so in general. Collins and Dorroh (1968) proved that (Cb(X), β)has the approximation property. Dorroh (1969) also showed that β isthe finest locally convex topology on Cb(X) which agrees with k on u-bounded sets.

The next major development in the study of β topology on Cb(X)has been its extension to the case of a completely regular space X , given,independently, by Van Rooij (1967), Giles (1971), Fremlin, Garling andHaydon (1972), Gulick (1972), Hoffman-Jφregensen (1972), and Sentilles(1972). In fact, Fremlin, Garling and Haydon (1972) and, independently,Sentilles (1972) introduced on Cb(X) three types of strict topologies: thesubstrict topology βo, the strict topology β and the superstrict topologyβ1 with β being equivalent to the strict topology ‘β’ of Buck (1958) inthe case of X a locally compact space. They also identified the topo-logical duals of (Cb(X), βo), (Cb(X), β) and (Cb(X), β1) with the spacesMt(X), Mτ (X) andMσ(X) of tight, τ -additive and σ-additive measures,respectively, on X (see LeCam (1957), Varadarajan (1965) and Wheeler(1983) for detail of these spaces of measures). Subsequently, several au-thors have further explored the properties of βo, β and β1 for both Cb(X)and Cb(X,E), where E is a normed space or a locally convex space; see,e.g. Mosiman and Wheeler (1972), Wheeler (1973), Summers (1972),Haydon (1976), Cooper (1971), Choo (1979), Fontenot (1974), Katsaras(1975), Khurana (1978a, 1978b), Morishita and Khan (1997), Zafarani(1986, 1988), and also the survey papers by Hirschfeld (1978), Collins(1976) and Wheeler (1983), and monographs by Cooper (1978), Prolla(1977), Schmets (1983), Singh and Manhas (1993).

In 1967, Nachbin studied in detail the more general notion of weightedtopology ωV on certain subspaces CVb(X) and CVo(X) of C(X) andβ, k, and u were shown to be the special cases of ωV for suitable choicesof V consisting of non-negative upper-semicontinuous functions on X .Further work on ωV has been done by Summers (1969, 1971), Prolla

x PREFACE

(1971a, 1971b), Bierstedt (1973, 1975), and others. Singh and Summers(1988) and Singh and Manhas (1991, 1992) made an extensive studyof composition and multiplication operators, respectively on CVb(X,E).Another useful topology on Cb(X) is the σ-compact-open topology σwhich was introduced by Gulick (1972) and further studied by Gulick andSchmets (1972). The strict and related topologies have also been studiedon not necessarily ‘function spaces’. For instance, Wiwegar (1961) andCooper (1971, 1978) defined it in the form of ‘mixed topology’ on anormed space; Busby (1968) considered it on the double centralizer (ormultiplier) algebra Md(A) of Banach algebra A; Sentilles and Taylor(1969) defined it on a Banach A-module; Ruess (1977) studied it on anarbitrary locally convex space (see the survey paper by Collins (1976)).

In all the above mentioned investigations about Cb(X,E), E hasbeen assumed to be the scalar field or a locally convex space. Thecase of E a general topological vector space has been first considered byShuchat in (1972a, 1972b) where he established several useful approxima-tion results for (Cb(X,E), u) and characterized the dual of (Cb(X,E), u),with X a compact space and without assuming the local convexity ofE (see also Bierstedt (1973) and Wealbroeck (1971, 1973)). In 1979,Khan defined the β and other related topologies on Cb(X,E), whereX is a Hausdorff space and E any Hausdorff topological vector space,and showed that β has almost all the properties of the “strict topol-ogy” studied by the above authors. Further work in this direction hasbeen done in later years by Khan (1980 through 2011), Khan-Rowlands(1981, 1991), Katsaras (1981, 1983), Kalton (1983), Nawrocki (1985,1989), Abel (1987, 2004), Prolla (1993b), Khan-Thaheem (1997, 2002),Manhas-Singh (1998), Khan-Mohammad-Thaheem (1999, 2005), Khan-Oubbi (2005), Katsaras-Khan-Khan (2011), Katsaras (2011) and others.The purpose of this monograph is to present some results of these au-thors regarding the strict, weighted, and related topologies on Cb(X,E)and CVo(X,E) in the non-locally convex setting. We also study the strictand related topologies on any topological modules. Several examples andcounter-examples are included in the text. .

The monograph consists of twelve chapters which are organized asfollows.

In Chapter 1, we introduce and study the strict topology β (and therelated ones such as the uniform topology u, compact-open topology k, σ-compact-open topology σ) and also the more general notion of weightedtopology ωV on the function spaces Cb(X,E) (resp. CVb(X,E)) with E

PREFACE xi

not necessarily a locally convex topological vector space. Here, we aremainly concerned with their linear topological properties.

Chapter 2 deals with completeness and quasi-completeness of functionspaces under the above mentioned topologies.

In Chapter 3, we establish the Arzela-Ascoli type theorems whichcharacterize the β-compact and ωV -precompact subsets of Cb(X,E) andCVb(X,E), respectively.

Chapter 4 contains various generalizations of the famous Stone-Weiers-trass theorem and related results. It is perhaps the most important onefrom the point of view of its applications.

Chapter 5 deals with characterization of maximal and closed ideals in(Cb(X,E), β) and (C(X,E), k), E a topological algebra, and also theirmaximal ideal spaces ∆((Cb(X,E), β)) and ∆((C(X,E), k)).

In Chapter 6, we present various versions of the M and S. Kreintype theorem for separability and trans-separability in the β, k, σ, and utopologies.

Chapter 7 deals with the problem of weak approximation in (Cb(X,E),β) and (CVo(X,E), ωV ), using some basic terminology of Measure The-ory.

In Chapter 8, we first study the integral of functions in Cb(X,E)with respect to certain E∗-valued measures, and then obtain the Rieszrepresentation type theorems for the dual of function spaces under varioustopologies.

In Chapter 9, we study necessary and sufficient conditions for cer-tain maps Mθ,Mψ, Cϕ and Wπ,ϕ to be the multiplication or compositionoperators on the weighted spaces CVo(X,E) and CVb(X,E). We also in-clude characterizations of compact weighted composition operators andcompact multiplication operators.

We have treated the non-commutative analogue of the strict and re-lated topologies in Chapter 10. In fact, we introduce and study thegeneral strict and related topologies on a topological left A-module Y .We also study properties of the topological module HomA(A, Y ) of con-tinuous homomorphisms.

Chapter 11 consists of some results on the mean value theorem andalmost periodicity for vector-valued functions.

In Chapter 12, we have considered the study of compact-open andstrict topologies on non-Archimedian function spaces.

Finally, in the Appendix A, we have included basic definitions and re-sults on Topological Spaces, Topological Vector Spaces, Non-Archimedean

xii PREFACE

Functional Analysis, Topological Algebras, Topological Modules, Mea-sure Theory and Uniform Spaces. Most of their proofs (with the excep-tion of some selected results) are omitted since they are available in anystandard book on these topics.

At the end of each chapter, the section ”Notes and Comments” con-tains references to the literature and gives some further information onthe results in the text. Some references are also included in the text toclarify the source of information. The bibliography lists some papers thatare not specifically referred but can be useful for supplementary mate-rial. We have treated the cases of strict topology and weighted topologyin separate sections in most of the chapters. Any one interested only inmore familiar results on uniform, strict and compact-open topologies caneasily skip the material on the more abstract notion of weighted topology.

Liaqat Ali Khan

Acknowledgements

I wish to express my sincere gratitude to Professors S. M. Khaleelulla,Mati Abel and Taqdir Husain for reading the first draft of manuscriptand offering several useful comments for improvement. I also wish tothank Keith Rowlands (the supervisor of my Ph.D. thesis of 1977), A.V.Arkhangel’skii, Surjit Singh Khurana, and my colleagues Abdul Latif,Naseer Shahzad and Lakhdar Meziani for discussions and consultationduring the preparation of this monograph. I thank Mr. M. Sharaz forhis skillful typing of the material.

Finally, the author wishes to acknowledge with thanks the Deanshipof Scientific Research, King Abdulaziz University, Jeddah, for their tech-nical and financial support.

xiii

CHAPTER 1

The Strict and Weighted Topologies

In this chapter, we study the uniform, strict, compact-open, σ-compact-open, weighted and other related topologies on Cb(X,E) or CVb(X,E)and their linear topological properties in the non-locally convex setting.

Throughout this monograph, we shall assume, unless stated other-wise, that X is a completely regular Hausdorff space and E a non-trivialHausdorff topological vector space (in short, a TVS) over the field K = (Ror C), and we let W denote a base of open balanced neighborhoods of 0in E. Most of the undefined terminology can be found in the AppendixA (A.1-A.8).

1

2 1. THE STRICT AND WEIGHTED TOPOLOGIES

1. Strict and related Topologies on Cb(X,E)

The notion of the strict topology β on Cb(X,E) was first introducedby Buck [Buc58] in 1958 in the case of X a locally compact space andE a locally convex TVS. Since then a large number of papers have ap-peared in the literature concerned with extending the results containedin Buck’s paper; see the Preface and the list of references. In this sec-tion, a general method for defining linear topologies on Cb(X,E), calledthe S-topologies, is considered first. Using the notion of an S-topology,the uniform topology u, strict topology β, compact-open topology k,σ-compact-open topology σ, countable-open-topology σo, and pointwisetopology p are introduced. We include here some results from [Kh79,KR91, Kat81], where E is a Hausdorff TVS but not necessarily locallyconvex.

Definition. A function f : X → E is said to bounded if f(X) is abounded subset of E, i.e. for each W ∈ W, there exists an r > 0 suchthat f(X) ⊆ λW for all λ ∈ K with |λ| ≥ r. f : X → E is said to vanishat infinity if, for each W ∈ W, the set

FW = x ∈ X : f(x) /∈ W

is relatively compact in X ; in particular, a function ϕ : X → K is saidto vanish at infinity if, for each ε > 0, the set

Fε = x ∈ X : |ϕ(x)| ≥ ε

is relatively compact in X . The support of f : X → E is defined as theclosure of the set x ∈ X : f(x) 6= 0 in X and we write

supp(f) = cl − x ∈ X : f(x) 6= 0.

Let C(X,E) be the vector space of all continuous E-valued functionson X and Cb(X,E) (resp. Co(X,E), Coo(X,E)) the subspace of C(X,E)consisting of those functions which are bounded (resp. vanish at infinity,have compact support. When E = K (= R or C), then these spaceswill be denoted by C(X), Cb(X), Co(X) and Coo(X). Let B(X) denotethe see of all bounded K-valued functions on X , and let Bo(X), (resp.Boo(X), Bσ(X), Bσo(X), Bp(X)) denote the subset of B(X) consisting ofthose functions which vanish at infinity (resp. have compact, σ-compact,countable, finite support). We shall denote by B(X)⊗E the vector spacespanned by the set of all functions of the form ϕ⊗ a, where ϕ ∈ B(X),a ∈ E, (ϕ ⊗ a)(x) = ϕ(x)a (x ∈ X). F (X,E) = EX denotes the vectorspace of all functions f : X → E.

1. STRICT AND RELATED TOPOLOGIES ON Cb(X,E) 3

Since we shall frequently use the notion of a function vanishing atinfinity, the following three lemmas are useful in this regard.

Lemma 1.1.1 Let ϕ : X → K be a function. Then:(a) ϕ vanishes at infinity iff, for each ε > 0, there exists a compact

set Kε ⊆ X such that

| ϕ(x) |< ε for all x ∈ X\Kε.

(b) If ϕ is non-negative upper semi-continuous (in particular, contin-uous), then ϕ vanishes at infinity iff for every ε > 0, the set Fε = t ∈X : |ϕ(t)| ≥ ε is compact iff for every ε > 0, there exists a compact setKε ⊆ X such that

| ϕ(x) |< ε for all x ∈ X\Kε.

Proof. (a) Suppose ϕ vanishes at infinity, and let ε > 0. Then theset Fε = x ∈ X : |f(x)| ≥ ε is relatively compact in X . Now the setKε = Fε is compact. If x ∈ X\Kε, then x ∈ X\Fε; hence | ϕ(x) |< ε.

Conversely, let ε > 0, and let Kε be the compact set such that |ϕ(x) |< ε for all x ∈ X\Kε. If x /∈ Kε, | ϕ(x) |< ε and so clearly x /∈ Fε.Hence Fε ⊆ Kε. Now Kε, being a compact subset of the Hausdorff spaceX, is closed. Therefore Fε ⊆ Kε = Kε. Then Fε, being a closed subset ofcompact set Kε, is compact.

(b) If ϕ is a non-negative upper semi-continuous, then the set x ∈X : ϕ(x) < ε = x ∈ X : |ϕ(x)| < ε is open, or that Fε is closed. Thenthe result follows from (a).

Analogously, we also obtain:

Lemma 1.1.2 Let f : X → E be a function. Then:(a) f vanishes at infinity iff, for each W ∈ W, there exists a compact

set KW ⊆ X such that

f(x) ∈ W for all x ∈ X\KW .

(b) If f is continuous, then f vanishes at infinity iff, for each W ∈W, the set FW = x ∈ X : f(x) /∈ W is compact iff for each W ∈ W,there exists a compact set KW ⊆ X such that

f(x) ∈ W for all x ∈ X\KW .

Lemma 1.1.3. Let f : X → E be a function and suppose ϕf ∈B(X,E) for each ϕ ∈ Bo(X). Then f ∈ B(X,E).

Proof. Suppose f is not bounded. Then there exist a sequence xn ⊆X , and a W ∈ W such that f(xn) /∈ n2W . Define ϕ : X → R by

ϕ(x) = 1/n if x = xn, and ϕ(x) = 0 if x 6= xn (n = 1, 2, ...).


Then ϕ ∈ Bo(X). [Clearly ϕ is bounded. To show that ϕ vanishes atinfinity, let ε > 0. Choose N ≥ 1 such that 1

N< ε. Then K = x1, ..., xN

is compact and if x /∈ K = x1, ..., xN, then

ϕ(x) ≤1

N + 1<

1

N< ε.

Alternatively,

x ∈ X : ϕ(x) ≥ ε = x ∈ X :1

N≤ ϕ(x) ≤ 1 = x1, ..., xN,

which, being a finite set, is compact in X .] Now, clearly

ϕ(xn)f(xn) =1

nf(xn) /∈ nW for all n = 1, 2, ....

This implies that ϕf is not bounded, which contradicts our hypothesis.Thus f ∈ B(X,E).

Remarks. (1) The above result need not hold if X is not Hausdorff.For example, let X = N with the topology

τ = N ∪ An : n ∈ N,

where An = i : i ∈ N, i < n for every n ∈ N. This is T0 but not T1(hence also not Hausdorff). In this case, the only closed compact subsetof X is ∅, so the only relatively compact set is ∅. Consequently, the onlymember of Bo(N) is the constant function with value 0. Now considerthe TVS E = R and the identity function f : N → R, f(n) = n, n ∈ N.Then ϕf is bounded for every ϕ ∈ Bo(N), because there is only the zerofunction ϕ to consider; but f is not bounded.

(2) Clearly, a finite subset A of a topological space X is always com-pact; but need not be closed. So it need not be relatively compact (e.g.if X is neither T1 nor regular).

Notation. For any ϕ ∈ B(X) and W ∈ W, we define

N(ϕ,W ) = f ∈ Cb(X,E) : ϕ(x)f(x) ∈ W for all x ∈ X.

If A ⊆ X and W ∈ W, then we write N(A,W ) to mean

N(A,W ) := N(χA,W ) = f ∈ Cb(X,E) : f(A) ⊆W,

where χA is the characteristic function of A.

Lemma 1.1.4. Let ϕ ∈ B(X) and W ∈ W. Then(a) If V ∈ W with V ⊆W, then N(ϕ, V ) ⊆ N(ϕ,W ).(b) If U, V ∈ W with U + V ⊆ W, then N(ϕ, U) + N(ϕ, V ) ⊆

N(ϕ,W ).(c) For any λ( 6= 0) ∈ K, N(ϕ, λW ) = λN(ϕ,W ).


(d) N(ϕ,W ) is absorbing.(e) If W is balanced (resp. convex), then N(ϕ,W ) is also balanced

(resp. convex).

Proof. (a) Let f ∈ N(ϕ, V ). Then, for any x ∈ X, ϕ(x)f(x) ∈ V ⊆Wand so f ∈ N(ϕ,W ).

(b) Let f ∈ N(ϕ, U) and g ∈ N(ϕ, V ). Then, for any x ∈ X,ϕ(x)f(x) ∈ U and ϕ(x)g(x) ∈ V, and so

[ϕ(f + g)](x) = ϕ(x)f(x) + ϕ(x)g(x) ∈ U + V ⊆W ;

that is, f + g ∈ N(ϕ,W ). Hence N(ϕ, V ) +N(ϕ, V ) ⊆ N(ϕ,W ).(c) If λ( 6= 0) ∈ K,

N(ϕ, λW ) = f ∈ Cb(X,E) : ϕ(x)f(x) ∈ λW for all x ∈ X

= λ1

λf ∈ Cb(X,E) : ϕ(x)f(x) ∈ λW for all x ∈ X

= λ1

λf ∈ Cb(X,E) : ϕ(x)(

1

λf)(x) ∈ W for all x ∈ X

= λg ∈ Cb(X,E) : ϕ(x)g(x) ∈ W for all x ∈ X

= λN(ϕ,W ).

(d) Let f ∈ Cb(X,E). Since (ϕf)(X) is bounded in E and W isabsorbing, there exists r > 0 such that (ϕf)(X) ⊆ λW for all | λ |≥ r.Hence, by (c),

ϕf ∈ N(ϕ, λW ) = λN(ϕ,W ) for all | λ |≥ r,

showing that N(ϕ,W ) is absorbing.(e)(i) Suppose W is balanced, and let λ ∈ K with |λ| ≤ 1. Then

λW ⊆W and so, by (a) and (c),

λN(ϕ,W ) = N(ϕ, λW ) ⊆ N(ϕ,W ).

Hence N(ϕ,W ) is balanced.(ii) Suppose W is convex, and let t ∈ R with 0 < t < 1 and f, g ∈

N(ϕ,W ). Then tW + (1− t)W ⊆ W , and so by (b) and (c),

tN(ϕ,W ) + (1− t)N(ϕ,W ) = N(ϕ, tW ) +N(ϕ, (1− t)W )

⊆ N(ϕ, tW + (1− t)W ) ⊆ N(ϕ,W ).

Hence N(ϕ,W ) is convex.

We now describe a general method of defining linear topologies onCb(X,E).


Definition. Let S be any subset of B(X). We define the S-topologyon Cb(X,E) to be the linear topology which has a subbase of neighbor-hoods of 0 consisting of all sets of the form N(ϕ,W ), where ϕ ∈ S andW ∈ W.

Definition. [Gil71] Let ϕ, ϕ1 ∈ B(X). Then ϕ1 is said to dominateϕ if there exists a t > 0 such that

|ϕ(x)| ≤ t |ϕ1(x)| for all x ∈ X.

Lemma 1.1.5. [Gil71, Kh79]

Let S and S1 be two subsets of B(X). If each element of S isdominated by an element of S1, then the S-topology on Cb(X,E) isweaker than the S1-topology.

Proof. Let N be any Sneighborhood of 0 in Cb(X,E), and suppose

N ⊇n⋂

i=1

N(ϕi,Wi),

where ϕ1, ..., ϕn ∈ S and W1, ...,Wn ∈ W. For each ϕi (i = 1, ..., n),choose a ti > 0 and a ψi ∈ S1 such that

|ϕi(x)| ≤ ti|ψi(x)| for all x ∈ X.

Let

N1 =n⋂

i=1

N(ψi,1

tWi),

where t = maxt1, ..., tn. Then N1 is an S1-neighborhood of 0 inCb(X,E). Further N1 ⊆ N . [Let g ∈ N1. Then, for any x ∈ X and1 ≤ i ≤ n,

ψi(x)g(x) ∈1

tWi;

since |ϕi(x)| ≤ ti|ψi(x)| ≤ t|ψi(x)| and Wi is balanced,

ϕi(x)g(x) ∈ tψi(x)Wi ⊆Wi.

Therefore g ∈ ∩ni=1N(ϕi,Wi) ⊆ N .] Hence N is an S1-neighborhood of0 in Cb(X,E).

Definition. Using the notion of an S-topology, we now introduce thestrict topology, the σ-compact-open topology, and other related topolo-gies on Cb(X,E), as follows.


S S-TopologyB(X) uniform topology uBo(X) strict Topology βBoo(X) compact-open topology kBσ(X) σ-compact-open topology σBσo(X) countable-open-topology σoBp(X) pointwise topology p

Remark 1.1.6. It easily follows from Lemma 1.1.5 that:(1) The u-topology on Cb(X,E) is the same as the 1-topology,

where 1 ∈ B(X) is the function identically 1 on X .(2) The topology k (resp. σ, σo, p) on Cb(X,E) is the linear topology

which has a subbase of neighborhoods of 0 consisting of all sets of theform N(K,W ), where K is any compact (resp. σ-compact, countable,finite) subset of X and W ∈ W.

(3) SinceBp(X) ⊆ Boo(X) ⊆ Bo(X) ⊆ B(X)

andBp(X) ⊆ Bσo(X) ⊆ Bσ(X) ⊆ B(X),

it is easy to verify that p 6 k 6 β 6 u and p 6 σo 6 σ 6 u.Further, note that if X is compact, then k = u; if X is discrete, then

p = k (since every compact set in a discrete space is finite).We mention that the topologies p and k can also be considered on

the larger space C(X,E).The following lemma gives us a convenient form for a base of neigh-

borhoods of 0 in Cb(X,E) for each of the topologies defined above.

Lemma 1.1.7. [Kh79]

Let S denote any one of the sets B(X), Bo(X), Boo(X), Bσ(X),Bσo(X), or Bp(X). Then the S-topology on Cb(X,E) has a base ofneighborhoods of 0 consisting of all sets of the form N(ϕ,W ), whereϕ ∈ S with 0 ≤ ϕ ≤ 1 and W ∈ W.

Proof. Let N be any Sneighborhood of 0, and suppose

N ⊇m⋂

i=1

N(ϕi,Wi),

where ϕ1, ..., ϕm ∈ S and W1, ...,Wm ∈ W. Let t = max1≤i≤m

‖ϕi‖. If

t > 0, choose a W ∈ W with tW ⊆ ∩mi=1Wi. Define

ϕ(x) = max1≤i≤m

|ϕi(x)|

t

(x ∈ X).


Then ϕ ∈ S, 0 ≤ ϕ ≤ 1. Further, N(ϕ,W ) ⊆ N . [Let g ∈ N(ϕ,W ).Since, for each x ∈ X and 1 ≤ i ≤ m, |ϕi(x)| ≤ tϕ(x) and ϕ(x)g(x) ∈ W,we have

ϕi(x)g(x) ∈ tW ⊆Wi;

that is, g ∈ ∩mi=1N(ϕi,Wi) ⊆ N .] If t = 0, then each ϕi = 0 and so

N(ϕi,Wi) = f ∈ Cb(X,E) : 0f(x) ∈ W for all x ∈ X = Cb(X,E).

Hence N = Cb(X,E), and so, if we take ϕo = 0, we have N ⊇ N(ϕo,W )for any W in W. Thus the Sneighborhoods of 0 have a base of therequired form.

We first study some basic properties of the topology β and its relationwith the topologies k and u. (For convenience, the properties of thetopologies σo and σ will be given separately in a later theorem.)

Theorem 1.1.8. [Buc58, Kh79]

(a) p 6 k 6 β 6 u.(b) If X is completely regular, then:

(i) u = β iff X is compact.(ii) β = k iff every σ-compact subset of X is relatively compact.

(c) u and β have the same bounded sets in Cb(X,E).(d) β = k on u-bounded subsets of Cb(X,E).(e) A sequence fn in Cb(X,E) is β-convergent iff it is u-bounded

and k-convergent.Proof. (a) Since every element of Bo(X) is dominated by 1 ∈ B(X),

it follows from Lemma 1.1.5 that the Bo(X)-topology is weaker than the1-topology, that is, β 6 u. Similarly, since Boo(X) ⊆ Bo(X), we havek 6 β.

(b) (i) Suppose u 6 β. Then, by Lemma 1.1.7, for anyW ∈ W, thereexist a ϕ ∈ Bo(X) with 0 ≤ ϕ ≤ 1 and a V ∈ W such that

N(ϕ, V ) ⊆ N(1,W ).

If E\W 6= ∅, let c ∈ E\W and choose t > 0 such that c ∈ tV . If X isnot compact, then X\F 6= ∅ for every compact set F in X . Since ϕ ∈B0(X), the set K = cl − x ∈ X : ϕ(x) > 1/t is compact in X . Letxo ∈ X\K. By complete regularity, there exists a ψ ∈ Cb(X) such that

0 ≤ ψ ≤ 1, ψ(xo) = 1 and ψ(K) = 0.

Let g = ψ ⊗ c. Then g ∈ N(ϕ, V ). [Let x ∈ X . If x ∈ K, then ψ(x) = 0and so

ϕ(x)g(x) = ϕ(x)ψ(x)c = 0 ∈ V ;


if x /∈ K, then tϕ(x) < 1 and so

ϕ(x)g(x) = ϕ(x)ψ(x)c ∈ ϕ(x)ψ(x)tV ⊆ V .]

But g /∈ N(1,W ) (since g(xo) = ψ(xo)c = c /∈ W ), which is a contra-diction. If W = E, choose a W0 in W such that W0 ⊆ E and then argueas above with W0 replacing W . Thus X is compact.

Conversely, if X is compact, then k = u and so, by (a), β = u.(b) (ii) If every σ-compact subset of X is relatively compact, then it

is easy to show that β 6 k.Conversely, let β 6 k, and suppose that there is a set G = ∪∞n=1Kn

(Kn compact in X) which is not relatively compact. Then, for eachcompact set F in X , G\F 6= ∅. Let

ϕ =∞∑

n=1

1

2nχKn .

Then ϕ ∈ Bo(X). [Clearly ϕ is bounded. To show that ϕ vanishes atinfinity, let ε > 0. Choose N ≥ 1 such that 1

N< ε. Now K = UN

n=1Kn iscompact. If x /∈ K, then χKn(x) = 0 for all n = 1, 2, ..., N and so

ϕ(x) =∑

n≥N+1

1

2nχKn(x) ≤

∑

n≥N+1

1

2n=

1

2N<

1

N< ε.]

Clearly, ϕ = 0 outside of G. Since β 6 k, for any balanced W ∈ W,there exist a compact set K in X and a V ∈ W such that

N(K, V ) ⊆ N(ϕ,W ).

If E\W 6= ∅, let d ∈ E\W , and yo ∈ G\K. Choose a ψ ∈ Cb(X) with

0 ≤ ψ ≤ (1/ϕ(yo)), ψ(yo) = 1/ϕ(yo), and ψ(K) = 0.

Let h = ψ ⊗ d. Then, since h(K) = 0 ⊆ V, h ∈ N(K, V ) but, since

ϕ(yo)h(yo) = d /∈ W,

we have h /∈ N(ϕ,W ), a contradiction. If W = E, choose a Wo in Wsuch that Wo ⊆ E and then argue as above with Wo replacing W .

(c) Since β 6 u, it follows that every u-bounded subset of Cb(X,E)is β-bounded.

Conversely, suppose there is a set A ⊆ Cb(X,E) which is β-boundedbut not u-bounded. Then there exist a sequence fn ⊆ A, xn ⊆ X ,and a W ∈ W such that

fn(xn) /∈ n2W.

Let ϕ(x) = 1/n if x = xn, and ϕ(x) = 0 if x 6= xn (n = 1, 2, ...). Then,as seen in the proof of Lemma 1.1.3, ϕ ∈ Bo(X) but ϕ(xn)fn(xn) /∈ nW ;


that is, fn and hence A is not β-bounded. This contradiction provesthe result.

(d) Let A be a u-bounded subset of Cb(X,E). Since k 6 β, it is suffi-cient to show that the identity mapping i : (A, k)→ (A, β) is continuous.Suppose f ∈ A and let fα be a net in A such that

k- limαfα = f.

Let ϕ ∈ Bo(X) with 0 ≤ ϕ ≤ 1, and let W ∈ W. We show that thereexists an index αo such that

ϕ(x)(fα(x)− f(x)) ∈ W for all x ∈ X and α > αo.

Choose a balanced V ∈ W such that V +V ⊆W . Since A is u-bounded,choose a t > 1 such that

g(x) ∈ tV for all x ∈ X and g ∈ A.

Since ϕ ∈ Bo(X), there exists a compact set K in X such that

ϕ(x) <1

tfor x /∈ K.

Let ϕ1 = ϕχK . Then ϕ1 ∈ Boo(X) and so, since k-limα fα = f , thereexists an index αo such that

ϕ1(x)(fα(x)− f(x)) ∈ V for all x ∈ X and α > αo.

Let y be any point of X . If y ∈ K, then ϕ1 = ϕ, so that

ϕ(y)(fα(y)− f(y)) ∈ V ⊆W for all α > αo.

If y /∈ K, then tϕ(y) < 1

ϕ(y)(fα(y)− f(y)) ∈ ϕ(y)tV − ϕ(y)tV ⊆ V + V ⊆W for all α.

Thus β-limα fα = f and so i : (A, k) → (A, β) is continuous; that is,β|A ⊆ k|A, as required.

(e) Suppose fn is β-convergent in Cb(X,E). Since a convergent se-quence in a TVS is always bounded, it follows that fn is β-bounded andhence u-bounded by (c). Moreover, since k 6 β, fn is k-convergent.

Conversely, suppose fn is u-bounded in Cb(X,E) and k- limn→∞

fn =

f , where f is in Cb(X,E). The f, fn : n = 1, 2.. is contained in au-bounded subset, say A, of Cb(X,E). By (d) β|A = k|A, and so, itfollows that β- lim

n→∞fn = f .

The following remarks contain some counterexamples related to theabove results.


Remark 1. Since k 6 β, every β-bounded set is k-bounded. Butthe converse is not true, in general; Hirschfeld ([Hir74], p.20) has giventhe following example of a k-bounded set which is not β-bounded.

Example. Let X = N, so that Cb(N) = ℓ∞. Define ϕn = nχ[n,∞);that is,

ϕn(m) =

n if m > n0 if m < n

(m ∈ X).

Let K ⊆ X = N be any compact set. Then K is finite, there exists aninteger no such that ϕn|K = 0 for all n > no. Since K is arbitrary, k-lim

n

ϕn = 0. So ϕn is k-bounded. On the other hand, supn‖ϕn‖ = ∞and so ϕn is not u-bounded or equivalently ϕn is not β-bounded.

Remark 2. It is easy to see that, if a net fα in Cb(X,E) is u-bounded and k-convergent, then fα is β-convergent. However, theconverse is not true; Gulick ([Gu72], p.166) has given an example of aβ-convergent net which is not u-bounded as follows.

Example. Let X = Ω, the space of ordinals less than the firstuncountable ordinal ω, with order topology. Let Ωo ⊆ Ω the collection ofnon-limit ordinals. Let fλ = nλχλ where λ is nλ(0 < nλ < ∞) greater

than a limit ordinal. The fλβ→ 0. But fλ : λ ∈ Ω\Ωo is not u-bounded.

We now discuss the relation of β with the Buck’s strict topology[Buc58]. Let β ′ denote the Co(X)-topology, which is the Buck’s originalstrict topology, on Cb(X,E).

Theorem 1.1.9. [Kh79] If X is locally compact, β ′ = β on Cb(X,E).Proof. It is clear that β ′ 6 β, since Co(X) ⊆ Bo(X). Now, let ϕ ∈

Bo(X). By Lemma 1.1.5, it is sufficient to show that there exists afunction ψ in Co(X) which dominates ϕ. For each n, the set

x ∈ X : |ϕ(x)| >1

2n

has a compact closure Kn say in X . Since X is locally compact, thereexist function ψn ∈ Coo(X) such that

0 ≤ ψn ≤ 1, ψn(Kn) = 1, and supp (ψn) ⊆ Kn.

Let ψ =∞∑n=1

12nψn. Clearly Kn ⊆ Kn+1 and ψ ∈ Co(X). To show |ϕ| ≤ ψ

on X , let y ∈ X . If y ∈∞⋃n=1

Kn, then y ∈ KN+1\KN for some N > 1, so


that

ψ(y) ≥∑

n>N

1

2n=

1

2N> |ϕ(y)|.

If y /∈∞⋃n=1

Kn, then ϕ(y) = 0 but ψ(y) > 0. Thus |ϕ(y)| ≤ ψ(y).

Consequently β 6 β ′.

Remark. If X is not locally compact, then Co(X,E) may be thetrivial vector space 0.

Counter-example. Let X = Q, the space of rationals and E = R.Then Co(Q,R) = 0, as follows. Suppose ϕ( 6= 0) ∈ Co(Q,R). Choosey ∈ Q with ϕ(y) 6= 0. We may suppose ϕ(y) > 0. Choose a, b > 0 suchthat a < ϕ(y) < b. Then the set

U = x ∈ Q : a < ϕ(x) < b = ϕ−1(a, b)

is non-empty (since y ∈ U) and open (since ϕ is continuous). Sinceϕ ∈ Co(Q,R), the set

V = x ∈ Q : |ϕ(x)| > a = x ∈ Q : ϕ(x) > a or ϕ(x) < −a

is relatively compact. Now, for any x ∈ U, ϕ(x) > a and so x ∈ V ; thatis, U ⊆ V . Then it follows that U is compact. Hence every point in Uhas a compact neighborhood. Upon translating U along Q, we arrive atthe contradiction that Q is locally compact.

Theorem 1.1.10. [Buc58, Kh79] Let E be any TVS. If X is locallycompact, then:

(a) Coo(X,E) is β-dense in Cb(X,E).(b) Coo(X,E) is k-dense in C(X,E).(c) Coo(X,E) is u-dense in Co(X,E).(d) Co(X,E) is u-closed in Cb(X,E).Conversely, if E is a non-trivial TVS, then each of the condition (a),

(b) implies that X is locally compact; if, in addition, Co(X,E) 6= 0,then (c) also implies that X is locally compact.

Proof. Suppose X is locally compact.(a) Let f ∈ Cb(X,E). Let ϕ ∈ Bo(X), 0 ≤ ϕ ≤ 1, and W ∈ W. Let

K ⊆ X be a compact set such that ϕ(x)f(x) ∈ W for x /∈ K. Sine X islocally compact, choose a function ψ ∈ Coo(X) such that 0 ≤ ψ ≤ 1 andψ(K) = 1. Let g = ψf . Then g ∈ Coo(X) and

ϕ(x)[g(x)− f(x)] = ϕ(x)[ψ(x)− 1]f(x)= 0 ∈ W if x ∈ K∈ (ψ(x)− 1)W ⊆W if x /∈ K.


Thus g − f ∈ N(ϕ,W ), and so f belongs to the β-closure of Coo(X,E);that is, Coo(X,E) is β-dense in Cb(X,E).

(b) Let f ∈ C(X,E). Let K be a compact subset of X . Choose aψ ∈ Coo(X) such that ψ = 1 on K. Then ψf ∈ Coo(X,E) and

(ψf)(x)− f(x) = 0 for all x ∈ K.

In particular, for any W ∈ W, ψf − f ∈ N(K,W ), and so f belongs tothe k-closure of Coo(X,E) Thus Coo(X,E) is k-dense in C(X,E).

(c) Let f ∈ Co(X,E). To show that f belongs to the u-closure ofCoo(X,E), let W ∈ W be balanced. Now K = x ∈ X : f(x) /∈ W iscompact. Choose ψ ∈ Coo(X) such that 0 ≤ ψ ≤ 1 and ψ = 1 on K.Then clearly g = ψf ∈ Coo(X,E). Let x ∈ X. If x ∈ K, then

f(x)− g(x) = f(x)− f(x) = 0 ∈ W.

If x ∈ X\K, then

f(x)− g(x) = [ψ(x)− 1]f(x) ∈ [ψ(x)− 1]W ⊆W.

Hence g − f ∈ N(X,W ). Thus Coo(X,E) is u-dense in Co(X,E).

(d) Let f ∈ F (X,E) with f ∈ Co(X,E)u. Then there exists a se-

quence fn ⊆ Co(X,E) such that fnu−→ f . We need to show that

f ∈ Co(X,E).f is continuous on X , as follows Let xo ∈ X and W ∈ W. We show

that there exists a neighborhoodG of xo inX such that, f(y)−f(xo) ∈ Wfor all y ∈ G.Choose a balanced V ∈ W such that V + V + V ⊆ W .Since fn

u−→ f, there exists n0 ≥ 1 such that

fn(x)− f(x) ∈ V for all n ≥ n0 and all x ∈ X. (1)

Since fn0is continuous at xo, there exists a neighborhood G of xo in X

such that

fn0(y)− fn0

(xo) ∈ V for all y ∈ G. (2)

Then, for any y ∈ G, using (1) and (2),

f(y)− f(xo) = (f(y)− fn0(y)) + (fn0

(y)− fn0(xo)) + (fn0

(xo)− f(xo))

∈ V + V + V ⊆W.

Hence f is continuous on X . Thus f ∈ Cb(X,E)We now need to verify that f vanishes at infinity. Let W ∈ W.

Choose a balanced V ∈ W such that V +V ⊆W . Since fnu−→ f , there

exists k0 ≥ 1 such that

fn(x)− f(x) ∈ V for all n ≥ N0 and x ∈ X .


Since fk0 ∈ C0(X,E), there exists a compact set K ⊆ X such that

fk0(x) ∈ V for all x ∈ X\K.

Then, for any x ∈ X\K,

f(x) = (f(x)− fk0(x)) + fk0(x) ∈ V + V ⊆W .

Therefore f ∈ Co(X,E). Thus Co(X,E) is u-closed in Cb(X,E).Conversely, suppose E is a non-trivial TVS.(i) Suppose Coo(X,E) is β-dense in Cb(X,E) but thatX is not locally

compact. There there exists a y ∈ X which has no compact neighbor-hood. Consequently g(y) = 0 for all g ∈ Coo(X,E). Now, let W ∈ Wbe balanced. Since E is non-trivial, we may assume W 6= E and choosea ∈ E\W. Consider the non-zero constant function h : X → E functiongiven by h(x) = a for all x ∈ X. Then clearly h ∈ Cb(X,E). Now,

N(y,W ) = f ∈ Cb(X,E) : f(y) ∈ W

is a pneighborhood of 0 in Cb(X,E) and clearly, for any g ∈ Coo(X,E),

h(y)− g(y) = a− 0 /∈ W ;

i.e., h− g /∈ N(y,W ). Therefore h does not belong to the p-closure ofCoo(X,E); since p ≤ β, h does not belong to the β-closure of Coo(X,E).Therefore Coo(X,E) is not β-dense in Co(X,E), a contradiction. ThusX is locally compact.

(ii) Suppose Coo(X,E) is k-dense in C(X,E) but that X is not locallycompact. There there exists a y ∈ X which has no compact neighbor-hood. Consequently g(y) = 0 for all g ∈ Coo(X,E). It follows (as in (i))that, if h is any non-zero constant function in C(X,E), then h does notbelong to the p-closure, and hence to the k-closure of Coo(X,E) (sincep ≤ k); that is Coo(X,E) is not k-dense in C(X,E).

(iii) Suppose Co(X,E) 6= 0 and that Coo(X,E) is u-dense in Co(X,E)but that X is not locally compact. There there exists a y ∈ X which hasno compact neighborhood. Consequently g(y) = 0 for all g ∈ Coo(X,E).It follows (as in (i)) that, if h is any non-zero function in Co(X,E), thenh does not belong to the p-closure, and hence not to the u-closure ofCoo(X,E); that is Coo(X,E) is not u-dense in Co(X,E).

If X is not assumed to be locally compact, then a result related tothe above theorem can be obtained by imposing a stronger condition onE, as follows.

Theorem 1.1.11. [Kh95] Suppose E is a locally bounded TVS. ThenCb(X,E) is k-dense in C(X,E).


Proof. Let f ∈ C(X,E), and let K be a compact subset of X andW ∈ W. By hypothesis, we can choose a bounded set V ∈ W. Let Sbe a closed shrinkable neighborhood of 0 in E with S ⊆ V (see SectionA.4). The Minkowski functional ρS of S is continuous and positivelyhomogeneous, and so, for any r > 0, the function hr : E → E defined by

hr(a) =

a if a ∈ rS

rρS(a)

a if a /∈ rS

is continuous with hr(E) ⊆ rS. This implies that the function hr f ∈Cb(X,E). [Clearly, hr f is continuous on X . To show that hr f isbounded on X , let U ∈ W. Choose λ > 0 with V ⊆ λU. Then

hr(f(X)) ⊆ hr(E)) ⊆ rS ⊆ rV ⊆ rλU.]

Choose t ≥ 1 with f(K) ⊆ tS. Then ht f ∈ Cb(X,E). Further, for anyx ∈ K, f(x) ∈ tS and so

ht(f(x))− f(x) = f(x)− f(x) = 0 ∈ W.

Thus Cb(X,E) is k-dense in C(X,E).

We next consider the properties of σ and σo topologies on Cb(X,E).

Lemma 1.1.12. [KR91] Suppose that A,B ⊆ X and that V,W ∈W are such that N(A, V ) ⊆ N(B,W ). Then:

(i) If W 6= E, then B ⊆ A.(ii) If B is non-empty, then V ⊆W .Proof. (i) Suppose B * A, and let x ∈ B\A. Since X is completely

regular, there exists a ϕ in Cb(X) such that ϕ(x) = 1 and ϕ(A) = 0.Then clearly supp (ϕ) ∩ A = ∅. Choose a ∈ E\W and let f = ϕ ⊗ a.Then f(A) = 0 ⊆ V , and so f ∈ N(A, V ); but f(x) = a /∈ W and sof /∈ N(B,W ). This contradiction implies that B ⊆ A.

(ii) Suppose V W . Choose b ∈ V \W . and let g = χX ⊗ b. IfA = ∅, then g(A) = ∅ ⊆ V ; if A 6= ∅, then g(A) = b ⊆ V . Henceg ∈ N(A, V ). Since B 6= ∅, g(B) = b * W and so g /∈ N(B,W ), acontradiction.

Theorem 1.1.13. [KR91] Let p, k, β, σ, u and σo be the topologies onCb(X,E) as defined above. Then

(a) p 6 k 6 β 6 σ 6 u and p 6 σo 6 σ.(b) σ = u iff X = A for some σ-compact subset A of X.(c) k = σ and σ = β iff every σ-compact subset of X is relatively

compact.(d) σo = u iff X is separable.(e) σo 6 k iff every countable subset of X is relatively compact.


(f) σo, σ and u have the same bounded sets in Cb(X,E).Proof. (a) Clearly, p 6 σo 6 σ 6 u (as noted in Remark 1.1.6) and,

by Theorem 1.1.8(a), p 6 k 6 β 6 u. Hence, we only need to verify thatβ 6 σ. Let ϕ ∈ Bo(X), with 0 ≤ ϕ ≤ 1, and W ∈ W. Then there existsa sequence Kn of compact sets in X such that

ϕ(x) < 1/n for x /∈ Kn,

which implies that ϕ = 0 outside of the set A = ∪∞n=1Kn and N(A,W ) ⊆N(ϕ,W ), giving β 6 σ.

(b) If X = A for some σ-compact subset A of X , then it is clear thatu 6 σ.

Conversely, suppose that u 6 σ and let W ∈ W, with W 6= E. Thenthere exist a σ-compact subset A of X and a V ∈ W such that

N(A, V ) ⊆ N(X,W ).

By Lemma 1.1.12, X = A, as required.(c) If every σ-compact subset of X is relatively compact, then clearly

σ 6 k.Conversely, suppose that σ 6 k and let A be any σ-compact subset

of X . If W ∈ W and W 6= E, then there exist a compact set B and a V∈ W such that

N(B, V ) ⊆ N(A,W ).

By Lemma 1.1.12, A ⊆ B, which implies that A is relatively compact.The second part of the assertion follows from part (b) and Theorem1.1.8(b)(ii).

(d) Since a separable set is the closure of a countable set, the σo-topology is equivalent to the topology of uniform convergence on theseparable subsets of X . Thus, if X is countable, it is clear that u 6 σo.

Conversely, if u 6 σo, then, for any W ∈ W, W 6= E, there exist aseparable set A and a V ∈ W such that

N(A, V ) ⊆ N(X,W ).

By Lemma 1.1.12, X = A and so X is separable.(e) This is similar to that of part (c).(f) Since σo 6 u, every u-bounded set is σo-bounded.Conversely, suppose that there exists a σo-bounded set H which is

not u-bounded. Then there exist sequences fn ⊆ H, xn ⊆ X and aW ∈ W such that

fnxn /∈ nW.


Let A = xn. Then H is not absorbed by N(A,W ), which is a contra-diction.

Recall that a TVS E is called ultrabornological if every bounded lin-ear map from E into any TVS F is continuous; E is called ultrabarrelledif every closed linear map from E into any complete metrizable TVSF is continuous; E is called quasi-ultrabarrelled if every bounded linearmap from E into any complete metrizable TVS F is continuous. Everymetrizable TVS is ultrabornological and every complete metrizable TVSis ultrabarrelled. Further, every ultrabornological and every ultrabar-relled TVS is quasi-ultrabarrelled.

Theorem 1.1.14. [Buc58, Kh79, KR91] Let t denote any one of thetopologies σo, β or σ on Cb(X,E).

(a) If t is ultrabornological, then t = u on Cb(X,E).(b) If E is metrizable, then the following are equivalent:

(i) t = u.(ii) t is metrizable.(iii) t is ultrabornological.

Proof. (a) The identity map i : (Cb(X,E), t)→ (Cb(X,E), u) is linearand bounded and so, since t is ultrabornological, i is continuous. Henceu 6 t.

(b) If E is metrizable, then (Cb(X,E), u) is also metrizable. Therefore(i) ⇒ (ii) ⇒ (iii) is clear and (iii) ⇒ (i) follows from part (a).

Remarks. (1) If E is a complete metrizable TVS, so is (Cb(X,E), u)and therefore in Theorem 1.1.14 (b), we have (i) ⇒ (iv) ⇒ (v), where:

(iv) t is ultrabarrelled(v) t is quasi-ultrabarrelled.

(2) In view of Theorem 1.1.14(a), the topologies σo, β and σ are notultrabornological or metrizable (even in the case of real-valued function).In fact, if X is not pseudocompact, then β 6= u on Cb(X).

The following theorem which gives an alternate form of β-neighborhoodsof 0 in Cb(X,E) is due to Katsaras [Kat81].

Theorem 1.1.15. [Kat81] β has a base of neighborhoods of 0 inCb(X,E) consisting of all sets of the form

N = N (Kn, an,W ) =n⋂

i=1

f ∈ Cb(X,E) : f(Kn) ⊆ anW, (∗)

where Kn is a sequence of compact subsets of X, an a sequence ofpositive numbers with an →∞ and W ∈ W.


Proof. Let N be of the form (∗) with W balanced. Then

ϕ := supn≥11

anχKn ∈ Bo(X).

Moreover, N(ϕ,W ) ⊆ N ,as follows. Let f ∈ N(ϕ,W ). Then

1

anχKn(x)f(x) = ϕ(x)f(x) ∈ W for all x ∈ X,

and so f(Kn) ⊆ anW for all n > 1. Therefore f ∈ N (Kn, an,W ). HenceN is a β-neighborhood of 0 in Cb(X,E).

Conversely, let No be a β-neighborhood of 0 in Cb(X,E). There existsa balanced neighborhood W of 0 in E and ϕ ∈ Bo(X) with 0 ≤ ϕ ≤ 1such that

N1 = N(ϕ,W ) ⊆ No.

For n > 1, there exists a compact subset Kn of X such that

x ∈ X : g(x) >1

n ⊆ Kn.

Set

N2 = N (Kn,n

2,W ) =

∞⋂

n=1

f ∈ Cb(X,E) : f(Kn) ⊆n

2W.

Then N2 ⊆ N1, as follows. Let f ∈ N2. Then f(Kn) ⊆n2W for all n > 1.

Let x ∈ X. If ϕ(x) 6= 0, then there exists n > 2 such that

1

n< ϕ(x) <

1

n− 1and so x ∈ Kn\Kn+1.

Then, since n < 2(n− 1),

f(x) ∈n

2W and ϕ(x) <

1

n− 1<

2

n.

Therefore, ϕ(x)f(x) ∈ n2(n−1)

W ⊆ W, and so f ∈ N1. Thus N2 ⊆ No, as

required.

Corollary 1.1.16. [Kat81] If N is a β-neighborhood of 0 in Cb(X,E),then there exists a balanced W ∈ W with the following property: For eachd > 0, there exist a compact set K ⊆ X and a positive number δ > 0such that

f ∈ Cb(X,E) : f(X) ⊆ dW, f(K) ⊆ δW ⊆ N .


Proof. Since N is a β-neighborhood of 0 in Cb(X,E), there existϕ ∈ Bo(X), 0 ≤ ϕ ≤ 1, and W ∈ W such that N(ϕ,W ) ⊆ N . For eachn > 1, let

Kn = x ∈ X : ϕ(x) >1

n.

Then each Kn is compact, Kn−1 ⊆ Kn, ϕ = 1 on K1,

1

n≤ ϕ(x) <

1

n− 1for x ∈ Kn\Kn−1(n ≥ 2),

and ϕ = 0 outside∞⋃n=1

Kn. Let now d > 0 be given. Choose an integer

N such that N > 2d. Let K =N⋃n=1

Kn = KN , δ =12. We now verify that

f ∈ Cb(X,E) : f(X) ⊆ dW, f(K) ⊆ δW ⊆ N(ϕ,W ).

Let f ∈ Cb(X,E) with f(X) ⊆ dW, f(K) ⊆ δW . Let x ⊆ X . Supposex ∈ K. If x ∈ K1, ϕ(x) = 1, and so

ϕ(x)f(x) ∈ δW =1

2W ⊆W.

If x ∈ Kj\Kj−1 (2 ≤ j ≤ N), and so

ϕ(x)f(x) ∈1

2(j − 1)W ⊆W.

Now, let x ∈ [∞⋃n=1

Kn]\KN . Then x ∈ Kp\Kp−1 for some p > N+1. Then

ϕ(x) <1

p− 1≤

1

N<

1

d,

and so

ϕ(x)f(x) ∈1

ddW =W.

Next, if x /∈∞⋃n=1

Kn, then ϕ = 0 and so ϕ(x)f(x) = 0 ∈ W. Thus, in each

case, f ∈ N(ϕ,W ).


2. Weighted Topology on CVb(X,E)

Let X be a completely regular Hausdorff space, E a Hausdorff TVSwith a baseW of balanced neighborhoods of 0 and V a Nachbin family onX . The purpose of this section is to consider the weighted topology on theweighted function space CVb(X,E) and study its various properties. Inparticular, we discuss the relation of weighted topology with the uniform,strict, compact-open and pointwise topologies.

Recall that, if X is a topological space, then a function ϕ : X → R issaid to be upper (resp. lower) semicontinuous on X if, for each r ∈ R,the set x ∈ X : ϕ(x) < r (resp. x ∈ X : ϕ(x) > r) is open. Clearly,ϕ : X → R is continuous iff it is both upper and lower semicontinuous.Further, the characteristic function χA of a subset A of X is upper (resp.lower) semicontinuous iff A is closed (resp. open). Every upper (resp.lower) semicontinuous function assumes its supremum (resp. infimum) ona compact set. In particular, every non-negative upper semicontinuousfunction on a compact set is bounded.

Definition. A Nachbin family V on X is a set of non-negative uppersemicontinuous functions onX such that, given u, v ∈ V and λ > 0, thereexists a w ∈ V such that λu, λv ≤ w (pointwise). Let CVb(X,E) (resp.CVo(X,E)) denote the subspace consisting of those f ∈ C(X,E) suchthat vf is bounded (resp. vanishes at infinity) for all v ∈ V . If U andV are two Nachbin families on X and, for every u ∈ U , there is a v ∈ Vsuch that u ≤ v (pointwise), then we write U ≤ V . If V is a Nachbinfamily on X such that, for every x ∈ X , there is a v ∈ V with v(x) 6= 0,then we write V > 0.

Lemma 1.2.1. [Pro71b, Kh85b] Let U and V be two Nachbin fami-lies on X with U ≤ V . Then CVb(X,E) ⊆ CUb(X,E) and CVo(X,E) ⊆CUo(X,E).

Proof. The first inclusion is immediate from the definition. Let f ∈CVo(X,E), and let u ∈ U and G ∈ W. Choose a v ∈ V with u ≤ v.There exists a compact set K ⊆ X such that

v(x)f(x) ∈ G for all x /∈ K.

Let y /∈ K ∩ supp(u). If y /∈ K, then u(y)f(y) ∈ G. If y /∈ supp(u), thenu(y) = 0, and so again u(y)f(y) ∈ G. Hence f ∈ CUo(X,E).

We now list some important classes of Nachbin families on X . Forany subset A of X , χA will denote its characteristic function.

(1) K+f (X) = λχF : λ > 0 and F ⊆ X, F finite.

(2) K+c (X) = λχK : λ > 0 and F ⊆ X, K compact.

2. WEIGHTED TOPOLOGY ON CVb(X,E) 21

(3) S+oo(X), the set of all non-negative upper semi-continuous func-

tions on X which have compact support,(4) S+

o (X), the set of all non-negative upper semi-continuous func-tions on X which vanish at infinity.

(5) K+(X), the set of all non-negative constant functions on X.Clearly,

K+f (X) 6 K+

c (X) 6 S+oo(X) 6 S+

o (X) 6 K+(X).

If V is a Nachbin family on X with V > 0, then it is easily seen thatK+f (X) 6 V.

Lemma 1.2.2. [Pro71b, Kh85b] If V = S+o (X), then

CVb(X,E) = CVo(X,E) = Cb(X,E).

Proof. Clearly, CVo(X,E) ⊆ CVb(X,E) and Cb(X,E) ⊆ CVb(X,E).Since every v ∈ S+

o (X) is bounded on a compact set, it follows that

CVb(X,E) ⊆ CVo(X,E).

To show that CVb(X,E) ⊆ Cb(X,E), let f ∈ CVb(X,E). Suppose f isnot bounded. Then there exist a G ∈ W and a sequence xn ⊆ X suchthat

f(xn) /∈ 22nG for all n > 1.

Let ϕ =∑∞

n=112nχxn. Then ϕ ∈ S

+o (X) and, for any n > 1,

ϕ(xn)f(xn) /∈ 2nG.

This implies that ϕf is not bounded, a contradiction. Hence f ∈ Cb(X,E).

We also note that:(1) If V = K+(X), then

CVb(X,E) = Cb(X,E) and CVo(X,E) = Co(X,E).

(2) If V = K+c (X) or V = S+

oo(X), then

CVb(X,E) = CVo(X,E) = C(X,E).

(3) If V = K+f (X), then also CVb(X,E) = CVo(X,E) = C(X,E).

Definition. [Pro71b, Kh85b] Let V be a Nachbin family on X . Theweighted topology ωV on CVb(X,E) is defined as the linear topology whichhas a base of neighborhoods of 0 consisting of all sets of the form

N(v,G) = f ∈ CVb(X,E) : (vf)(X) ⊆ G,

where v ∈ V and G ∈ W.


The assumption V > 0 implies that p ≤ ωV . Recall that p ≤ k onC(X,E) and k ≤ β ≤ u on Cb(X,E).

Theorem 1.2.3. [Pro71b, Kh85b] If V = S+o (X), then ωV = β, the

strict topology on CVb(X,E) = Cb(X,E).Proof. It is clear that ωV 6 β, since S+

o (X) ⊆ Bo(X). Now, letϕ ∈ Bo(X). By Lemma 1.1.5, it is sufficient to show that there exists afunction ψ in S+

o (X) which dominates ϕ. For each n, the set

x ∈ X : |ϕ(x)| >1

2n

has compact closure, K say, in X . Let ψ =∑∞

n=112nχKn . Clearly

Kn ⊆ Kn+1 and ψ ∈ S+o (X).

To show |ϕ| ≤ ψ on X , let y ∈ X . If y ∈∞⋃n=1

Kn, then y ∈ Kn+1\KN for

some N > 1, so that

ψ(y) ≥∑

n>N

1

2n=

1

2N> |ϕ(y)|.

If y /∈∞⋃n=1

Kn, then ϕ(y) = 0 but ψ(y) > 0. Thus |ϕ(y)| ≤ ψ(y).

Consequently β 6 ωV , as required.

Remark. Similarly, as above, we note that:(1) If V = K+(X), then ωV is the uniform topology u on

CVb(X,E) = Cb(X,E).

(2) If V = K+c (X), then ωV is the compact-open topology k on

CVb(X,E) = C(X,E).

(3) If V = K+f (X), then ωV is the pointwise topology p on

CVb(X,E) = C(X,E).

Lemma 1.2.4. Let U and V be Nachbin families on X with U 6 V .Then ωU 6 ωV on CVb(X,E).

Proof. This is clear.

Theorem 1.2.5. [Pro71b, Kh85b] Let V be a Nachbin family on Xsuch that χc(X) 6 V and all v ∈ V vanish at infinity. Then

(a) k 6 ωV 6 β 6 u on Cb(X,E).(b) k and ωV coincide on u-bounded subsets of Cb(X,E).Proof. (a) This follows immediately from Lemma 1.2.4.

2. WEIGHTED TOPOLOGY ON CVb(X,E) 23

(b) Let A be a u-bounded subset of Cb(X,E), and let f belong tothe k-closure of A in Cb(X,E). Take any v ∈ V and G ∈ W. Choose abalanced H ∈ W with H +H ⊆ G. Choose r > 0 such that f(X) ⊆ rHfor all g ∈ A. Let

K = x ∈ X : v(x) >1

r and s = supv(x) : x ∈ K.

There exists an h ∈ A such that (h−f)(K) ⊆ 1sH . Then, for any y ∈ X ,

v(y)[h(y)− f(y)] ∈

v(y)1

sH ⊆ G if y ∈ K

v(y)(rH − rH) ⊆ G if y /∈ K;

that is, h − f ∈ N(v,G). Hence f belongs to the ωV -closure of A, andso ωV 6 k on A.

The following example shows that CVb(X,E) may be trivial for rela-tively nice X .

Example. Let X = Q, the set of all rationals with the natural topol-ogy. This is of course a metrizable space. Consider on X the Nachbinfamily V = C+(X) consisting of all non-negative continuous functions.We claim that CVb(X,E) is reduced to 0 for every non-trivial TVS E.

[Indeed, assume that, for a given TVS E, CVb(X,E) 6= 0, andlet f( 6= 0) ∈ CVb(X ;E). Then f(xo) 6= 0 for some xo ∈ X . SinceE is a Hausdorff TVS, there exists some shrinkable neighborhood G ∈W so that ρG(f(xo)) 6= 0. With no loss of generality, we assume thatρG(f(xo)) = 1. Since ρG f : X → R is continuous at xo, taking ε =

12,

there exists δ > 0 such that if |x− xo| < δ,

|ρG(f(x)− ρG(f(xo)| <1

2;

in particular, ρG(f(x)) > ρG(f(xo)−12= 1

2. For an irrational t ∈ P with

|t− xo| < δ, the function vt : X → R+ given by

vt(x) =1

|t− x|, x ∈ X,

belongs to V = C+(X) and then, since f ∈ CVb(X ;E), vt must verify

supvt(x)ρG(f(x)) : x ∈ X < +∞.

But

supvt(x)ρG(f(x)) : x ∈ X ≥ vt(xo)ρG(f(xo)) =1

|t− xo|→ ∞ as t→ xo,

a contradiction.]


3. Notes and Comments

Section 1.1. A general method for defining linear topologies onCb(X,E), called the S-topologies, is considered first. Using the notionof an S-topology, the uniform topology u, strict Topology β, compact-open topology k, σ-compact-open topology σ, countable-open-topologyσo, and pointwise topology p are introduced. See also ([KN63], Section8, p. 68-82).

The topologies k and p on both C(X) and C(X, Y ), Y a uniformspace, have been extensively studied over the past sixty years and signif-icant contributions have been made in this field, among others, by Fox[Fox45], Arens [Are46a], Myers [My46], Hewitt [Hew48], Warner [war58],Wheeler [Whe76], Schmets [Schm83] and McCoy [Mc80]. In recent years,the Cp-theory has been actively investigated by several mathematicians(see the monographs by Arkhangel’skii [Ark92] and Tkachuk [Tka11],and their references).

The notion of the strict topology β on Cb(X,E) was first introducedby Buck [Buc58] in 1958 in the case of X a locally compact space andE a locally convex TVS. (In fact, this was also considered earlier in 1952by Buck himself [Buc52] on Cb(X) for the special case of X a group; itwas there called the ”strict topology” because of its resemblance to atopology used by Beurling [Beu45].) Since then a large number of papershave appeared in the literature concerned with extending the resultscontained in Buck’s paper; see the Preface and the list of references.Another useful topology on Cb(X,E) is the σ-compact-open topology σwhich was introduced by Gulick [Gu72] and studied further by Gulickand Schmets [GuSc72]. Most of these investigations have been concernedwith generalizing the space X and taking E to be the scalar field or alocally convex space. This section includes some results from [Kh79,KR91, [Kat81], where E is a Hausdorff TVS but not necessarily locallyconvex.

Section 1.2. This section is devoted to the study of weighted topol-ogy ωV on the weighted function space CVb(X,E), as given in [Kh85,Kh87]. In particular, the relation of weighted topology with the uniform,strict, compact-open and pointwise topologies is also given.

The fundamental work on weighted spaces had been mainly doneby Nachbin [Nac65, Nac67]. Further investigations have been made bySummers [SumW69, SumW72], Prolla [Pro71, Pro72, Pro77], Bierstedt[Bie73] and other authors (see the monograph [Pro77] and survey arti-cle [Col76] for more references). Most of these investigations have been

3. NOTES AND COMMENTS 25

concerned with generalizing the space X and taking E to be the scalarfield or a locally convex space. Later, Khan [Kh85, Kh87], Nawrocki[Naw89], Prolla [Pro93], Khan and co-authors [KT97, KT02, KMT05,KO05] and Manhas and Singh [MS98] have studied various aspects ofthe strict, weighted and other related topologies, where E is a HausdorffTVS. These results are given in later chapters.

CHAPTER 2

Completeness in Function Spaces

In this chapter, we study various completeness properties of the uni-form strict, weighted and other related topologies.

27

28 2. COMPLETENESS IN FUNCTION SPACES

1. Completeness in the uniform topology

In this section, we consider completeness of the spaces (B(X,E), u),(Cb(X,E), u) and (Co(X,E), u).

Let X be a topological space and E a TVS with a base W of neigh-borhoods of 0. Let B(X,E) denotes the vector space of all boundedfunctions f : X → E. The uniform topology u on B(X,E) is defined asthe linear topology which has a base of neighborhoods of 0 consisting ofall sets of the form

N(X,W ) = f ∈ B(X,E) : f(X) ⊆W,

where W over W. If E = (E, d) is metrizable, then (B(X,E), u) and(Cb(X,E), u) are also metrizable with respect to the metric given by:

ρ(f, g) = supx∈X

d(f(x), f(x)), f, g ∈ B(X,E) or Cb(X,E).

Theorem 2.1.1. [KN63]

Let X be a topological space and E a complete metrizable TVS.Then:

(a) (B(X,E), u) is complete, hence an F-space.(b) (Cb(X,E), u) is complete, hence an F-space.(c) If X is locally compact, then (C0(X,E), u) is complete, hence an

F-space.Proof. (a) Let fn be a u-Cauchy sequence in Cb(X,E). Since p

≤ u, fn is a p-Cauchy net in C(X,E); that is, for each x ∈ X , fn(x)is a Cauchy net in E. Since E is complete, for each x ∈ X, f(x) =

limn→∞ fn(x) exists in E. We need to show that fnu−→ f and that f is

continuous and bounded.Now, fn

u−→ f , as follows. Let W ∈ W be closed. Since fn is

u-Cauchy, there exists n0 ∈ I such that

fn(x)− fm(x) ∈ W for all n,m ≥ n0 and x ∈ X.

Fix any n ≥ n0. Since W is closed and, for each x ∈ X, fm(x) −→ f(x),we have

fn(x)− f(x) ∈ W for x ∈ X.

Thus fn(x)− f(x) ∈ W for all n ≥ n0 and all x ∈ X ; i.e.

fn − f ∈ N(X,W ) for all n ≥ n0.

This proves that fnu−→ f .

1. COMPLETENESS IN THE UNIFORM TOPOLOGY 29

Finally, f is bounded as follows. Let W ∈ W. Choose a balancedV ∈ W such that V + V ⊆ W . Since fn

u−→ f , there exists an n0 ≥ 1

such that

fn(x)− f(x) ∈ V for all n ≥ n0 and x ∈ X.

Since fn0is bounded, we can choose a constant r > 1 such that

fn0(x) ⊆ rV for all x ∈ X.

Then, for any x ∈ X ,

f(x) = [f(x)− fn0(x)] + fn0

(x) ∈ V + rV

⊆ rV + rV = r(V + V ) ⊆ rW .

Hence f is bounded on X . Then (Cb(X,E), u) is complete.(b) Let fn : n ∈ I be a u-Cauchy net in Cb(X,E). Since p ≤ u,

fn is a p-Cauchy net in Cb(X,E); that is, for each x ∈ X , fn(x) is aCauchy net in E. Since E is complete, for each x ∈ X, f(x) = limn fn(x)

exists in E. We need to show that fnu−→ f and that f is continuous

and bounded.As shown in part (a), f is bounded and that fn

u−→ f . Therefore, we

only need to show that f is continuous on X . Let x0 ∈ X and W ∈ W.We show that there exists a neighborhood G of x0 in X such that

f(y)− f(x0) ∈ W for all y ∈ G.

Choose a balanced V ∈ W such that V + V + V ⊆ W . Since fnu−→ f,

there exists and n0 ∈ I such that


Since fn0is continuous at x0, there exists a neighborhood G of x0 in X

such thatfn0

(y)− fn0(x0) ∈ V for all y ∈ G. (2)


f(y)− f(x0) = (f(y)− fn0(y)) + (fn0

(y)− fn0(x0)) + (fn0

(x0)− f(x0))

∈ V + V + V ⊆W.

Hence f is continuous on X . Thus f ∈ Cb(X,E), and so (Cb(X,E), u) iscomplete.

(c) Let fn be a u-Cauchy sequence in C0(X,E). Since p ≤ u, fnis a p-Cauchy sequence in C0(X,E); that is, for each x ∈ X , fn(x) isa Cauchy sequence in E. Since E is complete, for each x ∈ X, f(x) =limn→∞ fn(x) exists in E. Then as in part (a), f is continuous and that

fnu−→ f . We now need to verify that f ∈ C0(X,E). Let W ∈ W.



exists N0 ≥ 1 such that





f(x) = [f(x)− fk0(x)] + fk0(x) ∈ V + V ⊆W .

Therefore f ∈ C0(X,E). Thus (C0(X,E), u) is complete.

Theorem 2.1.2. Let X be a locally compact Hausdorff space and Eany TVS. Then:

(a) (C0(X,E), u) is closed in Cb(X,E).(b) If, in addition, E is complete, then (C0(X,E), u) is also complete.

Proof. (a) Let f ∈ F (X,E) with f ∈ C0(X,E)u. Then there exists

a sequence fn ⊆ F (X,E) such that fnu−→ f . We need to show that

f ∈ C0(X,E).To show that f is continuous on X . Let x0 ∈ X and W ∈ W. We

show that there exists a neighborhood G of x0 in X such that

f(y)− f(x0) ∈ W for all y ∈ G.


there exists and n0 ∈ I such that


Since fn0is continuous at x0, there exists a neighborhood G of x0 in X

such thatfn0

(y)− fn0(x0) ∈ V for all y ∈ G. (2)


f(y)− f(x0) = (f(y)− fn0(y)) + (fn0

(y)− fn0(x0)) + (fn0

(x0)− f(x0))

∈ V + V + V ⊆W.

Hence f is continuous on X . Thus f ∈ Cb(X,E)Next, f is bounded on X , as follows. LetW ∈ W. Choose a balanced

V ∈ W such that V + V ⊆ W . Since fnu−→ f , there exists an n0 ≥ 1

such that

fn(x)− f(x) ∈ V for all n ≥ n0 and x ∈ X.

Since fn0is bounded, we can choose a constant r > 1 such that

fn0(x) ⊆ rV for all x ∈ X.

1. COMPLETENESS IN THE UNIFORM TOPOLOGY 31

Then, for any x ∈ X ,

f(x) = [f(x)− fn0(x)] + fn0

(x) ∈ V + rV

⊆ rV + rV = r(V + V ) ⊆ rW .

Hence f is bounded on X .We now need to verify that f vanishes at infinity. Let W ∈ W.


exists N0 ≥ 1 such that





f(x) = [f(x)− fk0(x)] + fk0(x) ∈ V + V ⊆W .

Therefore f ∈ C0(X,E). Thus (C0(X,E), u) is closed in Cb(X,E).(b) If E is complete, then, by Theorem 2.1.1 (a), (Cb(X,E), u) is

complete. Hence, C0(X,E), being a closed subset of the complete spaceC0(X,E), is also complete.

Note. The above theorem gives an alternate proof of Theorem2.1.1(c).

Recall that a TVS F is called quasi-complete (or boundedly complete) if every bounded closed subset H of F is complete. Clearly complete-ness implies quasi-completeness. Further, every closed subset of a quasi-complete TVS is quasi-complete.

Theorem 2.1.3. Let E be a quasi-complete TVS. Then:(a) (Cb(X,E), u) and (B(X,E), u) and are quasi-complete.(b) If X is locally compact, then (Co(X,E), u) is quasi-complete.

Proof. (a) Let H be a closed and bounded set on (Cb(X,E), u), andlet fα be a u-Cauchy net inH. Clearly, for each x ∈ X , f(x) : f ∈ His a bounded set in E; in particular, for each x ∈ X , fα(x) is a boundednet in E. Further, since p ≤ u, for each x ∈ X , fα(x) is a Cauchynet in E. Since E is quasi-complete, for each x ∈ X, f(x) := limα fα(x)exists in E, and let

f(x) := limαfα(x), x ∈ X.


fαu−→ f : LetW ∈ W be closed. Since fα is u-Cauchy, there exists

αo ∈ I such that

fα(x)− fγ(x) ∈ W for all α, γ ≥ αo and x ∈ X.

Fix any α ≥ αo. Since W is closed and, for each x ∈ X, fγ(x) −→ f(x),we have

fα(x)− f(x) ∈ W for all x ∈ X.

Thus fα − f ∈ N(X,W ) for all α ≥ αo. This proves that fαu−→ f .

Finally, since fα ⊆ H and H is u-closed, f ∈ H. This shows that H iscomplete; hence (Cb(X,E), u) is quasi-complete. By the same argument,(B(X,E), u) is also quasi-complete.

(b) It is similar to part (a).

2. COMPLETENESS IN THE STRICT TOPOLOGY 33

2. Completeness in the Strict Topology

We consider completeness, sequential completeness and quasi-compl-eteness of the β, σ, σo on Cb(X,E) and ωV topologies on CVb(X,E) andof k topology on C(X,E).

Recall that a topological space (X, τ) is called:(i) a k-space if, for any U ⊆ X , U is closed in X whenever U ∩K is

closed in K for each compact K ⊆ X ;(ii) a kR-space if a function f : X → R is continuous on X whenever

f | K is continuous for every compact subset K of X ;Every locally compact space and every first countable (in particular,

metric) space is a k-space. If X is a k-space and Y any topological space,then a function f : X → Y is continuous on X iff f | K is continuous forevery compact subset K of X ; hence every k-space is a kR-space.

Theorem 2.2.1. [Buck58, Kh79] Let X be a k-space and E a com-plete TVS. Then:

(a) (Cb(X,E), β) is complete.(b) (C(X,E), k) is complete.Proof. (a) Let fα : α ∈ I be a β-Cauchy net in Cb(X,E). Since p ≤

β, fα is a p-Cauchy net in C(X,E); that is, for each x ∈ X , fα(x) is aCauchy net in E. Since E is complete, for each x ∈ X, f(x) := limα fα(x)

exists in E. We need to show that fαβ−→ f and that f is continuous and

bounded. Now, fαβ−→ f , as follows. Let ϕ ∈ Bo(X), 0 ≤ ϕ ≤ 1, and let

W ∈ W be closed. Since fα is β-Cauchy, there exists αo ∈ I such that

ϕ(x)[fα(x)− fγ(x)] ∈ W for all α, γ ≥ αo and x ∈ X.

Fix any α ≥ αo. Since W is closed and, for each x ∈ X, fγ(x) −→ f(x),we have

ϕ(x)[fα(x)− f(x)] ∈ W for x ∈ X.

Thus ϕ(x)[fα(x)− f(x)] ∈ W for all α ≥ αo and all x ∈ X ; i.e.

fα − f ∈ N(ϕ,W ) for all α ≥ αo.

This proves that fαβ−→ f . Next, f is continuous as follows. Since X

is a k-space, it suffices to show that, for any compact K ⊆ X , f |K iscontinuous. Let xo ∈ K and W ∈ W. We show that there exists aneighborhood G of xo in X such that

f(y) ∈ f(xo) +W for all y ∈ G ∩K.


[Choose a balanced V ∈ W such that V + V + V ⊆ W . Since fαβ−→ f

and k ≤ β, fαk−→ f and so there exists an αo ∈ I such that

fα(x)− f(x) ∈ V for all α ≥ αo and all x ∈ K. (1)

Since fαo is continuous at xo, there exists a neighborhood G of xo in Xsuch that

fαo(y) ∈ fαo(xo) + V for all y ∈ G. (2)

Then, for any y ∈ G ∩K, using (1) and (2),

f(y)− f(xo) = [f(y)− fαo(y)] + [fαo(y)− fαo(xo)] + [fαo(xo)− f(xo)]

∈ V + V + V ⊆W.]

Finally, f is bounded as follows. Suppose f is unbounded. Then thereexists a W ∈ W such that f(X) is not contained in nW for all n ≥ 1.Hence, for each n ≥ 1, there exists xn ∈ X such that f(xn) /∈ nW .Choose a balanced V ∈ W such that V + V ⊆W . Let

ψ =

∞∑

n=1

1

nχxn.

Then ψ ∈ Bo(X), and so, since fαβ−→ f , there exists an αo ∈ I such

thatψ(x)[fα(x)− f(x)] ∈ V for all α ≥ αo and x ∈ X.

In particular, since ψ(xn) =1n,

fαo(xn)− f(xo) ∈ nV for all n ≥ 1.

Since fαo is bounded, there exists an integer m > 1 such that fαo(X) ⊆mV . Then

f(xm) = [f(xm)− fαo(xm)] + fαo(xm) ∈ mV +mV ⊆ mW,

a contradiction. Hence f is bounded.(b) Let fα be a k-Cauchy net in C(X,E), and let


Using the arguments of part (a) with replacing ϕ by χK , K a compact

set in X , it easily follows that fαk−→ f and f is continuous. Hence

C(X,E) is k-complete.

Theorem 2.2.2. [KR91] Let X be a k-space and E a complete TVS.Then

(a) (Cb(X,E), σ) is complete.(b) If k ≤ σo, then (Cb(X,E), σo) is complete.


Proof. (a) Let fα be a σ-Cauchy net in (Cb(X,E)) and, for each x∈ X , let


As in the proof of Theorem 2.2.1, since X is a k-space and k 6 σ, it iseasily seen that fα

σ−→ f and that f is continuous on X . Suppose f is

not bounded. Then there exist a W ∈ W and a sequence xn in X suchthat

fxn /∈ nW for all n ≥ 1.

Let A = xn and choose a V ∈ W such that V + V ⊆W . There existsan index αo such that

fα(x)− fα(x) ∈ V for all α ≥ αo and x ∈ A.

Let λ be a positive number such that fα(X) ⊆ λV . Then, for n > λ,

f(xn) = [f(xn)− fαo(xn)] + fαo(xn) ∈ V + λV ⊆ nW,

a contradiction. Hence (Cb(X,E), σ) is complete.(b) The proof is almost identical to the one given for (a).

Theorem 2.2.3. [KR91] Let E be a sequentially complete TVS.Then (Cb(X,E), σ) and (Cb(X,E), σo) are sequentially complete.

Proof. Let fn be a σ-Cauchy sequence in Cb(X,E), and let


As in the proof of Theorem 2.2.2, we see that fnσ−→ f and that f is

bounded. Suppose that f is not continuous at some xo ∈ X . Then,without loss of generality, we may assume that there exist a net xλ :λ ∈ I in X , with xλ −→ xo, and a W ∈ W such that

f(xλ)− f(xo) /∈ W for all λ ∈ I. (1)

Choose a V ∈ W such that V + V + V ⊆ W . The functions fn (n =1, 2, ...) are continuous and so, for each positive integer n, there exists anindex λn such that

fn(xλ)− fn(xo) ∈ V for all λ ≥ λn.

Let A = xo, xλ1 , xλ2 , ..., a σ-compact set. Choose an integer m suchthat

fn − f ∈ N(A, V ) for all n ≥ m.

Then, for n ≥ m,

f(xλn)− f(xo) = [f(xλn)− fn(xλn)] + [fn(xλn)− fn(xo)]

+[fn(xo)− fn(xo)]


∈ V + V + V ⊆W,

a contradiction to (1). Hence (Cb(X,E), σ) is sequentially complete. Thesequential completeness of (Cb(X,E), σo) may be proved in the sameway.

Theorem 2.2.4. [KR91] Let E be a complete locally bounded TVS.If (Cb(X,E), σ) (resp. (Cb(X,E), σo)) is quasi-complete, then it is com-plete.

Proof. Suppose that (Cb(X,E), σ) is quasi-complete, and let fα bea σ-Cauchy net in Cb(X,E). Then, as in Theorem 2.1.3, there exists a

bounded E-valued function f on X such that fασ−→ f . We show that f

is continuous, as follows. Let V ∈ W be bounded. There exists a closedshrinkable neighborhood S of 0 in E such that S ⊆ V ; the Minkowskifunctional ρS is continuous (§ A.4). Since V is bounded, we may choosea real number λ ≥ 1, such that V + V ⊆ λS, and, since f is bounded,we may assume that

f(X) ⊆r

λS for some r ≥ 1.

We define a function hr : E −→ E by

hr(x) =

x if x ∈ rS,

rρS(x)

x if x ∈ E\rS.

Since ρS is continuous and ρS(x) = r if x ∈ ∂(rS), it follows from MapGluing Theorem (§ A.1) that hr is continuous. Now hr(E) ⊆ rS and sothe net hr fα is u-bounded. Furthermore, hr fα is σ-Cauchy, asfollows. Let A be σ-compact and let W ∈ W be any neighborhood of 0in E. Choose t ≥ 1 such that V + V ⊆ tW . There exists an index αosuch that

fα − f ∈ N(A,1

tλV ) for all α ≥ αo.

This implies that fα(A) ⊆ rS for α ≥ αo. Hence, for any x ∈ A andα, γ ≥ αo,

hr(fα(x))− hγ(fr(x)) = fα(x)− fα(x) ∈1

tλ(V + V ) ⊆W.

Since (Cb(X,E), σ) is quasi-complete there exists a function g in Cb(X,E)such that

hr fασ−→ g.

The above argument also shows that hr fασ−→ f , and so f = g. Thus

f is continuous.


Assuming quasi-completeness of (Cb(X,E), σo), its completeness maybe proved by using a similar argument as above.


3. Completion of (Cb(X,E), β)

We now consider the completion of (Cb(X,E), β), where X is not nec-essarily a k-space and E is not necessarily complete. Following Katsaras[Kat81], let

Bck(X,E) = f ∈ B(X,E) : f |K is continuous for each compact K ⊆ X

On Bck(X,E) we consider the linear topology βc which has as a base ofneighborhoods of 0 consisting of all sets of the form

f ∈ Bck(X,E) : (ϕf)(X) ⊆W,

where W is a neighborhood of 0 in E and ϕ ∈ Bo(X). Clearly thetopology induced on Cb(X,E) by βc is the strict topology β.

Theorem 2.3.1. [Kat81] If E is complete, then (Bck(X,E), βc) isalso complete.

Proof. Let (fα) be a βc-Cauchy net in Bck(X,E). Since p ≤ βc and Eis complete, there exists an E-valued function f on X such that f(x) =limα fα(x) for each x ∈ X . Now, by the argument similar to the oneused in the proof of Theorem 2.2.1, it is easy to verify that f ∈ Bck(X,E)and βc-limα fα = f.

Theorem 2.3.2. [Kat81] If (Cb(X,E), β) is complete, then E iscomplete.

Proof. Let sa be a Cauchy net in E. For each α, let fa : X → Ebe the constant function given by fa(x) = sa, x ∈ X . Clearly, the netfa is Cauchy in (Cb(X,E), β). Since (Cb(X,E), β) is complete, thereexists an f ∈ Cb(X,E) such that fa → f in the topology β. Hence, forany x ∈ X, limα sa = limα fa(x) = f(x).

Definition. Let∧

E be the unique (up to a topological isomorphism)Hausdorff completion of E. We may consider E as a dense subspace of∧

E. Let∧

βc denote the linear topology on Bck(X,∧

E) which has as a baseof neighborhoods of 0 consisting of all sets of the form

∧

U(ϕ,W ) = f ∈ Bck(X,∧

E) : (ϕf)(X) ⊆W,

where ϕ ∈ Bo(X) and W is a neighborhood of 0 in∧

E.Using Theorem 2.3.1, we have the following.

Theorem 2.3.3. [Kat81] With the above notations, the completion

of (Cb(X,E), β) is the∧

βc-closure of Cb(X,E) in Bck(X,∧

E) with theinduced topology.

3. COMPLETION OF (Cb(X,E), β) 39

Theorem 2.3.4. Let Bck(X,∧

E) and∧

βc be as above and suppose

that, for every compact subset K of X, the space C(K)⊗∧

E is u-dense

in C(K,∧

E). Then (Bck(X,∧

E),∧

βc) is the completion of (Cb(X,E), β).

Proof. Let f ∈ Bck(X,∧

E). We need to show that f belongs to the∧

βc-closure of Cb(X,E). Consider any∧

βc-neighborhood∧

U(ϕ,W ) of 0 in

Bck(X,∧

E), where ϕ ∈ Bo(X), 0 ≤ ϕ ≤ 1, and W is a neighborhood of

0 in∧

E. Choose an open balanced neighborhood V of 0 in∧

E such thatV + V + V + V ⊆ W. Let λ > 1 be such that f(X) ⊆ λV . There existsa compact subset K of X such that

ϕ(x) <1

λfor all x ∈ X \ K.

Since f |K ∈ C(K,∧

E), there exists h ∈ C(K)⊗∧

E such that (h−f)(K) ⊆1λV . We have

h(K) ⊆ f(K) +1

λV ⊆ λV + V ⊆ λV + λV.

There exists an extension h1 ∈ Cb(X) ⊗∧

E, h1 = h on K, h1(X) ⊆

λV + λV . Since E is dense in∧

E, there exists g ∈ Cb(X)⊗ E such that(g − h1)(X) ⊆ V . Now

(h1 − f)(X) ⊆ h1(X)− f(X) ⊆ λV + λV + λV.

Let x ∈ X. If x ∈ K,

ϕ(x)[g(x)− f(x)] = ϕ(x)[g(x)− h1(x)] + ϕ(x)[h1(x)− f(x)]

∈ ϕ(x)V + V ⊆ V + V ⊆W.

If x ∈ X\K,

ϕ(x)[g(x)− f(x)] = ϕ(x)[g(x)− h1(x)] + ϕ(x)[h1(x)− f(x)]

∈ ϕ(x)V +1

λ[λV + λV + λV ] ⊆W.

Therefore g−f ∈∧

U(ϕ,W ). Thus, by the above theorem, (Bck(X,∧

E),∧

βc)is the completion of (Cb(X,E), β).

Corollary 2.2.5. [Kat81] (Bck(X,∧

E),∧

βc) coincides with the comple-tion of (Cb(X,E), β) in each of the following cases:

(1) X has finite covering dimension.


(2) Every compact subset of X has finite covering dimension.(3) E is a complete metrizable TVS with a basis.(4) E is locally convex.(5) E is a complete TVS and has the approximation property.Proof. The proof follows from the above theorem and a later result,

corollary of the Stone-Weierstrass theorem (section 4.1), by which eachof the conditions (1)-(5) implies that, for every compact subset K of X,

the space C(K)⊗∧

E is u-dense in C(K,∧

E).

4. COMPLETENESS IN THE WEIGHTED TOPOLOGY 41

4. Completeness in the Weighted Topology

We now consider completeness in the weighted topology. Let V be aNachbin family on X , and let FVo(X,E) denote the set of all E-valuedfunctions f on X such that, for each v ∈ V , vf is bounded and vanishesat infinity. We may define the weighted topology ωV on FVo(X,E) inthe same way as in Section 1.2:

Definition. ωV is the linear topology FVo(X,E) which has a baseof neighborhoods of 0 consisting of all sets of the form

N(v,G) = f ∈ FVo(X,E) : (vf)(X) ⊆ G,

where v ∈ V and G ∈ W. Clearly, (FVo(X,E), ωV ) is Hausdorff if V > 0.

Theorem 2.4.1. [Pro71b, Kh85b] Let V be a Nachbin family on Xwith V > 0, and suppose E is complete. Then (FVo(X,E), ωV ) is com-plete; hence (CVo(X,E), ωV ) is complete iff it is closed in (FVo(X,E), ωV ).

Proof. Let fα be an ωV -Cauchy net in FVo(X,E). Since V > 0,ρ ≤ ωV and so fα is ρ-Cauchy. Since E is complete, for each x ∈ X ,

f(x) = limα fα(x) exists, and it is easily shown that fαωV−→ f . Let v ∈ V ,

G ∈ W. Choose a balanced H ∈ W with H +H ⊆ G. There exists anindex αo such that

v(x)(fα(x)− f(x)) ∈ H for all α ≥ αo and x ∈ X.

Choose r ≥ 1 with (vfα)(X) ⊆ rH . Then, for any x ∈ X ,

v(x)f(x) = v(x)(f(x)− fαo(x)) + v(x)fαo(x) ∈ H + rG ⊆ rG,

and so vf is bounded. Next, with v, G, H , and αo as above, choose acompact set K ⊆ X such that v(x)fαo(x) ∈ H for all x /∈ K. Then, forx /∈ K, we have v(x)f(x) ∈ G. Thus f ∈ FVo(X,E).

Theorem 2.4.2. [Pro71b, Kh85b] Suppose E is complete, and Uand V are Nachbin families on X with U ≤ V and V > 0.

(a) If CUo(X,E) is closed in (FUo(X,E), ωU), then (CVo(X,E), ωV )is complete.

(b) If (FUo(X,E), ωU) is complete (quasi-complete), so is

(CUo(X,E), ωV ).

Proof. (a) In view of Theorem 2.4.1, we need to show that CVo(X,E)is ωV -closed in FVo(X,E). Let f belong to the ωV -closure of CVo(X,E)

in FVo(X,E). There exists a net fα ⊆ CVo(X,E) with fαωV−→ f. Since

U ≤ V , ωU ≤ ωV and so fαωU−→ f. Since CVo(X,E) ⊆ CUo(X,E) and


CUo(X,E) is ωV -closed in FUo(X,E), we have f ∈ FUo(X,E). Hencef is continuous and so f ∈ CVo(X,E).

(b) If (CUo(X,E), ωU) is complete, then it follows immediately from(a) that (CVo(X,E), ωV ) is complete. Suppose now that (CUo(X,E), ωV )is quasi-complete, and let fα be a bounded Cauchy net in

(CUo(X,E), ωV ).

By Theorem 2.4.1, there exists an f ∈ (FUo(X,E) such that fαωV−→ f .

Since U ≤ V, fα ⊆ FVo(X,E) such that fαωU−→ f . Clearly, fα is ωU -

bounded. Then, by hypothesis, f ∈ CUo(X,E). Hence f ∈ CVo(X,E).

Corollary 2.4.3. [Pro71b, Kh85b] If X is a k-space and E iscomplete, then (Cb(X,E), β) is complete.

Proof. Let V = S+o (X). Then ωV = β on CVo(X,E) = Cb(X,E).

Further, for each x ∈ X , χx ∈ V , and so V > 0. Take U = K+c (X).

Then (CUo(X,E), ωU) = (C(X,E), k) which is complete since X is ak-space (see Theorem 2.2.1). So, by Theorem 2.4.2, (Cb(X,E), β) iscomplete.



Section 2.1. We present some classical results on completeness of thespaces (B(X,E), u), (Cb(X,E), u) and (Co(X,E), u), as given in [KN63]

Section 2.2. Various completeness properties (such as completeness,sequential completeness and quasi-completeness) of function spaces withthe topologies u, β, σ and k are considered. These are taken from [Kh79,KR91] which generalize earlier results given in [Buc58, GulD72].

Section 2.3. This includes some results, due to Katsaras [Kat81],on the completion of (Cb(X,E), β), where X is not necessarily a k-spaceand E is not necessarily complete.

Section 2.4. Completeness in the weighted spaces CVb(X,E) andCVo(X,E) is considered. These were obtained by Summers [SumW69]for E a scalar field, by Prolla [Pro71b] for E a locally convex TVS andby Khan [Kh85] for E any TVS.

CHAPTER 3

The Arzela-Ascoli type Theorems

In this chapter we study Arzela-Ascoli type theorems for the compact-open, strict and weighted topologies. The abstract version of the Arzela-Ascoli theorem was first given by Myers [My46] which was later improvedby Gale [Gal50]. The name k-space also first appeared in the paper[Gal50]. We include here some related results from [KN63, Kh79, RS88,KO05].

45

46 3. THE ARZELA-ASCOLI TYPE THEOREMS

1. Arzela-Ascoli Theorem for k and β Topologies

In this section, we present Arzela-Ascoli type theorems for the compact-open and strict topologies.

Definition. Let X be a topological space and E a TVS with a baseW of neighborhoods of 0. A subset A of C(X,E) is said to be equicontin-uous at xo ∈ X if, for each W ∈ W, there exists a neighborhood G(xo)of xo such that

f(x)− f(xo) ∈ W for all x ∈ G(xo) and f ∈ A.

A is said to be equicontinuous on X if it is equicontinuous at each pointof X .

The following version of the Arzela-Ascoli type theorem for the compact-open topology is given in [KN63, p. 81].

Theorem 3.1.1. [KN63, p. 81] Let X be a Hausdorff k-space andE a Hausdorff TVS , and let A ⊆ C(X,E). Then A is k-compact iff thefollowing conditions are satisfied.

(i) A is k-closed;(ii) A(x) = f(x) : f ∈ A is relatively compact in E for each

x ∈ X;(iii) A is equicontinuous on each compact subset of X.For the readers’ benefit and interest, we present here a well-known

version of above theorem in the more general setting of uniform spaces.Its proof is adapted from [Eng88, p. 440-443; Will70, § 43].

Recall that, if Y is a non-empty set, then, for any A,B ⊆ Y × Y, wedefine

A−1 = (y, x) : (x, y) ∈ A,

A B = (x, y) ∈ Y × Y : ∃ z ∈ Y such that (x, z) ∈ A and (z, y) ∈ B.

If A = B, we shall write A A = A2. Note that,

(x, z) ∈ A, (z, y) ∈ B ⇒ (x, y) ∈ A B.

A subset H of subsets of Y × Y is called a uniformity (or a uniformstructure) on Y if

(u1) (Y ) = (x, x) : x ∈ Y ⊆ A for all A ∈ H.(u2) A ∈ H =⇒ A−1 ∈ H, i.e. each A ∈ H is symmetric.(u3) If A ∈ H, ∃ some B ∈ H such that B2 ⊆ A.(u4) If A,B ∈ H, ∃ some C ∈ H such that. C ⊆ A ∩ B.(u5) If A ∈ H and A ⊆ B, then B ∈ H.

1. ARZELA-ASCOLI THEOREM FOR k AND β TOPOLOGIES 47

In this case, the pair (Y,H) is called a uniform space. The membersof H are called vicinities of H. For any x ∈ Y and A ∈ H, let

NA(x) = y ∈ Y : (x, y) ∈ A.

Then, for each x ∈ Y , the collection NA(x) : A ∈ H satisfies all theconditions of a neighborhood system at x. Consequently, there existsa unique topology τH on Y such that, for each x ∈ Y , the collectionNA(x) : A ∈ H forms a local base at x. In this case, the topology τHon Y is said to be the topology induced by (or derived from) the uniformityH. For any net yα ⊆ (Y,H), we say that yα → y ∈ Y if, for eachV ∈ H, there exists an αo ∈ I such that

(yα, y) ∈ V for all α ≥ αo.

Every TVS (E, τ) is a uniform space. [In fact, let W be a base ofneighborhoods of 0 in E. For each V ∈ W, let

AV = (x, y) ∈ E ×E : x− y ∈ V .

Then H = AV : V ∈ W is a uniformity on E. ]

Definition. Let X be a Hausdorff space and (Y,H) a uniform space.and let S = S(X) be a certain collection of subsets of X . For anyK ∈ S(X) and D ∈ H, let

M(K,D) = (f, g) ∈ C(X, Y )×C(X, Y ) : (f(x), g(x)) ∈ D for all x ∈ K.

Then the collection M(K,D) : K ∈ S(X) and D ∈ H form a subbasefor a uniformity, called the uniformity of uniform convergence on the setsin S(X) induced by H. The resultant topology τS on C(X, Y ) is calledthe topology of uniform convergence on the sets in S(X).

(1). If S(X) = X, then the τS -topology on C(X, Y ) is called theuniform topology and is denoted by u.

(2). If S(X) is the collection of all compact subsets of X , then the τS-topology on C(X, Y ) is called the compact-open topology and is denotedby k.

(3). If S(X) is the collection of all finite (or singleton) subsets of X ,then the τS-topology on C(X, Y ) is called the pointwise topology and isdenoted by p

Clearly p ≤ k ≤ u on C(X, Y ). and k = u if X is compact.

Alternate Definition. [Edw65, p. 33] Let X be a Hausdorff spaceand (Y,H) a uniform space having a base B of vicinities of H, and let


S = S(X) be a collection of subsets of X . For any f ∈ C(X, Y ),K ∈ S(X) and A ∈ B (or A ∈ H), let

N(f,K,A) = g ∈ C(X, Y ) : (g(x), f(x)) ∈ A for all x ∈ K.

Then the collection N(f,K,A) : f ∈ C(X, Y ), K ∈ S(X), A ∈ B forma subbase for a topology τS on C(X, Y ), called the topology of uniformconvergence on the sets in S(X).

Definition. [Eng88, p. 442] Let X be a Hausdorff space and (Y,H)a uniform space. A subset A of C(X, Y ) is said to be equicontinuous atxo ∈ X if, for each D ∈ H, there exists a neighborhood G(xo) of xo suchthat

(f(x), f(xo)) ∈ D for all x ∈ G(xo) and f ∈ A.

A is said to be equicontinuous on X if it is equicontinuous at each pointof X .

Lemma 3.1.2. Let X be a Hausdorff space and (Y,H) a uniformspace, and let A ⊆ C(X, Y ). If A is equicontinuous on X, then A

pis

also equicontinuous on X.Proof. Let xo ∈ X and D ∈ H. We show that there exists a neigh-

borhood G(xo) of xo such that

(g(y), g(xo)) ∈ D for all y ∈ G(xo) and g ∈ Ap. (1)

Choose a symmetric V ∈ H with V V V ⊆ D. Since A is equicontinuouson X , there exists a neighborhood G(xo) of xo such that

(f(y), f(xo)) ∈ V for all y ∈ G(xo) and f ∈ A. (2)

Let f ∈ Ap, and let fα : α ∈ I be a net in A such that fα

p→ f. Fix

any y ∈ G(xo). Choose α0 ∈ I such that

(fα0(y), f(y)) ∈ V and (fα0

(xo), f(xo)) ∈ V. (3)

Now, by (2) and (3),

(f(y), fα0(y)), (fα0

(y), fα0(xo)), (fα0

(xo), f(xo)) ∈ V,

and so(f(y), f(xo)) ∈ V V V ⊆ D.

This proves (1), Thus Apis equicontinuous on X.

Lemma 3.1.3 Let X be a Hausdorff space and (Y,H) a uniformspace, and let A ⊆ C(X, Y ). If A is equicontinuous on each compactsubset of X, then:

(a) The topologies k and p coincide on A.

(b) Ak= A

p.


Proof. The proofs of (a) and (b) are similar and so we shall only givethe proof of (b).

Since p ⊆ k, Ak⊆ A

pand so we need to prove that A

p⊆ A

k. Let

f ∈ Apand let fα : α ∈ I be a net in A such that p-lim

αfα = f.

Let K ⊆ X be compact and D ∈ H. Choose a symmetric V ∈ H withV V V ⊆ D. Since p-lim

αfα = f , for each x ∈ K, there exists αx ∈ I

such that

(fα(x), f(x)) ∈ V for all α ≥ αx. (1)

Since A is equicontinuous subset on K, by Lemma 3.1.2, Apis also

equicontinuous on K. Hence there exists a neighborhood G(x) of x suchthat

(g(y), g(x)) ∈ V for all y ∈ G(x) ∩K and g ∈ Ap. (2)

Choose a finite subcover G(xi) : i = 1, ..., m of G(x) : x ∈ K forK. Choose α0 ∈ I such that α0 ≥ αxi, i = 1, ..., m. Let y ∈ K. Theny ∈ G(xq) ∩K for some q (1 ≤ q ≤ m). Hence, for all α ≥ α0, it followsfrom (1) and (2) that

(fα(y), fα(xq)), (fα(xq), f(xq)), (f(xq), f(y)) ∈ V ;

consequently,

(fα(y), f(y)) ∈ V V V ⊆ D.

This implies that (fα, f) ∈M(K,D) for all α ≥ α0. Hence f ∈ Ak.

We now establish a version of the Arzela-Ascoli theorem for (C(X, Y ), k).

Theorem 3.1.4. Let X be a Hausdorff k-space and (Y,H) a Haus-dorff uniform space, and let A ⊆ C(X, Y ). Then A is k-compact iff thefollowing conditions are satisfied.

(i) A is k-closed;(ii) A(x) = f(x) : f ∈ A is relatively compact in Y for each

x ∈ X;(iii) A is equicontinuous on each compact subset of X.Proof. (⇒) Suppose A is k-compact in C(X, Y ). Then conditions (i)

- (iii) hold, as follows.(i) Since Y is Hausdorff, C(X, Y ), k) is also Hausdorff. So A, being

a compact subset of the Hausdorff space, is k-closed.(ii) Let x be any point in X . Consider the evaluation map δx :

C(X, Y )→ Y given by

δx(f) = f(x), f ∈ C(X, Y ).


Now δx is p-continuous, as follows. For any f ∈ C(X, Y ) and D ∈ H.Let

N(f, x, D) = g ∈ C(X, Y ) : (g(x), f(x)) ∈ D.

It is clear that N(f, x, D) is a pneighborhood of f in C(X, Y ) and

δx(N(f, x, D)) = g(x) : (g(x), f(x)) ∈ D ⊆ ND(f(x)).

Hence δx is p-continuous. Since p ≤ k and A is k-compact, it followsthat A is p-compact. Thus δx(A) = A(x) is relatively compact in Y .

(iii) Suppose K is any compact subset of X. let xo ∈ K and D ∈ H.Choose a symmetric V ∈ H with V V V ⊆ D. Now, for each f ∈ A,there exists a neighborhood G1(xo, f) of xo such that

(f(x), f(xo)) ∈ D for all x ∈ G1(xo, f).

Since K, being compact and Hausdorff, is regular, there exists a closedneighborhood G(xo, f) of xo in K such that G(xo, f) ⊆ G1(xo, f) ∩ K.Let

N(f,G(xo, f), V ) = g ∈ C(X, Y ) : (g(x), f(x)) ∈ V for all x ∈ G(xo, f).

a k-open neighborhood of f. Since A is k-compact, its open cover

N(f,G(xo, f), V ) : f ∈ A

has a finite subcover N(fi, G(xo, fi), V ), i = 1, ..., m. Let f ∈ A, sothat f ∈ N(fj , G(xo, fj) for some j (1 ≤ j ≤ m). Then, for any x ∈∩mi=1G(xo, fi),

(f(x), fj(x)), (fj(x), fj(xo)), (fj(xo), f(xo)) ∈ V.

and so(f(x), f(xo)) ∈ V V V ⊆ D.

So A is equicontinuous at xo and hence on K.[⇐] Suppose that A satisfies conditions (i) - (iii). Since A is equicon-

tinuous on each compact subset of X , it follows from Lemmas 3.1.2 and

3.1.3 that Apis equicontinuous, k = p on A

pand A

p= A

k. Since

A is k-closed, to show that A is k-compact, it suffices to show thatA is p-compact. We may regard C(X, Y ) as being embedded in theproduct space

∏x∈X Yx where each Yx = Y via the correspondence

f → (f(x))x∈X . The product topology on∏

x∈X Yx induces on C(X, Y )the topology p. In particular, we may regard A as being embedded in∏

x∈X A(x). By (ii) and the Tychonoff theorem,∏

x∈X A(x) is compact.

Hence it suffices to show that A is p-closed in∏

x∈X A(x).

Let g belong to the p-closure of A in∏

x∈X A(x). Since X is a k-space, to show that g is continuous on X , it is sufficient to show that g


is continuous on any compact subset K of X. Let xo ∈ K and D ∈ H.Choose a symmetric V ∈ H with V V V ⊆W. SinceA is equicontinuouson X , there exists a neighborhood G(xo) of xo such that

(f(y), f(xo)) ∈ V for all y ∈ G(xo) ∩K and f ∈ A.

Since g belongs to the p-closure of A in∏

x∈X A(x), for each x ∈ X, thereexists fx ∈ A such that

(fx(x), g(x)) ∈ V.

Then, for any y ∈ G(xo) ∩K,

(g(y), fy(y)), (fy(y), fy(xo)), (fy(xo), g(xo)) ∈ V.

and so(g(y), g(xo)) ∈ V V V ⊆ D.

Therefore g ∈ C(X, Y ). Hence g ∈ Ap. But A

p= A

k= A (since A is

k-closed). So g ∈ A, showing that A is p-closed in∏

x∈X A(x).

Note. In the case of Y = E, a TVS, the above theorem reduces toTheorem 3.1.1.

We now present an analogue of the above theorem for the β-topologyon Cb(X,E) with E a TVS. Here we use the terminology of Chapter 1.

Theorem 3.1.5. [Kh79] Let X be a Hausdorff k-space and E aHausdorff TVS. Then a subset A of Cb(X,E) is β-compact iff the fol-lowing conditions hold:

(i) A is β-closed and β-bounded;(ii) A(x) is relatively compact in E for each x ∈ X;(iii) A is equicontinuous on each compact subset of X.Proof. Suppose A is β-compact in Cb(X,E). Then conditions (i) -

(iii) hold, as follows.(i) Since E is Hausdorff, Cb(X,E), β) is also Hausdorff. So A, being

a compact subset of the Hausdorff space, is β-closed. Further, A isbounded since a compact set in a TVS is always bounded.

(ii) Let x be any point in X . Consider the map δx : Cb(X,E) → Egiven by

δx(f) = f(x), f ∈ Cb(X,E).

Now δx is p-continuous, as follows. For any f ∈ Cb(X,E) and W ∈ W.Let

N(f, x,W ) = g ∈ Cb(X,E) : g(x)− f(x) ∈ W.

It is clear that N(f, x,W ) is a pneighborhood of f in Cb(X,E) and

δx(N(f, x,W )) ⊆ f(x) +W.


Hence δx is p-continuous. Since p ≤ β and A is β-compact, it followsthat A is p-compact. Thus δx (A) = A(x) is relatively compact in E.

(iii) Since k ≤ β and A is β-compact, it follows by Theorem 3.1.1that A is equicontinuous on each compact subset of X .

Conversely, suppose that a subset A of Cb(X,E) satisfies conditions(i) - (iii). Since A, being β-bounded, is u-bounded, the topologies βand k coincide on A (Theorem 1.1.8, (c), (d)). Thus, to show that A isβ-compact, it is only necessary to show that A is k-compact. Now, byusing the same argument as the one use to prove Theorem 1.1.8 (c), wecan show that the β and k closures of A are the same. Consequently, Ais k-closed. In view of Theorem 3.1.1, this fact together with conditions(ii) and (iii) imply that A is k-compact.

2. ARZELA-ASCOLI THEOREM FOR WEIGHTED TOPOLOGY 53

2. Arzela-Ascoli Theorem for Weighted Topology

In this section, we present the Arzela-Ascoli type theorems whichcharacterize precompact subsets of weighted functions spaces CVo(X,E)and CVpc(X,E) in the non-locally convex setting. When the range spaceE happens to be quasi-complete, these characterizations become criteriafor relative compactness.

As in section 1.2, let X be a completely regular Hausdorff space, Eis non-trivial Hausdorff TVS with a baseW of closed balanced neighbor-hoods of 0 and V a Nachbin family of non-negative upper semicontinuousfunction on X . We also assume that V > 0; i.e., for each x ∈ X , thereexists a v ∈ V with v(x) > 0. Further, let

CVpc(X,E) = f ∈ C(X,E) : (vf)(X) is precompact in E for all v ∈ V .

Clearly, CVo(X,E) ⊆ CVpc(X,E) ⊆ CVb(X,E). The weighted topologyωV on CVb(X,E) has a base of neighborhoods of 0 consisting of all setsof the form

N(v,G) = f ∈ CVb(X,E) : (vf)(X) ⊆ G,

where v ∈ V and G ∈ W.Note that:(1) If V = K+(X), then CVpc(X,E) = Cb(X,E) and ωV is the uni-

form topology u.(2) If V = S+

o (X), then CVpc(X,E) = Cb(X,E) and ωV is the stricttopology βo.

(3) If V = K+c (X), then CVpc(X,E) = CVb(X,E) = CVo(X,E) =

C(X,E) and ωV is the compact-open topology k.(4) If V = K+

f (X),then CVpc(X,E) = CVb(X,E) = CVo(X,E) =C(X,E) and ωV is the pointwise topology p.

The assumption V > 0 implies that p ≤ ωV . Recall that p ≤ k onC(X,E) and k ≤ βo ≤ u on Cb(X,E).

Definition. Let E and F be TVSs. For any collection G of subsetsof E, CLG(E, F ) denotes the subspace of CL(E, F ) consisting of thoseT which are bounded on the members of G together with the topology tGof uniform convergence on the elements of G. This topology has a baseof neighborhoods of 0 consisting of all sets of the form

U(D,W ) := T ∈ CL(E, F ) : T (D) ⊆W,

where D ∈ G and W is a neighborhood of 0 in F (see § A.3).


Definition. If V is a Nachbin family on X , a net xα : α ∈ I iscalled a V -net if it is contained in Sv,1 := x ∈ X : v(x) ≥ 1 for somev ∈ V . We hence get the classical VR-spaces introduced by Bierstedt[Bie75b]: X is said to be a VR-space if a function f : X → R is continuouswhenever, for each v ∈ V, the restriction of f to Sv,1 is continuous. IfV = K+(X), then X is a VR-space means X is a kR-space.

Definition. For any x ∈ X , let δx : CVb(X,E)→ E denote the eval-uation map δx(f) = f(x), f ∈ CVb(X,E). Clearly, δx ∈ CL(CVb(X,E), E).Next, we define ∆ : X → CL(CVb(X,E), E) as the evaluation map givenby

∆(x) = δx, x ∈ X.

Let the subscript pc in CLpc(CVb(X,E), E) stand for the topology ofuniform convergence on precompact subsets of CVb(X,E).

Lemma 3.2.1. [RS88, KO05] The evaluation map

∆ : X → CLpc(CVb(X,E), E)

is continuous iff every precompact subset of CVb(X,E) is equicontinu-ous.

Proof. (⇒) Suppose ∆ : X → CLpc(CVb(X,E), E) is continuous,and let A be a precompact subset of CVb(X,E). To show that A isequicontinuous, let xo ∈ X and W ∈ W. Since ∆ is continuous at xo,there exists an open neighborhoodG(xo) of xo inX such that ∆(G(xo)) ⊆∆(xo) + U(A,W ); that is,

δx(f)− δxo(f) ∈ W for all x ∈ G(xo) and f ∈ A.

Hence f(G) ⊆ f(xo) +W for all f ∈ A, and so A is equicontinuous.(⇐) Suppose that every precompact subset of CVb(X,E) is equicon-

tinuous. To show that ∆ : X → CLpc(CVb(X,E), E) is continuous,let xo ∈ X and let A be a precompact subset of CVb(X,E) and W abalanced set in W. Since A, being precompact, is equicontinuous (byhypothesis), there exists an open neighborhood G(xo) of xo in X suchthat f(G) ⊆ f(xo) +W for all f ∈ A; that is

δx − δxo ∈ U(A,W ) for all x ∈ G.

Hence ∆(G(xo)) ⊆ ∆(xo) + U(A,W ).

Theorem 3.2.2. [RS88, KO05] Let X be a VR-space. Then everyprecompact subset of CVb(X,E) is equicontinuous.

Proof. In view of Lemma 3.2.1, it suffices to show that the evaluationmap ∆ : X → CLpc(CVb(X,E), E) is continuous. Since X is a VR-spaces,we only need to show that ∆ is continuous on each Sv,1 = x ∈ X :


v(x) ≥ 1, v ∈ V. Let v ∈ V and x ∈ Sv,1, and let A be a precompactsubset of CVb(X,E) and W ∈ W. Choose a balanced H ∈ W withH +H +H ⊆W . Since A is precompact, there exist h1, ..., hn ∈ A suchthat

A ⊆n⋃

i=1

(hi +N(v,H)).

Since each hi is continuous, there exists a neighborhood Gi of x in Xsuch that

hi(y)− hi(x) ∈ H for all y ∈ Gi, i = 1, ..., n.

Let G =n⋂i=1

Gi. Now, if y ∈ G ∩ Sv,1 and f ∈ A, then f = hi + g for

some 1 ≤ i ≤ 1 and g ∈ N(v,H). Hence

δy(f)− δx(f) = hi(y) + g(y)− hi(x)− g(x)

= hi(y)− hi(x) +1

v(y)v(y)g(y)−

1

v(x)v(x)g(x)

∈ H +1

v(y)H −

1

v(x)H ⊆ H +H −H ⊆W ;

that is, δy − δx ∈ U(A,W ) for all y ∈ G ∩ Sv,1. Hence ∆(G ∩ Sv,1) ⊆∆(x) + U(A,W ).

Theorem 3.2.3. [RS88, KO05] Let X be a VR-space. A subset A ofCVo(X,E) is precompact iff the following conditions hold.

(i) A is equicontinuous.(ii) A(x) = f(x) : f ∈ A is precompact in E for each x ∈ X.(iii) vA vanishes at infinity on X for each v ∈ V ; i.e., given v ∈ V

and W ∈ W, there exists a compact set K ⊆ X such that v(y)f(y) ∈ Wfor all f ∈ A and y ∈ X\K.

Proof. (⇒) Suppose A is a precompact subset of CVo(X,E). Weverify (i)-(iii), as follows.

(i) Since X is a VR-space, by Theorem 3.2.2, A is equicontinuous.(ii) Since V > 0, p ≤ ωV and so A is p-precompact. Hence, for each

x ∈ X, A(x) is precompact in E.(iii) Let v ∈ V and W ∈ W. Choose a balanced H ∈ W with

H +H ⊆W . Since A is precompact, there exist h1, ..., hn ∈ A such that

A ⊆n⋃

i=1

(hi +N(v,H)). (1)


Put K =n⋃i=1

y ∈ X : v(y)hi(y) /∈ H. Then K is compact (since each

hi ∈ CVo(X,E)). Now let f ∈ A and y ∈ X\K. By (1), there existsi ∈ 1, ..., n such that f ∈ hi +N(v,H). Hence

v(y)f(y) = v(y)[f(y)− hi(y)] + v(y)hi(y) ∈ H +H ⊆W.

Thus vA vanishes at infinity on X .(⇐) Suppose (i) - (iii) hold. Since CVo(X,E) ⊆ C(X,E), by (i), A

is an equicontinuous subset of C(X,E). Further, since, by (ii), A is p-precompact, it follows that A is a precompact subset of (C(X,E), k) (cf.Lemma 3.1.2). To show that A is precompact in CVo(X,E), let v ∈ Vand W ∈ W. Choose a balanced H ∈ W such that H + H ⊆ W . By(iii), there exists a compact K ⊆ X such that

v(y)f(y) ∈ H for all f ∈ A and y ∈ X\K. (2)

Since v is upper-semicontinuous, ‖v‖K = supv(y) : y ∈ K <∞. SinceA is precompact in (C(X,E), k), there exist h1, ..., hn ∈ A such that

A ⊆n⋃

i=1

(hi +N(χK ,1

‖v‖K + 1H)). (3)

We claim that

A ⊆n⋃

i=1

(hi +N(v,W )).

Let f ∈ A and y ∈ X . If y ∈ K, by (3), there exists j ∈ 1, ..., n suchthat

v(y)[f(y)− hj(y)] ∈ v(y)1

‖v‖K + 1H ⊆ H.

If y ∈ X\K, then, for any i ∈ 1, ..., n, (2) gives

v(y)[f(y)− hi(y)] = v(y)f(y)− v(y)hi(y) ∈ H −H ⊆ W.

This establishes our claim, and so A is precompact in CVo(X,E).

Following [RS88], we shall use the following notation: for any A ⊆CVpc(X,E), v ∈ V and W ∈ W, let

Tx(A, v,W ) = y ∈ X : v(y)f(y)−v(x)f(x) ∈ W for all f ∈ A, x ∈ X.

Theorem 3.2.4. [RS88, KO05] Let X be a VR-space. Then, for anyA ⊆ CVpc(X,E), the following are equivalent:

(a) A is precompact.(b) (i) A is equicontinuous;

(ii) A(x) is precompact in E for each x ∈ X ;


(iii) given v ∈ V and W ∈ W, there exists a compact set K ⊆ Xsuch that Tx(A, v,W ) : x ∈ K covers X.

(c) (i) vA(X) = v(x)f(x) : x ∈ X, f ∈ A is precompact in E eachv ∈ V ;

(ii) given v ∈ V and W ∈ W, there exists a finite set F ⊆ Xsuch that Tx(A, v,W ) : x ∈ F covers X.

(d) (i) A(x) is precompact in E for each x ∈ X ;(ii) given v ∈ V and W ∈ W, there exists a finite set F ⊆ X such

that Tx(A, v,W ) : x ∈ F covers X.

Proof. (a)⇒ (b) Since A is a precompact subset of CVpc(X,E), justas in the proof of Theorem 3.2.3, (b)(i) follow from Theorem 3.2.2 and(b)(ii) follows from the fact that V > 0. To prove (b)(iii), let v ∈ V andW ∈ W. Choose a balanced H ∈ W such that H +H +H ⊆ W . SinceA is precompact, there exist h1, ..., hn ∈ A such that

A ⊆n⋃

i=1

[hi +N(v,H)]. (1)

Now, consider the function h : X → En defined by

h(x) = (h1(x), h2(x), . . . , hn(x)).

This is a continuous function such that (vh)(X) is precompact, for it iscontained in the product

∏ni=1(vhi)(X) which is precompact. Hence for

the neighborhood Hn, there exists a finite subset F of X such that

(vh)(X) =⋃

x∈F

((vh)(x) +Hn) .

This gives

X ⊆⋃

x∈F

Tx(h1, . . . , hn, v, H). (2)

We now show that Tx(A, v,W ) : x ∈ F covers X . Fix y ∈ X. By(2), y ∈ Tx(hini=1, v, H) for some x ∈ F and so

v(y)hi(y)− v(x)hi(x) ∈ H for all i = 1, ..., n. (3)

Given f ∈ A, by (1), there exists i ∈ 1, ..., n such that f−hi ∈ N(v,H);i. e.,

v(z)(f(z)− hi(z)) ∈ H for all z ∈ X. (4)

So, by (3) and (4),

v(y)f(y)− v(x)f(x) = (v(y)f(y)− v(y)hi(y))

+(v(y)hi(y)− v(x)hi(x))

+(v(x)hi(x)− v(x)h(x)


∈ H +H −H ⊆W.

Hence y ∈ Tx(A, v,W ); i.e., (b)(iii) holds.(b) ⇒ (c) Suppose (b) holds. We first note that (b)(i) and (b)(ii)

together imply (as in the proof of Theorem 3.2.3) that A is a precompactsubset of (C(X,E), k). We now verify (c)(i) and (c)(ii).

(c)(i). Let v ∈ V and W ∈ W. Choose a balanced H ∈ W such thatH +H +H +H ⊆ W. By (b)(iii), there exists a compact K ⊆ X suchthat

Tx(A, v, H) : x ∈ K covers X. (1)

Since A is precompact in (C(X,E), k), there exist h1, ..., hn ∈ A suchthat

A ⊆n⋃

i=1

[hi +N(χK ,1

‖v‖K + 1H)]. (2)

Moreover, since each vhi(K) is precompact in E, there exist xij : 1 ≤j ⊆ ni ⊆ K such that

vhi(K) ⊆ni⋃

j=1

[v(xij)h(xij) +H ]. (3)

Now, fix any y ∈ X and f ∈ A. By (1), y ∈ Tx(A, v, H) for some x ∈ Kand so

v(y)f(y)− v(x)f(x) ∈ H for all f ∈ A. (1a)

By (2), there exists i ∈ 1, ..., n such that

(f − hi)(K) ⊆1

‖v‖K + 1H. (2a)

By (3), there exists j ∈ 1, ..., n such that

v(x)hi(x)− v(xij)hi(xij) ∈ H. (3a)

By (1a), (2a), (3a)

v(y)f(y)− v(xij)hi(xij) =[v(y)f(y)− v(x)f(x)] + v(x)[f(x)− hi(x)]

+ [v(x)hi(x)− v(xij)hi(xij)]

∈H + v(x)1

‖v‖K + 1H +H ⊆W. (4)

i.e. vA(X) ⊆n⋃i=1

ni⋃j=1

[v(xij)hi(xij) +W ] and vA(X) is precompact in E.

(c)

(ii


). Let v ∈ V and W ∈ W, and suppose these be same as in the proof

of (c)(i). Also, let hi and xij be same as above. Let F =n⋃i=1

xij : j =

1, ..., ni. Then, for any fixed y ∈ X and f ∈ A, (3a) and (4) give

v(y)f(y)− v(xij)f(xij) = [v(y)f(y)− v(xij)hi(xij)]

+[v(xij)hi(xij)− v(xij)f(xij)]

∈ (H +H +H) +H ⊆W ;

that is, y ∈ Tx(A, v,W ) with x ∈ F . Hence Tx(A, v,W ) : x ∈ F coversX .

(c) ⇒ (d) Since V > 0, p ≤ ωV and so, for any v ∈ V, vA(X) isprecompact implies that A(x) is precompact in E for each x ∈ X .

(d)⇒ (a) Suppose (d) holds. Fix any v ∈ V and W ∈ W. Choose abalanced H ∈ W with H +H +H ⊆ W . By (d)(ii) there exists a finiteset F ⊆ X such that Tx(A, v, H) : x ∈ F covers X . By (d)(i), A isp-precompact, and so there exists hi, ..., hn ∈ A such that

A ⊆n⋃

i=1

(hi +N(χF ,1

‖v‖F + 1H). (1)

We claim that A ⊆n⋃i=1

(hi + N(v,W )). Fix any f ∈ A. By (1), there

exists i ∈ 1, ..., n such that

(f − hi)(F ) ⊆1

‖v‖F + 1H).

Then, for any y ∈ X, y ∈ Tx(A, v, H) for some x ∈ F and so

v(y)f(y)− v(y)hi(y) = [v(y)f(y)− v(x)f(x)] + v(x)[f(x)− hi(x)]

+[v(x)hi(x)− v(y)hi(y)]

∈ H + v(x)1

‖v‖F + 1H −H

⊆ H +H +H ⊆W.

This proves our claim; i.e., (a) holds.

In the particular cases of V = K+c (X), K+(X) and S+

o (X), we men-tion that Theorems 3.2.3 and 3.2.4 give further improvements of Theo-rems 3.1.3 and 3.1.4 and are stated as follows:

Corollary 3.2.5. Let X be a kR-space and E a quasi-completeTVS . A subset A of (C(X,E), k) is relatively compact iff the follow-ing conditions hold:

(i) A is equicontinuous on each compact subset of X ,


(ii) A(x) is relatively compact in E for each x ∈ X.

Corollary 3.2.6. Let X be a locally compact space and E a quasi-complete TVS . A subset A of (Co(X,E), u) is relatively compact iff thefollowing conditions hold:

(i) A is equicontinuous,(ii) A(x) is relatively compact in E for each x ∈ X,(iii) A uniformly vanishes at infinity on X ; i.e., for any W ∈ W,

there exists a compact set K ⊆ X such that f(y) ∈ W for all f ∈ A andy ∈ X\K.

Corollary 3.2.7. Let X be a kR-space and E a quasi-completeTVS . A subset A of (Cb(X,E), βo) is relatively compact iff the followingconditions hold:

(i) A is equicontinuous on each compact subset of X,(ii) A(X) is relatively compact in E,(iii) A is uniformly bounded (i.e., A(X) is bounded in E).

Corollary 3.2.8. Let E be a quasi-complete TVS . A subset A of(Cpc(X,E), u) is relatively compact iff the following conditions hold:

(i) A(X) is relatively compact in E,(ii) given W ∈ W, there exists a finite open cover Ki : i = 1, ..., n

of X such that, for any i ∈ 1, ..., n and x, y ∈ Ki,

f(x)− f(y) ∈ W for all f ∈ A.



Section 3.1. Theorem 3.1.1. is a version of the Arzela-Ascoli Theo-rem for the k-topology, given in the book of Kelley and Namioka [KN63,p. 81] without a proof. For the readers’ benefit and interest, we present awell-known version of this theorem (Theorem 3.1.4) in the more generalsetting of uniform spaces. Its proof is adapted from [Eng88, p. 440-443;Will70, § 43]. Theorem 3.1.5 is an analogue of the Arzela-Ascoli theoremfor the β-topology, as given in [Kh79]. We mention that the abstractversion of the Arzela-Ascoli theorem was first given by Myers [My46]which was later improved by Gale [Gal50]. The name k-space also firstappeared in the paper [Gal50].

Section 3.2. Here the Arzela-Ascoli type theorems which charac-terize precompact subsets of weighted functions spaces CVo(X,E) andCVpc(X,E) are taken from [RS88, KO05]. When the range space Ehappens to be quasi-complete, these characterizations become criteriafor relative compactness. Theorem 3.2.2. is an analogue of Bierstedt[Bie75b], Ruess and Summers ([RS88], Lemma 2.3) and Oubbi ([Ou02],Proposition 9).

CHAPTER 4

The Stone-Weierstrass type Theorems

In this chapter, we are primarily concerned with presenting some gen-eralizations of the classical Stone-Weierstrass theorem to Cb(X,E) andCVb(X,E) endowed with various topologies for E a non-locally convexTVS. We also consider Weierstrass polynomial approximation, charac-terization of maximal submodules and the approximation property forvector-valued functions. Both X and E are always assumed to be Haus-droff..

63

64 4. THE STONE-WEIERSTRASS TYPE THEOREMS

1. Approximation in the Strict Topology

Various versions of the Stone-Weierstrass theorem for the spaces (Cb(X ,E), β), (Cb(X,E), u) and (C(X,E), k) are considered in this section.

Recall that M.H. Stone ([St37], p. 466) obtained the following gener-alization of the Weierstrass approximation theorem of 1885 (see [St48];[Simm62], p.157-167).

Theorem A. (Classical Stone-Weierstrass theorem for C(X)= C(X,C).) Let X be a compact space. Suppose a subset A of C(X)satisfies

(1) A is a self-adjoint subalgebra,(2) A separates points of X, i.e., for any x 6= y in X, there exists

an f ∈ A such that f(x) 6= f(y),(3) A does not vanish on X, i.e., for each x ∈ X. there exists a

function g ∈ A such that g(x) 6= 0.Then A is u-dense in C(X).

Note. If X is locally compact, then the above theorem holds forCo(X) in place of C(X).

We now consider the case of vector-valued functions. If E is a vectorspace, then C(X,E) need not be an algebra but is a Cb(X)-module. Anapparent analogue of Theorem A may be stated in the following form.

Theorem B. Let X be a compact space and E a normed space (ora TVS). Suppose a subset A of C(X,E) satisfies

(i) A is a C(X)-submodule,(ii) A separates points of X,(iii) A does not vanish on X.Then A is u-dense in C(X,E).Unfortunately, Theorem B is not true in general as we now show.

Counter-example: [Buc58] For any fixed xo ∈ X and M a properclosed vector subspace of a TVS E, let

S(xo,M) = f ∈ Cb(X,E) : f(xo) ∈M.

Then A = S(xo,M) satisfies conditions (i), (ii), (iii) as follows. Clearly,A is a C(X)-submodule of C(X,E). To verify (ii), let x 6= y ∈ X . SinceX , being a compact Hausdorff space, is normal, there exists a functionϕ ∈ Cb(X) such that 0 ≤ ϕ ≤ 1, ϕ(x) = 0 and ϕ(y) = 1. For anya( 6= 0) in M , (ϕ ⊗ a)(xo) = ϕ(xo)a ∈ M and so ϕ ⊗ a ∈ A. Further(ϕ⊗ a)(x) 6= (ϕ⊗ a)(y). To verify (iii), consider any b( 6= 0) in M . If 1 isthe unit function in Cb(X), then 1⊗ b is a non-zero constant function in

1. APPROXIMATION IN THE STRICT TOPOLOGY 65

A. Clearly, A = S(xo,M) is a proper u-closed subspace of C(X,E) andso cannot be u-dense in C(X,E).

Thus one must look for an appropriate replacement of conditions(i), (ii) and (iii) in Theorem B. Kaplansky ([Kap51], §3) had consideredsimilar conditions and obtained some non-commutative generalizationsof the classical Stone-Weierstrass theorem, replacing E by a family Ex :x ∈ X of C∗-algebras having units. But in our setting, the followingtheorem for the strict topology β on Cb(X,E) was established by R.C.Buck [Buc58].

Theorem 4.1.1. (Stone-Weierstrass theorem for (Cb(X,E), β)

). Let X be a locally compact metrizable space and E a finite dimen-sional normed space. Suppose a subset A of Cb(X,E) satisfies

(1) A is a Cb(X)-submodule,(2) for each x ∈ X, the set A(x) = g(x) : g ∈ A is dense in E.Then A is β-dense in Cb(X,E).The above result was later extended to the case of X any locally

compact space and E a locally convex TVS by Todd [Tod65] and Wells[Wel65], independently. In this chapter we first present a Stone-Weierstrasstype theorem, due to Khan [Kh79], with E any TVS but under the ad-ditional condition of finite covering dimension on X (see Section A.1).

The following lemma on “partition of unity”, due to Nachbin ([Nac67],Lemma 2, p. 69) plays a crucial role in the proof of Theorem 4.1.3 andalso in some later theorems.

Lemma 4.1.2. (Partition of Unity) Let X be a completely regularspace, and let M ⊆ C(X,K) be a Cb(X,K)-submodule. Let K be acompact subset of X such that

( ∗) M does not vanish on K.Then, given any open cover U1, ..., Un of K, there exist ϕ1, ..., ϕn ∈

M, such that

0 ≤ ϕi ≤ 1, ϕi = 0 outside Ui,

n∑

i=1

ϕi = 1 on K, and

n∑

i=1

ϕi ≤ 1 on X.

We mention that ifM = Cb(X,K), (∗) clearly holds.Proof. By (∗), for each t ∈ K, there is an ft ∈M such that ft(t) 6= 0.

We may assume that ft ≥ 0. [Indeed, let γ : K −→ K be defined by

γ(x) =

0 if x = 0,

|x|2 /x if 0 < |x| < 1,|x| /x if |x| ≥ 1.


Clearly, γ is bounded, continuous and satisfies γ(x) 6= 0 for x 6= 0 andγ(x)x ≥ 0 for x ∈ K. Set φ = γ ft ∈ Cb(X,K); then φ(x) 6= 0 andφft ≥ 0. We replace ft by φft ∈ M to achieve ft ≥ 0]. We may alsoassume that ft = 0 outside some Ui. [In fact, choose some Ui containingt and ψ ∈ C(X,K) such that

0 ≤ ψ ≤ 1, ψ(t) = 1 and ψ = 0 outside Ui.

Therefore, it would be sufficient to replace ft by ψft ∈M to assure thatft = 0 outside some Ui.]

By compactness of K, there exists a finite collection f1, ..., fm ⊆ft : t ∈ K ⊆ M such that

fj ≥ 0 on X , fj = 0 outside someUj (1 ≤ j ≤ m) and

m∑

j=1

fj(x) > 0

for allx ∈ K.

For each i = 1, ..., n, define Ji = j : fj = 0 outside Ui, 1 ≤ j ≤ m, andlet gi =

∑j∈Ji

fj. Then gi ∈M,

gi ≥ 0, gi = 0 outside Ui (1 ≤ i ≤ n) and

n∑

i=1

gi(x) > 0 for all x ∈ K.

By compactness of K, there is an h ∈ Cb(X,K) such that 0 ≤ h ≤1/∑n

i=1 gi on X , and that h = 1/∑n

i=1 gi on K. Setting ϕi = hgi, weclearly have

0 ≤ ϕi ≤ 1, ϕi = 0 outside Ui,

n∑

i=1

ϕi = 1 on K and

n∑

i=1

ϕi ≤ 1

on X.

Theorem 4.1.3. [Kh79] Let X be a completely regular space of fi-nite covering dimension and E a TVS. If A is a Cb(X)-submodule ofCb(X,E) such that, for each x ∈ A, A(x) is dense in E, then A isβ-dense in Cb(X,E).

Proof. Suppose X has covering dimension of order n, and let f ∈Cb(X,E). Let ϕ ∈ Bo(X), 0 ≤ ϕ ≤ 1, and W ∈ W. There exists abalanced V ∈ W such that V + V + ...+ V ((n+2)-terms) ⊆W . Let Kbe a compact subset of X such that ϕ(x)f(x) ∈ V for x /∈ K. For eachx ∈ X , choose a function gx in A and an open neighborhood G(x) of xsuch that

gx(y)− f(y) ∈ V for all y ∈ G(x).


The sets in G(x) : x ∈ K form an open cover of K, and so thereexists a finite open cover, G(xj): j = 1, ..., m say, of K. The sets inU = X\K,G(xj) : j = 1, ..., m form a finite open cover of X , andso, by hypothesis, there exists an open cover B of X such that B is arefinement of U and any point of X belong to at most n+ 1 members ofB. Since K is compact, a finite number of members of B, U1, ..., Ur say,will cover K. Moreover, since B is a refinement of U , for each 1 ≤ i ≤ r,there exists a ji, 1 ≤ m, such that Ui ⊆ G(xji). By Lemma 4.1.2, thereexist ϕ1, ..., ϕr ∈ Cb(X) such that 0 ≤ ϕi ≤ 1, ϕi = 0 outside of Ui,∑r

n=1 ϕi(x) = 1 for x ∈ K, and∑r

n=1 ϕi(x) ≤ 1 for x ∈ X . We define anE-valued function g on X by

g(x) =r∑

n=1

ϕi(x)gxji (x), x ∈ X,

where gxji is the function in A chosen as indicated earlier. Then g ∈ A.Let y be any point in X . If Iy = i : y ∈ Ui, then Iy has at most(n+ 1)-members and ϕi(y) = 0 if i /∈ Iy. Consequently, if y ∈ K, then

ϕ(y)(g(y)− f(y)) = ϕ(y)

r∑

n=1

ϕi(x)(gxji (y)− f(y))

= ϕ(y)

∑

i∈Iy

ϕi(y)(gxji (y)− f(y))

∈ V + V + ...+ V (at most (n+1)-times)

⊆ W.

If y /∈ K, we have

ϕ(y)(g(y)− f(y)) = ϕ(y)

r∑

i=1

ϕi(y)[gxji(y)− f(y)])

+

r∑

i=1

ϕi(y)− 1

ϕ(y)f(y) (∗)

∈ V + ... + V (at most (n+1)-times) + V

⊆ W.

Thus g − f ∈ N(ϕ,W ), and so f belongs to the β-closure of A; that is,A is β-dense in Cb(X,E), as required.

An equivalent version of Theorem 4.1.3 is as follows.


Theorem 4.1.4. [Buc58, Kh79] Let X and E be as in Theorem4.1.3, and let A be a Cb(X)-submodule of Cb(X,E) and f ∈ Cb(X,E).Then the following conditions are equivalent:

(1) f belongs to the β-closure of A.(2) For each x ∈ X, f(x) ∈ A(x).Proof. (1) ⇒ (2) Suppose that f belongs to the β-closure of A, and

let x be any point in X . Let fα be a net in A such that f = β-limα fα.Since p ≤ β, f = p-limα fα, and so, in particular, fα(x) −→ f(x). Since

fα(x) ⊆ A(x), it follows that f(x) ∈ A(x).

(2) ⇒ (1) Suppose that, for each x ∈ X , f(x) ∈ A(x). If N(ϕ,W )is any β-neighborhood of 0 in Cb(X,E), then, in the same way as inTheorem 4.1.3, we can find a function h in A such that h−f ∈ N(ϕ,W ).Thus h belongs to the β-closure of A.

We next state two versions of the Stone-Weierstrass theorem for theu and k topologies. Their proofs are omitted as they are similar to thatof Theorem 4.1.3.

Theorem 4.1.5. [Kh80] Let X be a locally compact space of finitecovering dimension and E a Hausdorff TVS. If A is a Co(X)-submoduleof Co(X,E) such that, for each x ∈ X, A(x) is dense in E, then A isu-dense in Co(X,E).

Theorem 4.1.6. [Kh95] Let X be a completely regular space suchthat every compact subset of X has a finite covering dimension, and letE be a TVS. If A is a C(X)-submodule of C(X,E) such that, for eachx ∈ X, A(x) is dense in E, then A is k-dense in C(X,E).

Corollary 4.1.7. Let X be a completely regular space and E a TVS.(a) If X has finite covering dimension, then Cb(X) ⊗ E is β-dense

in Cb(X,E).(b) If X is locally compact and has finite covering dimension, then

Co(X)⊗ E is u-dense in Co(X,E).(c) If each compact subset of X has finite covering dimension, then

C(X)⊗ E is E is k-dense in C(X,E).Proof. (a) Since Cb(X) contains the unit function 1, it is easy to see

that (Cb(X)⊗ E)(x) = E for each x ∈ X . It follows from the Theorem4.1.3 that Cb(X)⊗E us β-dense in Cb(X,E).

(b) Its proof is similar to part (a) by using Theorem 4.1.5.(c) Its proof is similar to part (a) by using Theorem 4.1.6.

Remark 4.1.8. If E is a locally convex TVS, then, with slightmodification in the proofs, Theorems 4.1.3-4.1.6 and Corollary 4.1.7 holdwithout the hypothesis of finite covering dimension.


Let Cpc(X,E) (resp. Crc(X,E)) denote the subspace of Cb(X,E) con-sisting of those functions f such that f(X) is precompact (resp. relativelycompact). Note that if E is complete, then Cpc(X,E) = Crc(X,E).

Recall that a TVS E is said to have the approximation property if theidentity map on E can be approximated uniformly on precompact setsby continuous and linear maps of finite rank; E is said to be admissible ifthe identity map on E can be approximated uniformly on compact setsby continuous maps of finite rank. Every locally convex TVS and everyF -space with a basis (e.g. ℓp, 0 < p < 1) is admissible.

Theorem 4.1.9. [Shu72, Kh79, Kh95] Let X be a completely regularspace and E a TVS.

(a) E is admissible iff for all topological spaces X, Cb(X) ⊗ E isu-dense in Crc(X,E).

(b) If E has the approximation property, then Cb(X)⊗E is u-densein Cpc(X,E).

(c) If X is a normal space of finite covering dimension and E anyTVS, then Cb(X)⊗ E is u-dense in Crc(X,E).

(d) If E is an admissible TVS, then(i) C(X)⊗E is k-dense in C(X,E).(ii) Cb(X)⊗ E is β-dense in Cb(X,E).

Proof. (a) Suppose E is admissible, and let f ∈ Crc(X,E) and W ∈

W a balanced set. Since f(X) is compact in E, there exists a continuous

map u : f(X)→ E with range contained in a finite dimensional subspace

of E such that u(a)−a ∈ W for all a ∈ f(X). Then h = uf ∈ Cb(X)⊗Eand h(x)− f(x) ∈ W for all x ∈ X .

Conversely, let K ⊆ E be a compact set and W ∈ W. Since, byhypothesis, Cb(K) ⊗ E is u-dense in Cb(K,E), there exists some v =∑n

i=1 vi⊗ai (vi ∈ Cb(K), ai ∈ E) in Cb(K)⊗E such that v(a)−a ∈ W forall a ∈ K. Note that the range of v is contained in the finite dimensionalsubspace spanned by a1, ..., an. Thus E is admissible.

(b) Suppose E has the approximation property, and let f ∈ Cpc(X,E)

and W ∈ W a balanced set. Since f(X) is precompact in E, there exists

a continuous linear map u : f(X) → E with range contained in a finite

dimensional subspace of E such that u(a) − a ∈ W for all a ∈ f(X).Then h = u f ∈ Cb(X)⊗E and h(x)− f(x) ∈ W for all x ∈ X .

(c) Since X is normal, its Stone-Cech compactification βX also hasfinite covering dimension [GJ60]. So, by Corollary 4.1.7, Cb(βX) ⊗ Eis u-dense inCb(βX,E). Note that each function in Crc(X,E) has acontinuous extension to all of βX . Hence Cb(X)⊗ E and Crc(X,E) are


linearly isomorphic to Cb(βX) ⊗ E and Cb(βX,E), respectively, and sothe result follows.

(d) (i) Let f ∈ C(X,E), and let K be a compact subset of X andW ∈ W a balanced set. By hypothesis, there exists a continuous mapu : f(K) → E with range contained in a finite dimensional subspace ofE such that u(f(x))− f(x) ∈ W for all x ∈ K. We can write

u f =m∑

i=1

(ui f)⊗ ai, where ui f ∈ C(K) and ai ∈ E.

By the famous Tietze extension Theorem, there exist h1, ..., hm ∈ C(X)such that hi = ui f on K. Let g =

∑mi=1 hi ⊗ ai. Then g ∈ C(X)⊗ E

and g − f ∈ N(K,W ).(d) (ii) Let f ∈ Cb(X,E), and let ϕ ∈ Bo(X) and W ∈ W. Choose

an open balanced neighborhood V ∈ W such that V + V + V ⊆ W .Choose r > ‖ϕ‖ with f(x) ⊆ rV , and let

K = x ∈ X : ϕ(x) ≥ 1/r.

Then f(K) is a compact subset of E and so, by hypothesis, there ex-ists a continuous map u : f(K) → E with range contained in a finitedimensional subspace of E such that

u(f(x))− f(x) ∈ (1/r)G for all x ∈ K.

We can write

u f =

n∑

i=1

(ui f)(x)ai (ui f ∈ C(K) and ai ∈ E).

Again there exist θi(1 ≤ i ≤ n) in Cb(X) such that θi = ui f on K. Leth =

∑ni=1 θi ⊗ ai. Then

K ⊆ h−1(rV + rV ) = F (say),

which is open inX , and so there exists a θ ∈ Cb(X) with 0 ≤ θ ≤ 1, θ = 0on X\F . Let g = θh. Then g ∈ Cb(X) ⊗ E and g = h = u f onK. Further, g(x) ⊆ rV + rV . It is now easily verified that g − f ∈N(ϕ,W ).

Next, we give a brief account of an interesting and more general ver-sion of the Stone-Weierstrass theorem due to Prolla [Pro93b]. Most ofthe preceding versions of this theorem describe the closures of “Cb(X)-submodules” of Cb(X,E) under various topologies. Prolla [Pro93b] hasestablished a result that describes the u-closure of certain “subsets” ofCb(X,E) which are not necessarily linear subspaces (see also [Pro93a,


PN97]). Its proof uses two results of Jewitt ([Je63], Lemma 2 and The-orem 1) and a “lemma” of Prolla ([Pro93b], Lemma 3) on “approximat-ing the characteristic functions of open neighborhoods”. (This lemmais an analogue of Lemma 1.3 of Machado’s proof [Mac77] of the Stone-Weierstrass theorem, and of Lemma 1 of Brosowski and Deutsch [BD81].We shall not reproduce the proofs here since they can also be found inthe book ([Pro93a], Chapter 11).

We first introduce some notations and terminology. Let X be a com-pact space and E a TVS. (In this case C(X,E) = Cb(X,E) and β = u.)Let I = [0, 1], and let C(X, I) denote the subset of C(X,R) consistingof those functions from X into I. A subsetM of C(X, I) is said to haveproperty (V ) [Je63] if

(1) ϕ ∈ M implies 1− ϕ ∈M;(2) ϕ, θ ∈ M implies ϕθ ∈M.If A ⊆ C(X,E), then a function ϕ ∈ C(X, I) is called a convex

multiplier of A if

ϕf + (1− ϕ)g ∈ A for every pair f, g ∈ A.

Let MA denote the set of all multipliers of a subset A of C(X,E).Clearly, ϕ ∈ MA implies 1 − ϕ ∈ MA. Further, if ϕ, θ ∈ M, thenthe identity

(ϕθ)f + (1− ϕθ)g = ϕ(θf + (1− θ)g] + (1− ϕ)g

for f, g ∈ A shows that ϕθ ∈ MA. Hence MA has property V . Wemention that property (V ) is considered as a hypothesis in Theorem 1 of[Je63].

The main results of [Pro93a] are stated as follows.

Theorem 4.1.10. Let X be a compact space of finite covering di-mension and E a TVS. Let A be a subset of C(X,E) such that set MA

of all multipliers of A separates the points of X. Let f ∈ C(X,E) begiven. The following are equivalent:

(1) f belongs to u-closure of A.

(2) For each x ∈ X, f(x) ∈ A(x).

Corollary 4.1.11. Let X and E be as given above, and let A be asubset of C(X,E) such that

(a) the set MA of multipliers separates points of X;(b) for each x ∈ X, A(x) is dense in E.Then A is u-dense in C(X,E).

Corollary 4.1.12. Let X and E be as given above. Then C(X)⊗Eis u-dense in C(X,E).


2. Approximation in the Weighted Topology

We next consider the weighted approximation problem. Let V ba Nachbin family on X , M ⊆ C(X) a Cb(X)-submodule, and A ⊆CVo(X,E) an M-submodule. The ‘weighted approximation problem’,formulated originally by Nachbin [Nac65], consists in finding a charac-terization of the ωV -closure of A in CVo(X,E).

ForM⊆ C(X) or C(X,E), we shall denote

z(M) = x ∈ X : g(x) = 0 for all g ∈M.

Theorem 4.2.1. [Pr71a, Kh85] Let X be a completely regular spaceof finite covering dimension, E a TVS, V > 0 a Nachbin family on X,and M ⊆ C(X) a Cb(X)-submodule. Let A ⊆ CVo(X,E) be an M-submodule such that z(M) ⊆ z(A), and let f ∈ CVo(X,E). Then f

belongs to the ωV -closure of A iff, for each x ∈ X, f(x) ∈ A(x).

Proof. (⇒) Suppose f belongs to the ωV -closure of A. Since V > 0,ρ ≤ ωV and so f belongs to the ρ-closure of A; that is, or each x ∈ X ,f(x) ∈ A(x).

(⇐) Suppose X has covering dimension n, and let v ∈ V and G ∈ W.Choose a balanced H ∈ W with

H +H + ...+H((n+ 2)-terms) ⊆ G.

Then K = x ∈ X : v(x)f(x) /∈ H is compact. By hypothesis, for eachx ∈ K, we can choose a gx ∈ A such that f(x)− gx(x) ∈ H . Note thatgx(x) 6= 0 and so, since z(M) ⊆ z(A), K ⊆ X\z(M). Choose openneighborhoods S ′(x) and S ′′(x) of x in X such that

v(y) ≤ 1 for all y ∈ S ′(x) and f(y)− gx(y) ∈ H for all y ∈ S ′′(x).

Put S(x) = S ′(x) ∩ S ′′(x). Then open cover S(x) : x ∈ K has a finitesubcover S(xj) : 1 ≤ j ≤ m (say). Let U be an open refinement ofX\K,S(xj) : 1 ≤ j ≤ m such that any point of X belongs to at mostn + 1 members of U . Choose Ti : 1 ≤ i ≤ r ⊆ U which covers K.Since K ⊆ X\z(M), we can apply Lemma 4.1.2 to obtain a collectionϕi : 1 ≤ i ≤ r ⊆ M such that 0 ≤ ϕi ≤ 1, ϕi = 0 outside Ti,∑r

i=1 ϕi = 1 on K, and∑r

i=1 ϕi ≤ 1 on X , Then g =∑r

i=1 ϕigji ∈ Aand, for any y ∈ X ,

v(y)[g(y)− f(y)] =v(y)r∑

i=1

ϕi(y)[gji(y)− f(y)] + [r∑

i=1

ϕi(y)− 1]v(y)f(y)

∈H + ... +H (at most (n+1)-terms) +H ⊆ G

2. APPROXIMATION IN THE WEIGHTED TOPOLOGY 73

(by considering the cases y ∈ K and y /∈ K); that is, g − f ∈ N(v,G).Thus f belongs to the ωV -closure of A.

Corollary 4.2.2. Let X, E, V and A be as in Theorem 4.2.1and M = Cb(X). Then A is ωV -dense in CVo(X,E) iff for eachx ∈ X, A(x) is dense in E. In particular, CVo(X) ⊗ E is ωV -densein CVo(X,E).

Theorem 4.2.3. [Kh87] (a) CVo(X,E) is ωV -closed in CVb(X,E).(b) If X is locally compact, then Coo(X,E) is ωV -dense in CVo(X,E).Proof. (a) Let f belong to the ωV -closure of CVo(X,E) in CVb(X,E).

To show that f ∈ CVo(X,E), let v ∈ V . Then, for any W ∈ W, the setF = x ∈ X : v(x)f(x) /∈ W is compact, as follows. Choose a balancedG ∈ W with G + G ⊆ W . There exists a function g ∈ CVo(X,E) suchthat

g − f ∈ N(v,G).

Now the set K = x ∈ X : v(x)g(x) /∈ G is compact. If x ∈ X\K, then

v(x)f(x) ∈ v(x)g(x) +G ⊆ G+G ⊆W,

and so x ∈ X\F. Therefore, F ⊆ K which implies that F is compact.Thus f ∈ CVo(X,E).

(b) Let f ∈ CVo(X,E). To show that f belongs to the ωV -closureof Coo(X,E), let v ∈ V and W ∈ W be balanced. Now K = x ∈ X :v(x)f(x) /∈ W is compact. Since X is locally compact, there existsa ψ ∈ Coo(X) such that 0 ≤ ψ ≤ 1 and ψ = 1 on K. Then clearlyg = ψf ∈ Coo(X,E). Let x ∈ X. If x ∈ K, then

v(x)[f(x)− g(x)] = v(x)[f(x)− f(x)] = 0 ∈ W ;

If x ∈ X\K, then

v(x)[f(x)− g(x)] = [ψ(x)− 1]v(x)f(x) ∈ [ψ(x)− 1]W ⊆W ;

that is g − f ∈ N(v,W ). Thus Coo(X,E) is ωV -dense in CVo(X,E).

Theorem 4.2.4. [Kh87] Let X be a completely regular Hausdorffspace. Suppose E is a locally bounded TVS. Then Cb(X,E)∩CVo(X,E)is ωV -dense in CVo(X,E).

Proof. Let B be a bounded neighborhood of 0 in E. There exists aclosed shrinkable neighborhood S of 0 in E with S ⊆ B. The Minkowskifunctional ρ = ρS of S is continuous and positively homogeneous, andS = a ∈ E : ρ(a) ≤ 1.Consequently, for any t > 0, the function


ht : E −→ E defined by

ht(a) =

a if a ∈ tS,t

ρ(a)a if a /∈ tS,

is continuous. Further, ht(E) ⊆ tS. [Let a ∈ E. If a ∈ tS, clearlyht(a) = a ∈ tS; if a /∈ tS, then ρ(a) > t > 0 and 1

ρ(a)a ∈ S, and so

ht(a) =t

ρ(a)a ∈ tS.]

Now, for any f ∈ CVo(X,E), let Ft = ht f. We claim that Ft ∈Cb(X,E) ∩ CVo(X,E). Clearly Ft is continuous on X ; further, Ft isbounded since

Ft(X) = ht(f(X)) ⊆ ht(E) ⊆ tS ⊆ tB

and B is bounded. Hence Ft ∈ Cb(X,E). Now, let v ∈ V. To show thatvFt vanishes at infinity, we need to show that, for any balanced G ∈ W,the set A = x ∈ X : v(x)Ft(x) /∈ G is compact. [Since f ∈ CVo(X,E),the set B = x ∈ X : v(x)f(x) /∈ G is compact. Let x ∈ X\B, so thatv(x)f(x) ∈ G. Then, if f(x) ∈ tS,

v(x)Ft(x) = v(x)ht(f(x)) = v(x)f(x) ∈ G

and if f(x) /∈ tS, ρ(f(x)) > t and

v(x)Ft(x) = v(x)ht(f(x)) =t

ρ(f(x))v(x)f(x) ∈

t

ρ(f(x))G ⊆ G;

that is, x ∈ X\A. Therefore A ⊆ B, which implies that A is compact.]Finally, let f ∈ CVo(X,E), and let v ∈ V and W ∈ W be balanced.

We show that, for a suitable t > 0, the function Ft = ht f. satisfiesFt − f ∈ N(v,W ). [Choose r ≥ 1 such that B ⊆ rW and B ⊆ rS. Theset K = x ∈ X : v(x)f(x) /∈ 1

rS is compact and so we choose t ≥ 1

with f(K) ⊆ trB. Now, let x ∈ X . If f(x) ⊆ tS, then

v(x)[Ft(x)− f(x)] = v(x)[ht(f(x))− f(x)] = v(x)[f(x)− f(x)] = 0 ∈ W.

If f(x) /∈ tS, then ρ(f(x)) > t and also x /∈ K (since f(K) ⊆ trB ⊆ tS)

and so v(x)f(x) ∈ 1rS; hence

v(x)[Ft(x)− f(x)] = v(x)[t

ρ(f(x))f(x)− f(x)] = [

t

ρ(f(x))− 1]v(x)f(x)

∈ [t

ρ(f(x))− 1](

1

r)S ⊆ [

t

ρ(f(x))− 1]W ⊆W.

Thus Ft − f ∈ N(v,W ).]

2. APPROXIMATION IN THE WEIGHTED TOPOLOGY 75

Theorem 4.2.5. [Kh87] Let X be a completely regular space. Sup-pose E is an admissible TVS and V ⊆ S+

o (X). Then Cb(X) ⊗ E isωV -dense in CVo(X,E).

Proof. Let f ∈ Cb(X,E), and let v ∈ V andW ∈ W. Choose an openbalanced G ∈ W such that G + G + G ⊆ W . Choose r ≥ max1, ‖v‖with f(X) ⊆ rG, and let K = x ∈ X : v(x) ≥ 1

r. Then f(K) is a

compact subset of E and so, by hypothesis, there exists a continuous mapϕ : f(K) −→ E with range contained in a finite dimensional subspaceof E such that ϕ(f(x)) − f(x) ∈ 1

rG for all x ∈ K. We can write

ϕ f =∑n

i=1(ϕi f) ⊗ ai, where ϕi f ∈ C(K) and ai ∈ E. By theTietze extension Theorem, there exist ψi (1 ≤ ψi ≤ 1) in Cb(X) suchthat ψi = ϕi f on K. Let h =

∑ni=1 ψi ⊗ ai. For any x ∈ K,

h(x) = (ϕ f)(x) ∈ f(x) +1

rG ⊆ rG+ rG;

hence K ⊆ h−1(rG + rG) = F (say). Clearly F is open in X , and sothere exists a ψ ∈ Cb(X) with

0 ≤ ψ ≤ 1, ψ = 1 on K and ψ = 0 on X\F.

Let g = ψh. Then g ∈ Cb(X) ⊗ E and g = h = ϕ f on K. Further,g(X) ⊆ rG+ rG. [Let x ∈ X. If x ∈ F, then

g(x) = ψ(x)h(x) ∈ ψ(x)(rG+ rG) ⊆ (rG+ rG);

if x /∈ F, then g(x) = 0 ∈ rG+ rG.] Finally, g− f ∈ N(v,W ), as follows.Let x ∈ X. If x ∈ K, then

v(x)[g(x)− f(x)] = v(x)[ϕ(f(x))− f(x)] ∈ v(x)1

rG ⊆ G ⊆W ;

if x /∈ K, then v(x) < 1rand so

v(x)[g(x)− f(x)] ∈ v(x)[rG+ rG− rG] ⊆ v(x)rW ⊆W.

Corollary 4.2.6. If E is locally bounded and admissible and V ⊆S+o (X), then Cb(X)⊗E is ωV -dense in CVo(X,E).We do not know whether, for any Nachbin family V and E admissible,

CVo(X)⊗E is ωV -dense in CVo(X,E). However, under some restrictionson X , this is true for E any TVS.

Theorem 4.2.7. [Kh87] Let X be a locally compact space of finitecovering dimension. Then Coo(X)⊗ E is ωV -dense in CVo(X,E).

Proof. By Theorem 4.2.3 (b), Coo(X,E) is ωV -dense in CVo(X,E).Therefore, it suffices to show that Coo(X)⊗E is ωV -dense in Coo(X,E).Let f ∈ Coo(X,E), and let v ∈ V and W ∈ W be balanced. There exists


a compact set K ⊆ X such that f(x) = 0 for x /∈ K. Choose r ≥ 1with v(x) ≤ r for all x ∈ K. Since X is locally compact and E is offinite covering dimension, by Corollary 4.1.7, Co(X) ⊗ E is u-dense inCo(X,E); hence Coo(X) ⊗ E is u-dense in Coo(X,E). Then there existsa function g ∈ Coo(X)⊗ E with g = 0 outside K and such that

g(x)− f(x) ∈1

rW for all x ∈ X.

Now, let x ∈ X. If x ∈ K, then

v(x)[g(x)− f(x)] ∈v(x)

rW ⊆W ;

if x /∈ K, then

v(x)[g(x)− f(x)] = v(x)[0 − 0] = 0 ∈ W.

Thus g − f ∈ N(v,W ).

3. WEIERSTRASS POLYNOMIAL APPROXIMATION 77

3. Weierstrass Polynomial Approximation

In this section, we present an infinite dimensional version of theWeier-strass polynomial approximation theorem, due to Bruno [Bru84].

Let E and F be real TVSs and C(E, F ) the space of all continuousfunctions from E into F , and let

P (E, F ) = span(ϕ(x))ny : ϕ ∈ E∗, x ∈ X, y ∈ F, n ≥ 1,

the set of all continuous polynomials of finite type from E into F .

Definition. A sequence Sn of continuous linear operators froma TVS E into itself is said to have the Grothendieck approximationproperty (GAP) if it satisfies:

(i) each Sn has finite rank,(ii) Sn is equicontinuous,(iii) Sn converges uniformly on compact subsets to the identity on

E.For example, in a complete metric TVS with a Schauder basis en,

the sequence of projections

Sn :

∞∑

i=1

aiei → anen

has the GAP.

Example. For 0 < p < 1, the spaces ℓp(R), given by

ℓp(R) = a = ai ⊆ R:∞∑

i=1

|ai|p <∞,

with metric d(a, b) =∑∞

i=1 |ai − bi|p, are complete metric vector spaces

with Schauder bases and are not locally convex [Simo65].

Theorem 4.3.1. [Bru84] Let E and F be real TVS. Suppose E hasa sequence Sn of projections with the GAP and F has a sequence Tnwith the GAP. Then the space P (E, F ) of continuous polynomials fromE into F of finite rank is k-dense in C(E, F ).

Proof. The proof is divided into three steps.

Step I. We show that, for any f ∈ C(E, F ) and n ≥ 1, TnfSnk−→ f .

Let K ⊆ E be a compact set and W a neighborhood of 0 in F . Choosea balanced neighborhood V of 0 in F with V + V ⊆ W . Since Tn isequicontinuous, there exists a neighborhood U of 0 in E such that

Tn(U) ⊆ V for all n ≥ 1.


Since f is uniformly continuous on K and Sn converges uniformly tothe identity of E on K, it easily follows that fSn converges uniformlyto f on K. So there exists an integer n1such that

fSn(x)− f(x) ∈ U for all x ∈ K and n ≥ n1.

Since Tn converges uniformly to the identity of F on f(K), there existsan integer n2 such that

Tnf(x)− f(x) ∈ V for all x ∈ K and n ≥ n2.

Hence, if n ≥ maxn1, n2 and x ∈ K,

TnfSn(x)− f(x) = Tn(fSn(x)− f(x)) + (Tnf(x)− f(x))

∈ Tn(U) + V ⊆ V + V ⊆W .

Step II. We show that, for any f ∈ C(E, F ) each TnfSn belongsto the k-closure of P (E, F ). Put fn = TnfSn. Let K ⊆ E be a com-pact set and V a balanced neighborhood of 0 in F . Since E and F areHausdorff TV Ss and Sn, Tn are of finite rank, Sn(E) and Tn(F ) arelinearly homeomorphic to finite-dimensional Euclidean spaces. By theclassical Stone-Weierstrass theorem [St48], P (Sn(E), Tn(F )) is k-densein C(Sn(E), Tn(F )). Since f ′n = fn|Sn(E) belong to C(Sn(E), Tn(F ))and Sn(K) is compact in Sn(E), there exists a p′n ∈ P (Sn(E), Tn(F ))such that

f ′n(x)− p′n(x) ∈ V for all x ∈ Sn(K).

We define an extension pn of p′n(x) from Sn(K) to E by

pn(x) = p′nSn(x), x ∈ E.

Clearly pn ∈ P (E, F ) and for any x ∈ K,

fn(x)− pn(x) = fnSn(x)− p′nSn(x) ∈ V.

Step III. We now verify that P (E, F ) is k-dense in C(E, F ). Letf ∈ C(E, F ), and let K ⊆ E be compact and W a neighborhood of 0 inF . Choose a balanced neighborhood V of 0 in F with V + V ⊆ W . Bystep I, there exists an integer n such that

f(x)− TnfSn(x) ∈ V for all x ∈ K.

By step II, there exists a pn ∈ P (E, F ) such that

TnfSn(x)− pn(x) for all x ∈ K.

Hence, for any x ∈ K,

f(x)− pn(x) = [f(x)− TnfSn(x)] + [TnfSn(x)− pn(x)]

∈ V + V ⊆W.

4. MAXIMAL SUBMODULES IN (Cb(X,E), β) 79

4. Maximal Submodules in (Cb(X,E), β)

This section includes characterizations of maximal and closed Cb(X)-submodules of (Cb(X,E), β) and CVo(X,E),as given in [Kh84, Kh85b].

Throughout this section, X will denote a completely regular Haus-dorff space and E a Hausdorff TVS with non-trivial topological dual E∗.

Definition. A TVS F is said to have the separation property (or,equivalently, the Hahn-Banach extension property) if, for any closed vec-tor subspace S of F and a ∈ F\S, there exists a maximal (proper) closedvector subspace M of F such that S ⊆M and a /∈M . F is said to havethe weaker separation property if every proper closed vector subspace ofF is contained in some maximal closed subspace of F . Clearly, the sep-aration property implies the weaker separation property. Every locallyconvex TVS has the separation property [Scha66], and it is shown byGregory and Shapiro [GrSh70] that if 〈F, F ′〉 is a dual pair of a vectorspace F with the weak topology on F not equal to the Mackey topology,then there is a non-locally convex linear topology between them havingthe separation property.

We shall require the following version of the Stone-Weierstrass theo-rem.

Theorem 4.4.1. [Kh84] For any Cb(X)-submodule A of Cb(X,E),consider the following conditions.

(1) A is β-dense in Cb(X,E).(2) For each x ∈ X, A(x) is dense in E.(3) For each x ∈ X and maximal closed subspace M of E, there

exists an f ∈ A such that f(x) /∈M .Then (1) ⇒ (2) ⇒ (3). If X has finite covering dimension, then (2)

⇒ (1); if E has the weaker separation property, then (3) ⇒ (2).Proof. (1) ⇒ (2) is evident since ρ ≤ β.(2) ⇒ (1) is same as Theorem 4.1.3.(2) ⇒ (3). Let x ∈ X and M a maximal closed subspace of E,

and let a ∈ E\M . Choose an open neighborhood G of a in E suchthat G ⊆ E\M . Since A(x) is dense in E, there exists an f ∈ A withf(x) ∈ G. Hence f(x) /∈M .

(3)⇒ (2). Suppose E has the weaker separation property. If A(x) 6=E for some x ∈ X , there exists a maximal closed subspace M of E suchthat A(x) ⊆ M . But, by (3), there exists a g ∈ A with g(x) /∈ M , and

so A(x) ⊆M . This gives us the desired contradiction.


Corollary 4.4.2. [Kh84] Suppose X has finite covering dimensionand E has the separation property. Let A be a Cb(X)-submodules ofCb(X,E), and f ∈ Cb(X,E). Then the following are equivalent:

(1) f belongs to the β-closure of A.(2) For each x ∈ X, f(x) ∈ A(x).(3) For each x ∈ X and maximal closed subspace M of E, A(x) ⊆ M

implies that f(x) ∈M .We now characterize the maximal β-closed Cb(X)-submodules of Cb(X ,

E). We begin with the following.

Lemma 4.4.3. [Kh84] For any x ∈ X and M a maximal closedsubspace of E,

S(x,M) = f ∈ Cb(X,E) : f(x) ∈M

is a maximal β-closed Cb(X)-submodule of Cb(X,E).Proof. First note that, since M is a vector subspace, S(x,M) is a

Cb(X)-submodule of Cb(X,E). There exists a non-zero ϕ ∈ E∗ such thatM = ϕ−1(0) (§ A.2) Given any x ∈ X , define a map Tx : (Cb(X,E), β)→E by

Tx(f) = f(x), f ∈ Cb(X,E).

If W is any neighborhood of 0 in E, then

N(x),W ) = f ∈ Cb(X,E) : f(x) ∈ W

is a β-neighborhood of 0 in Cb(X,E) such that

Tx(N(x,W )) ⊆W.

Hence Tx is continuous. It is easy to see that

S(x,M) = (ϕ Tx)−1(0),

and so S(x,M) is maximal and β-closed.

Theorem 4.4.4.[Kh84] Suppose X has finite covering dimension.(1) If E has the weaker separation property, then every maximal

β-closed Cb(X)-submodule of Cb(X,E) is of the form S(x,M), wherex ∈ X and M is a maximal closed subspace of E.

(2) If E has the separation property, then every β-closed Cb(X)-submodule of Cb(X,E) is the intersection of all maximal β-closed Cb(X)-submodules which contain it.

Proof. (1) LetA be a maximal β-closed Cb(X)-submodule of Cb(X,E),and let f ∈ Cb(X,E) with f /∈ A. Since A is β-closed, by Theorem 4.4.1,

there exists an xo ∈ X such that f(xo) /∈ A(xo). By the weaker sepa-ration property of E, there exists a maximal closed subspace M of E

4. MAXIMAL SUBMODULES IN (Cb(X,E), β) 81

such that A(xo) ⊆ M . Clearly, A ⊆ S(xo,M). Hence, by Lemma 5.5.3,A = S(xo,M).

(2) Let A1 be a β-closed Cb(X)-submodule of Cb(X,E), and letg ∈ Cb(X,E) with g /∈ A1. Then, as in the above proof and usingCorollary 4.4.2, there exist an x1 ∈ X and a maximal closed subspaceM1 of E such that

A ⊆ S(x1,M1) and g /∈ S(x1,M1).

Remarks. (1) If E is assumed to be locally convex, then Theo-rems 4.4.1 and 4.4.4, hold without restricting X to have finite coveringdimension (see Remark 4.1.8).

(2) In general, a maximal Cb(X)-submodule of Cb(X.E) need notbe β-closed. For instance if E admits a discontinuous linear functionalϕ and M = ϕ−1(0), then, for any x ∈ X, S(x,M) is maximal but notβ-closed by Theorem 4.4.1 (since M is not closed in E).

Finally, we give a characterization of the maximal ωV -closed Cb(X)-submodules of CVo(X,E).

Theorem 4.4.5. [Kh85b] Let X be of finite covering dimension,E a TVS, and V > 0 a Nachbin family on X. Suppose that E has theseparation property. Then:

(a) Every maximal ωV -closed Cb(X)-submodule of CVo(X,E) is ofthe form

SV (x,M) = f ∈ CVo(X,E) : f(x) ∈M,

where x ∈ X and M is a maximal closed subspace of E.(b) Every ωV -closed Cb(X)-submodule of CVo(X,E) is the intersec-

tion of all maximal ωV -closed Cb(X)-submodules which contain it.Proof. (a) LetA be a maximal ωV -closed Cb(X)-submodule of CVo(X ,

E), and let f ∈ CVo(X,E) with f /∈ A. By Theorem 4.2.1, there exists

an x ∈ X such that f(x) /∈ A(x). By the separation property of E, thereexists a maximal closed subspace M of E such that

A(x) ⊆M and f(x) /∈M.

It can be easily shown that S(x,M) is a maximal ωV -closed Cb(X)-submodule of CVo(X,E), and thus A = SV (x,M).

(b) The proof is similar to that of the above part and Theorem 4.4.4(2).


5. The Approximation Property

Recall that a TVS E is said to have the approximation property ifthe identity map on E can be approximated uniformly on precompactsets by continuous linear maps of finite rank (i.e. with range containedin finite dimensional subspaces of E).

Theorem 4.5.1. [Kat81] Let X be a completely regular space of finitecovering dimension and E a Hausdorff TVS having the approximationproperty. Then (Cb(X,E), β) has the approximation property.

Proof. Suppose X has finite covering dimension n. Let A be a β-precompact subset of Cb(X,E), and let ϕ ∈ Bo(X) with 0 ≤ ϕ ≤ 1 andW ∈ W a balanced set. We show that there exists a continuous linearmapping T : (Cb(X,E), β)→ (Cb(X,E), β) of finite rank such that

Tf − f ∈ N(ϕ,W ) for all f ∈ A. (1)

Choose a balanced W1 ∈ W with W1 + ... +W1(n + 2 terms) ⊆ W .Also choose an open balanced W2 ∈ W with

W2 +W2 +W2 +W2 ⊆W1.

Since A, being β-bounded, is u-bounded, there exists an r ≥ 1 suchthat f(X) ⊆ rW2 for all f ∈ A. Choose a compact K ⊆ X such thatϕ(x) < 1/r for all x ∈ X\K. Put B = ∪f(K) : f ∈ A. It is easy tosee that B is a precompact subset of E. Since E has the approximationproperty, there exists a continuous linear mapping S : E → E of finiterank such that

sa− a ∈ (1/r)W2 for all a ∈ B. (2)

Choose an open balanced W3 ∈ W such that

W3 ⊆W2 ∩ S−1(W2) =W2 ∩ a ∈ E : Sa ∈ W2. (3)

Since A is β-precompact, there exists a finite set A1 ⊆ A such that

A ⊆ A1 +N(ϕ,W3). (4)

For each x ∈ X , the set

G(x) =⋂

g∈A1

(y ∈ X : g(y)− g(y) ∈ W3 (5)

is an open neighborhood of x. Since K is compact, there exist x1, ..., xp ∈K such that K ⊆ ∪pi=1G(xi). The collection G(x1), ..., G(xp), X\Kforms an open cover of X . Since X has finite covering dimension n andK is compact, there exists an open cover H1, ..., Hm of K which isa refinement of the cover G(x1), ..., G(xp) and such that each pointof K belongs to at most n + 1 of the Hi’s. For each Hi, choose an

5. THE APPROXIMATION PROPERTY 83

index ji such that Hi ⊆ G(yi), yi = xji. By Lemma 4.1.2, there existh1, ..., hm ∈ Cb(X) with

0 ≤ hi ≤ 1, hi = 0 on X\Hi,m∑

i=1

hi(x) = 1 on K,

and∑m

i=1 hi(x) ≤ 1 for each x ∈ X .Define T : Cb(X,E)→ Cb(X,E) by

T (f) =

m∑

i=1

hi ⊗ Sf(yi), f ∈ Cb(X,E).

It is easy to verify that T is a β-continuous linear mapping of finite rank.To complete the proof, we need to verify (1). Let f ∈ A, and let x ∈ X .By (4), there exists a g ∈ A such that

ϕ(f − g)(X) ⊆ W3. (6)

Note that Ix = i : x ∈ Hi contains at most n+ 1 indices. Now

Tf(x)− f(x) =m∑

i=1

hi(x)Sf(yi)− f(x).

For any i ∈ Ix, x ∈ Hi ⊆ G(yi) and so, using (2), (3) and (6)

ϕ(x)[Sf(yi)− f(x)] = S[ϕ(x)(f(yi)− g(yi)] + ϕ(x)[Sg(yi)− g(yi)]

+S[ϕ(x)(g(yi)− g(x)] + ϕ(x)[g(x)− f(x)]

∈ S(W3) + (ϕ(x)/r)W2 + ϕ(x)W3 +W3

⊆ W2 +W2 +W2 +W2 ⊆W1.

Case I. Suppose x ∈ ∪mi=1Hi. If x ∈ K, then∑m

i=1 hi(x) = 1 and so

ϕ(x)[Tf(x)− f(x)] = ϕ(x)

[m∑

i=1

hi(x)Sf(yi)−m∑

i=1

hi(x))f(x)

]

=m∑

i=1

hi(x)ϕ(x)[Sf(yi)− f(x)]

∈m∑

i=1

hi(x)W1 ⊆W1 + ...+W1(n+1 terms)

⊆W.


If x ∈ (∪mi=1Hi)\K, then∑n

i=1 hi(x) ≤ 1 and ϕ(x)r < 1, and so

ϕ(x)[Tf(x)− f(x)] =ϕ(x)

[m∑

i=1

hi(x)Sf(yi)−m∑

i=1

hi(x))f(x)

+

m∑

i=1

hi(x)f(x)− f(x)

]

=∑

i∈Ix

hi(x)ϕ(x)[S(f(yi))− f(x)]

+

[∑

i∈Ix

hi(x)− 1

]ϕ(x)f(x)

∈∑

i∈Ix

hi(x)W1 + [∑

i∈Ix

hi(x)− 1]ϕ(x)rW2

⊆m∑

i=1

hi(x)W1 +W2

⊆W1 + ...+W1 (n+ 1)-terms+W2 ⊆W.

Case II. Suppose x ∈ X\∪mi=1Hi. Then each hi(x) = 0 and ϕ(x)r < 1and so

ϕ(x)[Tf(x)− f(x)] = ϕ(x)[0 − f(x))] ∈ ϕ(x)rW2 ⊆W.

Thus (1) holds.

Theorem 4.5.2. [Kh95] Suppose every compact subset of X hasfinite covering dimension and E has the approximation property. Then(C(X,E), k) has the approximation property.

Proof. Let A be a k-precompact subset of C(X,E), and let K be acompact subset of X and W ∈ W. Suppose K has covering dimensionn, and choose a balanced set V ∈ W such that

V + V + ...+ V ((n+ 1)-terms) ⊆W.

Now choose an open balanced set U ∈ W such that U +U +U +U ⊆ V .Since A is precompact in C(X,E), it is easy to verify that D = ∪f(K) :f ∈ A is a precompact subset of E. By hypothesis, there exists acontinuous linear map S : E → E of finite rank such that

Sa− a ∈ U for all a ∈ D.

5. THE APPROXIMATION PROPERTY 85

Choose an open balanced set U1 ∈ W such that

U1 ⊆ U ∩ a ∈ E : Sa ∈ U.

Since A is precompact, there exists a finite subset A1 of A such that

A ⊆ A1 +N(K,U1).

For each x ∈ K, the set G(x) = ∩g∈Ay ∈ X : g(y) − g(x) ∈ U1 is an

open neighborhood of x in X . Then there exists a finite open cover U= G(x1), ..., (G(xp) (say) of K. By hypothesis, we can choose an opencover V = Hi, ..., Hm of K which is a refinement of U and such thateach point of K belongs to at most n + 1 members of V. For each Hi,choose an index ji such that Hi ⊆ G(yi), yi = xji . Put Hm+1 = X\K.Then there exist ϕ1, ..., ϕm+1 ∈ Cb(X) with

0 ≤ ϕi ≤ 1, ϕi = 0 on X\Hi,

m+!∑

i=1

ϕi(x) = 1 for x ∈ K,

and∑m+1

i=1 ϕi(x) ≤ 1 for x ∈ X . Define T : C(X,E)→ C(X,E) by

Tf =

m+!∑

i=1

ϕi ⊗ Sf(y1), f ∈ C(X,E).

It is easy to verify that T is a continuous linear map of finite rank. Tocomplete the proof, we show that

Tf − f ∈ N(K,W ) for all f ∈ A.

[Let f ∈ A, and let x ∈ K. Choose g ∈ A1 such that (g− f) ⊆ U1. Notethat Ix = i : x ∈ Hi contains at most n+ 1 indices. Now

Tf(x)− f(x) =m+!∑

i=1

ϕi(x)[Sf(yi)− f(x)] =∑

i∈Ix

ϕi(x)[Sf(yi)− f(x)].

For each i ∈ Ix,

Sf(yi)− f(x) = S[f(yi)− g(yi)] + [Sg(yi)− g(yi)]

+g(yi)− g(x)] + [g(x)− f(x)]

∈ SU1 + U1 + U1 ⊆ V.

Hence

Tf(x)− f(x) ∈∑

i∈Ix

ϕi(x)V ⊆W.]


Remark. We mention, as before, that if E is a locally convex TVS,then with slight modification of proofs, Theorems 4.5.1 and 4.5.2 holdwithout the hypothesis of finite covering dimension.



Section 4.1. The Stone-Weierstrass theorem was first proved byM.H. Stone [St37] in 1937 for the space (Cb(X), u) as a generalization ofthe classical Weierstrass’s approximation theorem of 1885 (see also [St48]for its refinement). Several authors have later generalized it or given itsnew proof; see, e.g. Hewitt [Hew47], Kaplansky ([Kap51], §3), de Branges[Bra59], Bishop [Bis61], Jewitt [Je63], Machado [Mac77], Brosowski andDeutsch [BD81], and Ransford [Ran84].

The Stone-Weierstrass theorems for (Cb(X), β) and (Cb(X,E), β) werefirst established by Buck [Buc58] in 1958 for E finite dimensional. Its ex-tension to (Cb(X,E), β), E a locally convex TVS, was later given by Todd[Tod65] and Wells [Wel65], independently. For E any TVS, a special caseof the theorem (that Cb(X)⊗E is u-dense in Cb(X,E)) was establishedby Shuchat [Shu72a] for (Cb(X,E), u), assuming X a compact space offinite covering dimension (see also Waelbroeck [Wae73], §. 8.1)). Moregeneral forms of the theorem were later obtained by Khan [Kh79, Kh80,Kh95], Katsaras [Kat81], Abel [Ab87] and Prolla [Pro93a, Pro93b] for(Cb(X,E), β), (Cb(X,E), u) and (C(X,E), k), assuming the hypothesisof finite covering dimension on X . The present section contains resultsof these latter authors.

Section 4.2. The ‘weighted approximation problem’, formulated byNachbin [Nac65] and further studied by Summers [SumW71] for E areal or complex field, consists in finding a characterization of the ωV -closure of A in CVo(X,E), where A ⊆ CVo(X,E) an M-submodulewith M ⊆ C(X) a Cb(X)-submodule. Prolla [Pro71a, Pro71b, Pro72]extended some results of above authors to the case of E a locally convexTVS. This section contains some results from [Kh85, Kh87] when E is notnecessarily locally convex. In particular, these results give extension of([Nac67], Propositions 1 and 2, p.64) and ([Nac67], Proposition 5, p.66)to E a locally bounded TVS.

Section 4.3. The main result is an infinite dimensional version of theWeierstrass polynomial approximation theorem, due to Bruno [Bru84],for P (E, F ), the set of all continuous polynomials of finite type from aTVS E into a TVS F . Earlier results in this direction were obtained byPrenter [Pre70] for E = F real separable Hilbert space and by Machado-Prolla [MP72] for E and F real locally convex spaces. The hypotheseson E and F involve the existence of sequences of projections having theGrothendieck approximation property (GAP) (cf. [Scha71], pp.108-115).


Section 4.4. C. Todd [Tod65] has used the Stone-Weierstrass theo-rem to characterize the maximal β-closed Cb(X)-submodules of Cb(X,E)for E a locally convex TVS. In this section, we present the extension ofthese characterizations to a non-locally convex setting, as given in [Kh84].These require the additional assumptions on E to have the separationproperty (or, equivalently, the Hahn-Banach extension property) or theweaker separation property considered by Gregory and Shapiro [GrSh70].

Section 4.5. The approximation property for locally convex spaceswas defined by Grothendieck in 1955, who also raised “the approximationproblem” (i.e. the problem whether every locally convex TVS E has theapproximation property). This problem remained unresolved until 1973when Enflo [Enf73] gave an example of a separable Banach space withoutthe approximation property. However, despite this development, therestill exist some concrete Banach spaces about which it is not yet knownwhether they have the approximation property (see Singer ([Sin70], Vol.I-II).

As regards the function spaces, Grothendieck [Gro55] was the firstto show that if X is a compact Hausdorff space, then (C(X), ‖·‖) hasthe approximation property. In the case of the space (Cb(X), β), thisresult was proved by Collins and Dorroh [CD68] for X locally compactand by Fremlin, Garling and Haydon [FGH72] for X completely regu-lar. Bierstedt [Bie75] obtained a similar result for the weighted space(CVb(X), ωV ). In the case of vector-valued functions, Fontenot [Fon74]has shown that if X is completely regular and E a normed space withthe metric approximation property, then (Cb(X,E), β) has the approxi-mation property. Further extensions of this result have been obtained byProlla [Pro77] for E a locally convex TVS and by Katsaras [Kat81] forE any TVS but assuming X to have a finite covering dimension. Thissection contains results from [Kat81, Kh95].

CHAPTER 5

Maximal Ideal Spaces

In this chapter we give characterization of maximal closed ideals inthe algebras (Cb(X,A), β) and (C(X,A), k) and also of the maximal idealspaces of these algebras.

89

90 5. MAXIMAL IDEAL SPACES

1. Maximal Ideals in (Cb(X,A), β)

We first show that, if A is topological algebra with jointly continuousmultiplication, then (Cb(X,A), β) and (C(X,A), k) become topologicalalgebras with jointly continuous multiplication, as follows.

Theorem 5.1.1. Let X be a topological space and A a topologicalalgebra with jointly continuous multiplication. Then (Cb(X,A), β) and(C(X,A), k) are topological algebras with jointly continuous multiplica-tion.

Proof. Let W be a base of τneighborhoods of 0 in A consisting ofabsorbing and balanced sets. Let f, g ∈ Cb(X,A), and let fα and gα

be nets in Cb(X,A) such that fαβ→ f and gα

β→ g. We need to show

that fαgαβ→ fg. Let ϕ ∈ Bo(X), 0 ≤ ϕ ≤ 1, and W ∈ W. Choose a

V ∈ W such that V + V + V ⊆ W . Since A is a topological algebrawith jointly continuous multiplication, there exists a U ∈ W such thatU2 ⊆ V. Define ϕ′(x) =

√ϕ(x), x ∈ X. Then ϕ′ ∈ Bo(X). Choose r ≥ 1

such that

(ϕ′f)(X) ⊆ rU, (ϕ′g)(X) ⊆ rU. (1)

There exists an index αo such that

fα − f, gα − g ∈ N(ϕ′,1

rU) for all α ≥ αo. (2)

Now, for any x ∈ X and α ≥ αo, using (1) and (2),

ϕ(x)[fα(x)gα(x)− f(x)g(x)] = ϕ′(x)[fα(x)− f(x)].ϕ′(x)[gα(x)− g(x)]

+ϕ′(x)f(x).ϕ′(x)[gα(x)− g(x)]

+ϕ′(x)[fα(x)− f(x)].ϕ′(x)g(x)

∈ U.U + rU.1

rU +

1

rU.rU

⊆ V + V + V ⊆W ;

that is, fαgα − fg ∈ N(ϕ,W ) for all α ≥ αo. Hence the multiplication isjointly continuous in (Cb(X,A), β).

Replacing both ϕ, ϕ′ by χK , K any compact set in X , in the aboveproof, we conclude that (C(X,A), k) is also topological algebras.

We shall require the following version of the Stone-Weierstrass The-orem from Chapter 4 for the characterization of β-closed and k-closedideals.

1. MAXIMAL IDEALS IN (Cb(X,A), β) 91

Theorem 5.1.2. Let X be a completely regular space of finite cov-ering dimension and E a TVS, and let A be a Cb(X)-(resp. C(X))-submodule of Cb(X,E) (resp. C(X,E)). Then the following conditionsare equivalent:

(1) f belongs to the β-closure (resp. k-closure) of A.

(2) For each x ∈ X, f(x) ∈ A(x).In the remainder of this section, we shall assume that A is a topologi-

cal algebra with jointly continuous multiplication and identity e. Further,all ideals considered are the proper ideals.

Theorem 5.1.3. [Kh86] For any x ∈ X and any closed left (resp.right, two-sided) ideal M of A,

S(x,M) = f ∈ Cb(X,A) : f(x) ∈M

is β-closed left (resp. right, two-sided) ideal in Cb(X,A); if M is maxi-mal, so is S(x,M).

Proof. The first part is straightforward. Let M be a maximal closedleft ideal in A and x ∈ X , and let J be a left ideal in Cb(X,A) such thatS(x,M) ⊆ J . Suppose there is a function g ∈ J such that g(x) /∈ M .Then g(x) has a left inverse c (say) in E. Since J is a left ideal, (1⊗c)g ∈J , where 1 ∈ Cb(X) is the function identically 1 on X . But,

[(1⊗ c)(1⊗ g(x))](y) = cg(x) = (1⊗ e)(y) for all y ∈ X,

and so the identity function 1 ⊗ e ∈ J . This contradiction shows thatS(x,M) = J , as required.

The following theorem gives us a characterization of β-closed (left,right) ideals in Cb(X,A).

Theorem 5.1.4. [Kh86] Suppose X has finite covering dimension.Then every β-closed left (resp. right) ideal J in Cb(X,A) is of the formJ = ∩

x∈XS(x, Jx), where, for each x ∈ X, Jx represents a closed left

(resp. right) ideal in A.Proof. Let J be a β-closed left ideal in Cb(X,A). It is easily seen

that, for each x ∈ X , Jx = J(x) is a closed left ideal in A. Now, letf ∈ J and ϕ ∈ Cb(X). Then, for any y ∈ X ,

ϕ(y)f(y) = ϕ(y)[ef(y)] = (ϕ⊗ e)(y).

Since (ϕ⊗ e)f ∈ J , we have ϕf ∈ J . Hence J is a Cb(X)-submodule ofCb(X,E) and so, by Theorem 6.1.2,

J = f ∈ Cb(X,E) : f(x) ∈ J(x) for all x ∈ X = ∩x∈X

S(x, Jx).

We next consider the β-closed two-sided ideals.


Theorem 5.1.5. [Kh86] Suppose X has finite covering dimensionand A is a simple topological algebra. Then any β-closed two-sided idealJ in Cb(X,A) is of the form

JF = f ∈ Cb(X,A) : f(x) = 0 for all x ∈ F,

where F ⊆ X is a closed subset.Proof. First note that, for any closed subset F ⊆ X, JF is a β-closed

two-sided ideal in Cb(X,A). Now, let J be a β-closed two-sided idealin Cb(X,A). Then the set F = x ∈ X : f(x) = 0 for all f ∈ J isclosed in X , and clearly J ⊆ JF . Let g ∈ JF , and suppose that g /∈ J .Since J is β-closed, by Theorem 5.1.2, there exists a y ∈ X such thatg(y) /∈ Jy = J(y). Therefore, Jy 6= A and so, by hypothesis, Jy = 0.Hence y ∈ F . But g(y) /∈ Jy implies that g(y) 6= 0, a contradiction. ThusJ = JF , as required.

Notice that, in the notation of Theorem 5.1.4, we can write the aboveJ as J = ∩

x∈FS(x, 0).

We now characterize the maximal β-closed ideals in Cb(X,A).

Theorem 5.1.6. [Kh86] Suppose X has finite covering dimensionand A is a Q-algebra. Then any maximal β-closed left (resp. right, two-sided) ideal in Cb(X,A) is of the form S(x,M), where x ∈ X and M isa maximal left (resp. right, two-sided) ideal in A.

Proof. Let J be a maximal β-closed left ideal in Cb(X,A), and letf ∈ Cb(X,A) with f /∈ J . By Theorem 5.1.2, there exists an x ∈ X

such that f(x) /∈ Jx = J(x). Since A is a Q-algebra, there exists amaximal (closed) proper left ideal M in A such that Jx ⊆ M . Clearly,J ⊆ S(x,M), and so, by the maximality of J and Theorem 5.1.3, J =S(x,M).

We next state, without proof, the analogues of Theorems 5.1.3-5.1.6for the algebra (C(X,A), k).

Lemma 5.1.7. For any x ∈ X and any closed left (resp. right,two-sided) ideal M of A,

S(x,M) = f ∈ C(X,A) : f(x) ∈M

is k-closed left (resp. right, two-sided) ideal in C(X,A); if M is maxi-mal, so is S(x,M).

The following theorem gives us a characterization of k-closed (left,right) ideals in C(X,A).

Theorem 5.1.8. [Kh86] Suppose X has finite covering dimension.Then every k-closed left (resp. right) ideal I in C(X,A) is of the form

1. MAXIMAL IDEALS IN (Cb(X,A), β) 93

I = ∩x∈X

S(x, Ix), where, for each x ∈ X, Ix represents a closed left (resp.

right) ideal in A.

Theorem 5.1.9. [Kh86] Suppose X has finite covering dimensionand A is a simple topological algebra. Then any k-closed two-sided idealJ in C(X,A) is of the form

J = JF = f ∈ C(X,A) : f(x) = 0 for all x ∈ F,

where F ⊆ X is a closed subset.Notice that, in the notation of Theorem 5.1.8, we can write the above

I as I = ∩x∈F

S(x, 0).

Theorem 5.1.10. [Kh86] Suppose X has finite covering dimensionand A is a Q-algebra. Then any maximal k-closed left (resp. right, two-sided) ideal in C(X,A) is of the form S(xo,Mo), where xo ∈ X and Mo

is a maximal left (resp. right, two-sided) ideal in A.

Remarks. (1) In general, a maximal ideal in Cb(X,A) need not beβ-closed. For instance, if A admits a discontinuous multiplicative linearfunctional ϕ andM = ϕ−1(0), then, for any x ∈ X, S(x,M) is a maximalideal but not β-closed. Thus, if A admits no discontinuous multiplicativelinear functional, then every maximal ideal in Cb(X,A) is β-closed. Thisremark also applies to (C(X,A), k).

(2) If the algebra A is locally convex, then Theorems 5.1.4 - 5.1.6 and5.1.8 - 5.1.10 hold without restricting X to have finite covering dimension.


2. Maximal Ideal Space of (Cb(X,A), β)

Let X be a completely regular space and A be a topological alge-bra with jointly continuous multiplication over C and with non-trivialdual A

∗

Recall that ∆(A) (resp. ∆c(A)) denotes the set of all non-zero multiplicative (multiplicative continuous) linear functionals on A.The set ∆c(A) equipped with the Gelfand topology, (i.e., the relativew

∗

= w(A∗

, A)-topology induced from A∗

) is called the maximal idealspace (or the carrier space) of A. We shall assume that ∆c(A) is non-empty. In the literature, ∆c(A) is also called the space of continuoushomomorphisms of A, or equivalently the space of closed maximal idealswith codimension 1 of A. For each x ∈ A, let x : ∆c(A) → C be theGelfand function defined by x(ϕ) = ϕ(x) for each ϕ ∈ ∆c(A) and let

A = x : x ∈ A. The mapping x→ x, for each x ∈ A, will be called theGelfand transform. It is clear that the Gelfand transform is an algebrahomomorphism from A into C(∆c(A)).

It is well-known (see e.g. [Simm62]) that if X is a compact (resp.completely regular) Hausdorff space, then the maximal ideals in Cb(X)are in one-one correspondence with the points in X (resp. βX); i.e.,∆c(Cb(X)) ∼= X (resp. ∆c(C(X)) ∼= βX). For the study of the idealstructures of vector-valued algebras (Cb(X,A), β) and (C(X,A), k), thesituation is not so simple. However, the space ∆c(C(X,A)) has beenstudied in many papers under various kinds of topological assumptionson X, A and C(X,A). Further, results similar in spirit to ∆c(C(X,A)) ∼=X ×∆c(A) have also been obtained by various authors.

In this section, we present results concerning ∆c((Cb(X,A), β) ∼= X×∆c(A). We follow the approach of Prolla [Pro81], but with no convexityassumptions is made about A.

We begin with the following classical result which gives a characteri-zation of ∆c(C(X), k).

Theorem 5.2.1. (i) The k-closed ideals of C(X) are of the form

IF = f ∈ C(X) : f |F = 0,

where F is a closed ideal in X.(ii) For any x ∈ X, define δx : C(X) → K by δx(f) = f(x),

f ∈ C(X). Then δx ∈ ∆c(C(X)) and the map G : x → δx is a homeo-morphism. Thus X ∼= ∆c((C(X), k)).

Proof. ([Diet69], p. 203) If I is an ideal in C(X), let

F = ∩z(f) : f ∈ I = x ∈ X : f(x) = 0 for all f ∈ I.

2. MAXIMAL IDEAL SPACE OF (Cb(X,A), β) 95

Then F is closed in X and I is contained in the closed ideal IF . Toshow that IF ⊆ I, it is sufficient to show thatI is k-dense in IF . [Leth ∈ IF , K ⊆ X a compact set and ε > 0. Then I|K = f |K : f ∈ I is asubalgebra of C(X). It is actually an ideal, since if g ∈ C(K), g extendsto a g1 ∈ C(X) and g(f |K) = g1f |K ∈ I|K . Using the Tietze extensiontheorem and the Urysohn lemma, as in ([Simm62], pp.329-330), I|K isnorm dense in f ∈ C(K) : f |F ∩ K = 0, since F ∩K = ∩z(f |K) :f ∈ I. Since h|F ∩K = 0, there is a g ∈ I with pK(g − h) ≤ ε. Thus hbelongs to the k-closure of I; that is, k-cl(I) = IF .] So, if I is k-closed,I = IF .

(ii) Clearly δx : C(X) → K is a non-zero continuous multiplicativelinear functional. We first show that the map G : x→ δx is one-one. [Letx 6= y ∈ X. Since X is completely regular, there exist an f ∈ C(X) suchthat f(x) = 0 and f(y) = 1. Then G(x)(f) = 0 and G(y)(f) = 1, and soG(x) 6= G(y).] We next show that G is onto. [Let ϕ ∈ ∆c(C(X)). Sincekerϕ is a closed maximal ideal of C(X), by (i), there is a closed set Fsuch that kerϕ = IF . Since kerϕ is maximal (and proper), F reduces toa point x ∈ X. So kerϕ = ker δx and thus ϕ = δx ([Simm62], p. 321).]So G is a bijection.

Finally, G is continuous and open, as follows. [For f ∈ C(X), define

f : ∆c(C(X))→ K by

f(ϕ) = ϕ(f), ϕ ∈ ∆c(C(X)).

The relative topology w(C(X)∗, C(X)) on ∆c(C(X)) is generated by

f : f ∈ C(X), and so

f−1(V ) : V open in K, f ∈ C(X)

is a subbase for the topology on ∆c(C(X)). Since X is completely regular,its topology equals the weak topology generated by C(X), and so

f−1(V ) : V open in K, f ∈ C(X)

is a subbase for the topology on X ([Simm62], p. 134). But

fG(x) = f(δx) = δx(f) = f(x), so G−1(f−1(V )) = fG−1(V ) = f−1(V );

that is, G is continuous and open.] Since G is a bijection, it is a homeo-morphism.

Definition. ∆c(A) is called locally equicontinuous if each h ∈ ∆c(A)has an equicontinuous neighborhood; this always happens in the case ofA a normed algebra, since in this case ∆c(A) is a subset of the unit ballof E∗.


Notation. If A is a topological algebra, we can now define a mappingG : X ×∆c(A)→ ∆c(C(X,A)) by

G(x, ϕ) = ϕ δx, (x, ϕ) ∈ X ×∆c(A), (∗)

where δx : C(X,A)→ A is the algebra homomorphism given by

δx(f) = f(x), f ∈ C(X,A).

Theorem 5.2.2. Let X be a locally compact space of finite coveringdimension and A a topological algebra with identity, and let G be themapping from X×∆c(A) into ∆c((Cb(X,A), β),as defined in (∗). Then:

(a) G is a bijection.(b) The inverse mapping G−1 is continuous.(c) If ∆c(A) is locally equicontinuous, then G itself is continuous.Proof. Obviously, by (∗), G(x, ϕ) ∈ ∆c(Cb(X,A)) if (x, ϕ) ∈ X ×

∆c(A).(a) To show that G is one-one, let (x, h) 6= (y, k) be given in X ×

∆c(A). If x = y, then h 6= k. Choose u ∈ A such that h(u) 6= k(u).Since X is locally compact, Cb(X,E) is essential in the sense that we canchoose f ∈ Cb(X,E) such that f(x) = u. Then

G(x, h)(f) = h(f(x)) = h(u) 6= k(u) = G(x, k)(f).

If x 6= y, choose ϕ ∈ Cb(X) such that ϕ(x) = 0 and ϕ(y) = 1. Choosenow u ∈ A with k(u) = 1. Choose f ∈ Cb(X,E), say f = ϕ ⊗ u, suchthat f(y) = u. Then g = ϕf ∈ Cb(X,E) and we have

G(x, k)(g) = h(g(x)) = h(0) = 0, but G(y, k)(g) = k(g(y)) = k(u) = 1.

To show that G

is onto, let H ∈ ∆c(Cb(X,A)). Then I = kerH is a proper closedtwo-sided maximal ideal of (Cb(X,A), β). By Theorem 5.2.1, there existsa unique x ∈ X such that the ideal Ix = cl(I(x)) is proper. Choosefo ∈ Cb(X,A) such that H(fo) = 1. Then fo is a modular identity for I.For each u ∈ A, uf ∈ Cb(X,A). Define h = hfo : A→ K by setting

h(u) = hfo(u) := H(ufo), u ∈ A.

Clearly, h ∈ A∗

. Now, if u, t ∈ A, then uf ∈ I. Choose g ∈ C(X,E) withg(x) = u. Then, since H(fo) = 1,

h(ut) = H(utfo) = H(fo)H(utfo) = H(foutfo) = H(fou)H(tfo)

= H(fou)H(fo)h(t) = H(foufo)h(t) = H(fo)H(ufo)h(t)

= h(u)h(t);

2. MAXIMAL IDEAL SPACE OF (Cb(X,A), β) 97

hence h is multiplicative. Let J = ker h.We claim that J ⊆ I(x). [Indeed,if u ∈ J, then uf ∈ I. Choose g ∈ Cb(X,A) such that g(x) = u. Thengfo−g ∈ I, and therefore ufo(x)−u ∈ I(x). Now ufo(x) = ufo(x) ∈ I(x),so u ∈ I(x).] Since Ix is proper, it now follows that h 6= 0, i.e., h ∈ ∆c(A).Let L = ker(h δx). We claim that I ⊆ L. [Indeed, let g ∈ Cb(X,A) besuch that g /∈ L. Then g(x) /∈ J. On the other hand, J is a maximal idealand J ⊆ I(x) ⊆ Ix. Since Ix is proper, it follows that J = Ix. Henceg(x) /∈ Ix. By Theorem 5.2.1, g /∈ I.] Since I is maximal and L is closedand proper, I = L. This shows that H and h δx have the same kernel.Since both are multiplicative,

H = h δx = G(x, h).

(c) Using the assumption that ∆c(A) is locally equicontinuous, G

is continuous, as follows. Let (xo, ho) ∈ X × ∆c(A), ε > 0 andg ∈ Cb(X,E) be given. Choose an equicontinuous neighborhood N of hoin ∆c(A) such that

N ⊆ h ∈ ∆c(A) : |(h− ho)(g(xo))| < ε/2.

Let W be a neighborhood of g(x) in A such that

|h(w − g(xo))| < ε/2 for all w ∈ W,h ∈ N.

Next, choose a neighborhood U of xo in X such that g(x) ∈ W for allx ∈ U. Then (x, h) ∈ U ×N implies g(x) ∈ W and h ∈ N. Therefore

|h(g(x)− g(xo))| < ε/2 and |(h− ho)(g(xo))| < ε/2.

It follows that

|G(x, h)−G(xo, ho)(g)| = |h(g(x))− ho(g(xo))|

≤ |(h− ho)(g(xo))|+ |h(g(x)− g(xo))| < ε

for all (x, h) ∈ U ×N .(b) To show that

G−1

is continuous, let Hα → H in ∆c(Cb(X,A)). Since G is onto, thereexist nets xα in X and hα in ∆c(A) such that Hα = G(xα, hα), andpoints x in X and h in ∆c(A) such that H = G(x, h). Since H 6= 0,there is some f ∈ Cb(X,A) such that H(f) = 1. Choose αo such thatHα(f) 6= 0 for all α ≥ αo. Take any g ∈ Cb(X). Then gf ∈ Cb(X,A)and, for all α ≥ αo,

g(xα) =g(xα)hα(f(xα))

hα(f(xα))=Hα(gf)

Hα(f).


Now

g(xα)→H(gf)

H(f)

= H(gf) = h(g(x))f)x) = g(x)h(f(x)) = g(x)H(f) = g(x).

Since X is a completely regular Hausdorff space and g ∈ Cb(X) wasarbitrary, xα → x. Next, for any u ∈ A, uf ∈ Cb(X,A) and so forα ≥ αo,

hα(u) =hα(u)hα(f(xα))

hα(f(xα))=Hα(uf)

Hα(f).

Hence

hα(u)→H(uf)

H(f)= H(uf) = h(uf(x)) = h(u)h(f(x)) = h(u)H(f) = h(u),

i.e., hα → h in the relative weak topology of ∆c(A). Thus

G−1(Hα) = G−1[G(xα, hα)] = (xα, hα)→ (x, h) = G−1(H),

and so G−1 is continuous.

Corollary 5.2.3. Let X be a completely regular space of finitecovering dimension and A topological algebra with identity, and let G bethe mapping from X × ∆c(A) into ∆c((Cb(X,A), β), as defined in (∗).If ∆c(A) is locally equicontinuous, then G is a homeomorphism.

Theorem 5.2.4. Let X be a completely regular space of finitecovering dimension and A topological algebra with identity, and let G bethe mapping from X ×∆c(A) into ∆c((C(X,A), k), as defined in (∗). If∆c(A) is locally equicontinuous, then G is a homeomorphism.

Theorem 5.2.5. Let X be a locally compact space of finite coveringdimension and A topological algebra with identity, and let G be the map-ping from X ×∆c(A) into ∆c((Co(X,A), u), as defined in (∗). If ∆c(A)is locally equicontinuous, then G is a homeomorphism.

Note. If the algebra A is locally convex, then 5.2.2-5.2.5 hold withoutrestricting X to have finite covering dimension.



Section 5.1. The classical results on characterization of maximaland closed ideals in (C(X,A), k) and (Cb(X,A), u) were obtained byKaplansky [Kap47] in the case of A a topological ring. These results arefurther studied in ([Nai72], Ch.V, §26.2) for A a Banach algebra and in([Pro77], Ch.1, § 6) for A a locally convex algebra.

In 1958, Buck [Buc58] showed that there is a one-one correspondencebetween the maximal β-closed ideals in Cb(X) and the points of X . Thissection contains characterizations of the maximal and β-closed (resp.k-closed) ideals in Cb(X,A) (resp. C(X,A) where X is a completelyregular Hausdorff space of finite covering dimension and A a Hausdorfftopological algebra (not necessarily locally convex) with jointly continu-ous multiplication, as given in [Kh86].

Section 5.2. Concerning the results similar in spirit to ∆c(C(X,A)) ∼=X ×∆c(A), Hausner ([Hau57], p. 248) showed that if X is compact andA a commutative Banach algebra, then

∆c((C(X,A), u)) ∼= X ×∆c(A).

This was an improvement of an earlier result of Yood [Y51], Theorem3.1, Lemma 5.1), where A is a commutative B∗ algebra with identity (orA a commutative Banach algebra with identity and assuming that everymaximal ideal in C(X,E) was of the form S(x,M) = f ∈ C(X,E) :f(x) ∈M, with x ∈ X and M a maximal ideal in A).

Using tensor products, Mallios ([Mal66], Theorem 5.1), showing

∆c((Co(X,A), u)) ∼= X ×∆c(A),

extended the Hausner-Yood theorem to the case of X a locally compactspace and A a complete locally m-convex algebra Q-algebra for which∆c(A) is locally equicontinuous; and (again using tensor products) Di-etrich ([Diet69], Theorem 4) showed that ∆c((C(X,A, k)) ∼= X ×∆c(A)if X is a completely regular k-space and A is a complete locally convexalgebra with ∆c(A) locally equicontinuous. Prolla [Pro81] obtained asimilar result for the weighted function algebra (CVo(X,A), ωV ),withoutusing tensor product techniques. Hery [Her76] extended some results ofthe above authors to the case of A a certain commutative topological al-gebras, not necessarily locally convex, with identity (see also Arhippainen[Arhi92, Arhi96]. This section contains results about ∆c((Cb(X,A), β) ∼=X×∆c(A), following the approach of Prolla [Pro81] without the convex-ity assumptions on A.


Some generalizations of Hery’s results to the case when X is realcom-pact and A is a metrizable algebra or X is a completely regular Haus-dorff space and A is a realcompact metrizable algebra are considered inMR (99k:46089) and MR (2002b:46084) and in the case of topologicalmodule-algebras in MR (87c:46056) (here are given conditions when allideals (not necessarily two-sided) are fixed).

Moreover, descriptions of ∆c(Cb(X,A), β),∆c(Co(X,A), u),∆c(CVo(X ,A), u) and ∆c(Crc(X,A), u) are given in [Ab04] as well and, when A islocally convex, in MR (58#30268) and (for closed (maximal, but notmaximal closed) one-sided ideals) in MR (84d:46084).

In addition, the description of all closed maximal (not maximal closed)modular left (right and two-sided) ideals in subalgebras of C(X ;A;S)(the algebra of A-valued continuous functions f on X for which f(S)is relatively compact in A for each S ∈ S, S is a cover of X which isclosed with respect to finite unions) is given in MR (2005c:46064) andin MR (2004a:46045) (the description ∆c(A) of A, where A is a sub-algebra of C(X ;A;S), is included) by Mart Abel in case when X isa completely regular Hausdorff space, S is a compact cover of X andA is a locally m-pseudoconvex Hausdorff algebra, a locally pseudocon-vex Waelbroeck Hausdorff algebra or an exponentially galbed Hausdorffalgebra with bounded elements. Similar results for real topological al-gebras are given by Olga Panova in MR 2326371 and in [O. Panova,Real Gelfand-Mazur algebras. Dissertation, University of Tartu, Tartu,2006, Dissertationes Mathematicae Univesitatis Tartuensis, 48. TartuUniversity Press, Tartu, 2006, 84 pp.].

Most recently, Oubbi [Oub07] has given several interesting resultsrelated to the above ones.

CHAPTER 6

Separability and Trans-separability

In this chapter, we consider various characterizations of separabilityand trans-separability of the function spaces (Cb(X,E), β), (C(X,E), k),(Cb(X,E), σ), and (Cb(X,E), u).

101

102 6. SEPARABILITY AND TRANS-SEPARABILITY

1. Separability of Function Spaces

Recall that X is called submetrizable if it can be mapped by a one-to-one continuous function onto some metric space Z (or, equivalently, if itstopology is stronger than a metrizable topology on X). If, in addition,Z is separable, X is called separably submetrizable. A submetrizablespace is Hausdorff but not necessarily regular. A topological space X isseparably submetrizable iff there exists a sequence gn ⊆ Cb(X) whichseparates points of X . The space ℓ∞(N) is submetrizable under its weaktopology w(ℓ∞(N),M(N)), where M(N) is the dual of C(N). The spaceM [0, 1] is a non-separable Banach space, although the weak topology ofM [0, 1] is submetrizable. A TVS is submetrizable iff its origin is a Gδ.

We state the following three theorems without proof for referencepurpose.

Theorem 6.1.1. [KK40, [SZ57] (i) (Cb(X), u) is separable iff X iscompact metric space.

(ii) If X is locally compact, then (Co(X), u) is separable iff X isσ-compact metric space.

Theorem 6.1.2. [War58, GuSc72, SumW72] The following state-ments are equivalent:

(a) (Cb(X), β) is separable.(b) (C(X), k) is separable.(c) X is separably submetrizable.

Theorem 6.1.3. [GuSc72] The following statements are equivalent:(a) (Cb(X), t) is separable, where t is either σo, σ or u.(b) X is a compact metric space.We now present the extension of above results to the case of E-valued

functions. We first obtain an extension of Theorem 7.1.2.

Theorem 6.1.4. [Kat81, Kh86b] The following are equivalent:(a) (Cb(X)⊗ E, β) is separable.(b) (C(X)⊗E, k) is separable.(c) X is separably submetrizable and E is separable.Proof. (a)⇒ (b) If (Cb(X)⊗E, β) is separable, then (Cb(X)⊗E, k) is

also separable. Since Cb(X) is k-dense in C(X), it follows that Cb(X)⊗Eis k-dense in C(X)⊗E. Thus (C(X)⊗E, k) separable.

(b) ⇒ (c) We first note the fact that both (C(X), k) and E areisomorphic to subspaces of (C(X)⊗ E, k) via the maps g → g ⊗ a (0 6=a ∈ E fixed) and b → 1X ⊗ b, respectively. Hence, by (b), (C(X), k)is separable and so, by Theorem 6.1.2, X is separably submetrizable.Further, E is also separable.

1. SEPARABILITY OF FUNCTION SPACES 103

(c)⇒ (a) Suppose X is separably submetrizable. Then, by Theorem6.1.2, (Cb(X), β) is separable. Let ϕm and an be countable densesubsets of (Cb(X), β) and E, respectively. Let A be the countable sub-space generated by ϕm ⊗ an : m,n = 1, 2, ... over rationals. We showthat A is β-dense in Cb(X) ⊗ E. Let f ∈ Cb(X) ⊗ E, ϕ ∈ Bo(X) with0 ≤ ϕ ≤ 1, and W ∈ W. We can write f =

∑pi=1 θi ⊗ bi (θi ∈ Cb(X),

bi ∈ E). Let V ∈ W be balanced with V + V + ...+ V (2p-terms) ⊆ W .Choose r ≥ 1 such that each bi ∈ rV . Let s = max||θi|| : i ≤ i ≤ p.For each i = 1, ..., p, there exist ϕmi

∈ ϕm and ani∈ an such that

||ϕ(ϕmi− θi)|| < 1/r(s + 1) and ani

− bi ∈ (1/r(s + 1))V . Note that||ϕϕmi

|| < s + 1. Let g =∑n

i=1 ϕmi⊗ ani

. Then g ∈ A and, for anyx ∈ X ,

ϕ(x)[g(x)− f(x)] =

p∑

i=1

ϕ(x)ϕmi(x)[ani

− bi]

+

p∑

i=1

ϕ(x)[ϕmi(x)− θi(x)]bi

∈1

r[V + ... + V (p-terms)] +

1

s+ 1[V + ...V (p-terms)]

⊆ W.

Hence g − f ∈ N(ϕ,W ), as required.

Corollary 6.1.5. Let X be a locally compact σ-compact space. Then(Cb(X)⊗ E, β) is separable iff X is metrizable and E is separable.

Corollary 6.1.6. Suppose either X has finite covering dimension orE is locally convex, or E is complete metrizable with a basis. Then thefollowing are equivalent:

(a) (Cb(X,E), β) is separable.(b) (C(X,E), k) is separable.(c) X is separably submetrizable and E is separable.In fact, each of the above restrictions on X or E implies that Cb(X)⊗

E is β-dense in Cb(X,E) and that C(X)⊗E is k-dense in C(X,E) (seeChapter 5). It is not known whether or not these ‘density’ results holdfor E a locally bounded space. However, we can prove

Theorem 6.1.7. [Kh86b] Let X be any Hausdorff space and E anylocally bounded TVS. Then (Cb(X,E), β) is separable iff (C(X,E), k) isso.

Proof. Let fn be a countable k-dense subset of C(X,E). Let Vbe a balanced bounded neighborhood of 0 in E, and let S be a closed


shrinkable neighborhood of 0 with S ⊆ V . The Minkowski functionalρ = ρS of S is continuous (see Section A.4) and positive homogeneousand, consequently, for each m = 1, 2, ..., the function hm : E → E definedby

hm(a) =

a if a ∈ mSmρ(a)

a if a ∈ E\mS

is continuous. Further, hm(E) ⊆ mS ⊆ mV, which shows that thefunctions hm fn ∈ Cb(X,E). We show that hm fn : m,n = 1, 2, ...is β-dense in Cb(X,E). Let f ∈ Cb(X,E), ϕ ∈ Bo(X) with 0 ≤ ϕ ≤ 1,and W ∈ W. Choose r ≥ 1 such that V + V ⊆ rSV + V ⊆ rW . Choosean integer M ≥ 1 with f(X) ⊆ (M/r)V . Let K be a compact subset ofX such that ϕ(x) < 1/rM for x ∈ X\K. There exists an integer N suchthat (fN − f)(K) ⊆ (1/r)V . Let y ∈ X . If y ∈ K, then fN(y) ∈ MSand so

ϕ(y)(hN fN (y)− f(y)) = ϕ(y)(fN(y)− f(y)) ∈ W.

If y ∈ X\K, then

ϕ(y)[hM fN(y)− f(y)] ∈ ϕ(y)[MS −M

rV ]

⊆M

rM[V −

1

rV ] ⊆ V + V ⊆ W.

Thus hM fN − f ∈ N(ϕ,W ).Conversely, suppose (Cb(X,E), β) is separable. Then (Cb(X,E), k)

is also separable. So we only need to show that Cb(X,E) is k-densein C(X,E) (cf. Theorem 1.1.11). Let f ∈ C(X,E) and K a compactsubset of X . If V is a bounded neighborhood of 0 in E, let S be a closedshrinkable neighborhood of 0 with S ⊆ V . Choose r ≥ 1 with f(K) ⊆ rS.Then, as in the above proof, we get a function hr f ∈ Cb(X,E) suchthat hr f(x)− f(x) = 0 for all x ∈ K.

Theorem 6.1.8. [KR91] The following statements are equivalent:(a) (Cb(X)⊗ E, t) is separable, where t is either σo, σ, or u.(b) X is a compact metric space and E is separable.Proof. (a) ⇒ (b) We first note the fact that both (Cb(X), t) and E

are isomorphic to subspaces of (Cb(X) ⊗ E, t) via the maps g → g ⊗ a(0 6= a ∈ E fixed) and b→ 1X⊗b, respectively. Hence, by (a), (Cb(X), t)is separable and so, by Theorem 6.1.3, X is a compact metric space.Further, E is also separable.

(b) ⇒ (a) By Theorem 6.1.1 (i), (Cb(X), u) is separable. Let gm :m = 1, 2, ... and gn : n = 1, 2, ... be countable dense subsets of(Cb(X), u) and E, respectively and let A consist of finite sums of the

1. SEPARABILITY OF FUNCTION SPACES 105

elements gm ⊗ an : m,n = 1, 2, .... Let f ∈ Cb(X) ⊗ E and W ∈ Wany neighborhood of 0 in E. The function f may be written as f =∑p

i=1 fi ⊗ bi, where fi ∈ Cb(X) and bi ∈ E. Choose a V ∈ W such thatV + V + ... + V (2p-terms) ⊆ W and choose r ≥ 1 such that bi ∈ rV(1 ≤ i ≤ p). Let s = max‖fi‖ : 1 ≤ i ≤ p. Then, for each 1 ≤ i ≤ p,there exist gmi

∈ gm and ani∈ an such that

‖fi − gmi‖ <

1

r(s+ 1)and ani

− bi ∈1

r(s+ 1)V.

Let g =∑p

i=1 gmi⊗ ani

. Then g ∈ A and, for any x ∈ X ,

g(x)− f(x) =

p∑

i=1

gmi(x)[ani

− bi] +

p∑

i=1

gmi(x)− f(x)]bi

∈1

r(V + ...+ V (p-terms)) +

1

(s+ 1)(V + ...+ V (p-terms))

⊆ W.

Thus g − f ∈ N(X,W ), and so A is u-dense in Cb(X)⊗ E.

Corollary 6.1.9. [KR91] Suppose (Cb(X,E), u) is metrizable andCb(X) ⊗ E is u-dense in Cb(X,E). Then the following statements areequivalent:

(a) (Cb(X,E), t) is separable, where t is either σo, σ or u.(b) X is a compact metric space and E is separable.

Theorem 6.1.10. (Cb(X,E), u) is separable iff (Cb(X,E), σo) isseparable.

Proof. If (Cb(X,E), u) is separable, then clearly (Cb(X,E), σo) is sep-arable. Suppose that (Cb(X,E), σo) is separable but that (Cb(X,E), u)is not. Then, for any sequence fn : n = 1, 2, ... in Cb(X,E), thereexist an f ∈ Cb(X,E), W ∈ W, and a sequence xn in X such thatfn(xn)−f(xn) /∈ W for all n. Thus, if A = xn, then fn−f /∈ N(χA,W ).This contradicts the separability of (Cb(X,E), σo).

Next, using Theorem 6.1.1(ii) instead of Theorem 6.1.3, we can easilyobtain:

Theorem 6.1.11. Let X be locally compact. Then (Co(X) ⊗ E, u)is separable iff X is a σ-compact metric space and E is separable.

Corollary 6.1.12. [KR91] Suppose that either X has finite coveringdimension or E locally convex, or E has the approximation property.Then

(i) (Cb(X,E), t) is separable iff X is compact metric space and Eis separable, where t is either σ, σo or u.


(ii) If X is locally compact, (Cb(X,E), u) is separable iff X is σ-compact metric space and E is separable.

Proof. Each of the above restrictions on X or E implies that, in case(i), Cb(X)⊗E is u-dense in Cb(X,E) and that, in case (ii), Co(X) ⊗Eis u-dense in Cb(X,E). The result now follows by applying Theorems6.1.8 and 6.1.11.

Finally we give some examples concerning the above results.

Examples 6.1.13. For 0 < p < 1, let ℓp = ℓp(N) denote the usualspace of scalar sequences, and let hp = hp(D) denote the Hardy spaceof certain harmonic functions on the unit disc D of the complex plane(see [Shap85]). ℓp and hp are not locally convex but are locally bounded(hence metrizable) and their duals separate points; ℓp is separable whilehp is not since it contains a copy of ℓ∞ ([Shap85], Theorem 3.5). Let ω bethe first uncountable ordinal and α < ω a limit ordinal. Let Y = [0, ω],Yo = [0, ω), Z = [0, α], and Zo = [0, α), each endowed with the ordertopology ([Will70, [StSe78). Then

1. Cb(X) ⊗ ℓp is ( a) β-dense in Cb(X, ℓp) (b) k-dense in C(X, ℓp),and (c) u-dense (hence also σ-, σo-dense) in Crc(X, ℓp) for any completelyregular Hausdorff space X since ℓp is admissible (see Chapter 4).

2. (Cb(Z, ℓp), u) is separable since Z is compact and metrizable, but(Cb(Y, ℓp), u) and (Cb(Yo, ℓp), u) are not separable since Y and Yo are notmetrizable.

3. (Co(Zo, ℓp), u) is separable since Zo is σ-compact and metrizable;(Co(Yo, ℓp), u) is not separable since Yo is not σ-compact and also notmetrizable.

4. (Cb(Zo, E), β) and (C(Zo, E), k) are separable for E = ℓp but notfor E = hp, since Zo is separable and metrizable, ℓp is separable, and hpis not separable.

2. TRANS-SEPARABILITY OF FUNCTION SPACES 107

2. Trans-separability of Function Spaces

In this section we present some results on trans-separability, as givenin [Kh99, Kh04, Kh08b].

Recall that a locally convex TVS L is called seminorm-separable if,for each continuous seminorm p on L, (L, p) is separable. The followingtwo classical results are stated for reference purpose.

Theorem 6.2.1. [GuSc72] The following statements are equivalent:(a) (Cb(X), β) is seminorm-separable.(b) (C(X), κ) is seminorm-separable.(c) Every compact subset of X is metrizable.

Theorem 6.2.2. [GuSc72] The following statements are equivalent:(a) (Cb(X), σ) is seminorm-separable.(b) (Cb(X), σo) is seminorm-separable.(c) The closure in βX of each σ-compact subset of X is metrizable

in βX.

Definition. A uniform space L is called trans-separable if everyuniform cover of L admits a countable subcover. In particular, a TVS Lis trans-separable if, for each neighborhood W of 0 in L, the open covera+W : a ∈ L of L admits a countable subcover.

For TVSs, a generalized notion of separability, namely, the neighborhood-separability may be defined, as follows.

Definition. Let L be a TVS, and let V be a neighborhood of 0 inL. A subset H of L is said to be V -dense in L if, for any z ∈ L andδ > 0, there exists an element y ∈ H such that y − z ∈ δV . L is calledneighborhood-separable if, for each neighborhood V of 0, there exists acountable V -dense subset of L.

Another notion of generalized separability may also be considered, asfollows.

Definition. Let (L, τ) be a TVS whose topology is generated bya family Q(τ) of continuous F-seminorms. Then (L, τ) is called F-seminorm-separable if (L, q) is separable for each q ∈ Q(τ). Clearly, sepa-rability implies F-seminorm-separability; the converse holds in metrizablespaces.

The following result establishes the equivalence of all the above no-tions of generalized separability.

Theorem 6.2.3. [Kh04, Kh08b] For a TVS (L, τ), the following areequivalent:


(1) (L, τ) is trans-separable.(2) (L, τ) is neighborhood-separable.(3) (L, τ) is F-seminorm-separable.

Proof. (1)⇒ (2) Suppose L is trans-separable, and let V be a neigh-borhood of 0. For each n ≥ 1, Un = x+ 1

nV : x ∈ L is a uniform cover

of L, and so it has a countable subcover U∗n = x(n)k + 1nV : k ∈ N. Let

D = ∪∞n=1x(n)k : k ∈ N. To show that D is V -dense in L, let y ∈ L

and δ > 0. Choose N ≥ 1 such that 1N< δ. Since U∗N is a cover of

L, y ∈ x(N)K + 1

NV for some K ∈ N. Then y − x(N)

K ∈ δV . Hence L isneighborhood-separable.

(2)⇒ (1) Suppose L is neighborhood-separable, and let U be a neigh-borhood of 0. Choose a balanced neighborhood V of 0 with V + V ⊆ U .Let zn be a countable V -dense subset of L. Since L = ∪x∈L(x+V ), toeach zn ∈ L, there exists some xn ∈ L such that zn− xn ∈ V . Let y ∈ L.Choose zk such that y − zk ∈ V. Then

y − xk = (y − zk) + (zk − xk) ∈ V + V ⊆ U.

Hence L = ∪n≥1(xn + U), and so L is trans-separable.(1) ⇒ (3) Suppose L is trans-separable, and let q ∈ Q(τ). For each

n ≥ 1, let Vn = x ∈ L : q(x) < 1/n, a balanced neighborhood of 0 inL. Then, for each n ≥ 1, Un = x + Vn : x ∈ L is a uniform cover of

L, and so it has a countable subcover U∗n = x(n)k + Vn : k ∈ N. Let

D = ∪∞n=1x(n)k : k ∈ N. To show that D is dense in (L, q), let y ∈ L

and ε > 0. Choose N ≥ 1 such that 1/N < ε. Since U∗N is a cover of

L, y ∈ x(N)K + VN for some K ∈ N. Then q(y − x(N)

K ) < 1/N < ε. Hence(L, q) is separable.

(3)⇒ (1) Suppose (L, q) is separable for each q ∈ Q(τ). Let x+U :x ∈ L be any uniform cover of L, where U is neighborhood of 0 in L.Choose a balanced neighborhood V of 0 in L with V + V ⊆ U. Chooseq ∈ Q(τ) such that W = x ∈ L : q(x) < 1 ⊆ V . Let zn bea countable dense subset in (L, q). Since L = ∪x∈L(x + W ), to eachzn ∈ L, there exists some xn ∈ L such that zn − xn ∈ W . Let y ∈ L.Choose zk such that q(y − zk) < 1. Then

y − xk = (y − zk) + (zk − xk) ∈ W +W ⊆ U,

and so L = ∪n≥1(xn + U).

Theorem 6.2.4. [Kh08b] Let E be any non-trivial TVS. Then thefollowing statements are equivalent:

(a) (Cb(X)⊗ E, β) is trans-separable.


(b) (C(X)⊗E, k) is trans-separable.(c) Every compact subset of X is metrizable and E is trans-separable.Proof. (a)⇒ (b) This follows from the fact that k 6 β on Cb(X)⊗E

and that Cb(X)⊗ E is k-dense in C(X)⊗ E.(b) ⇒ (c) This follows from Theorem 6.2.1 and the fact that both

(C(X), k) and E are isomorphic to subspaces of (C(X) ⊗ E, k) via themaps g → g ⊗ a (0 6= a ∈ E fixed) and a→ 1X ⊗ a, respectively.

(c)⇒ (a) By Theorem 6.2.1, (Cb(X), β) is trans-separable. Fix a ϕ ∈Bo(X), 0 ≤ ϕ ≤ 1 and a balanced W ∈ W. We need to show that thereis a countable set H ⊆ Cb(X)⊗E such that Cb(X)⊗E = H+N(ϕ,W ).

For every pairm,n ∈ N choose a balanced Um,n ∈ W so that, denotingVm,n = Um,n +mUm,n + Um,n, one has

Vm,n + · · ·+ Vm,n (n-summands) ⊆ W.

Also, choose a countable set Dm,n in E so that E = Dm,n +Um,n. Let Dbe the union of all these sets Dm,n (m,n ∈ N).

Next, for each k ∈ N denote Bk = f ∈ Cb(X) : ‖f‖ϕ ≤ 1/k andchoose a countable set Gk in Cb(X) so that Cb(X) = Gk +Bk. Let G bethe union of all these sets Gk (k ∈ N).

We are going to show that the countable setH = Hϕ,W of all functionsin Cb(X) ⊗ E of the form h =

∑ri=1 gi ⊗ di, where gi ∈ G and di ∈ D

(i = 1, . . . , r, r ∈ N), is as required.Take any f ∈ Cb(X)⊗E. Then f =

∑ni=1 fi⊗ai for some f1, . . . , fn ∈

Cb(X) and a1, . . . , an ∈ E. Let m ∈ N be such that ‖fi‖ϕ ≤ m fori = 1, . . . , n, and next choose k ∈ N so that k−1ai ∈ Um,n for i = 1, . . . , n.By the definitions of Dm,n and Gk, there are d1, . . . , dn ∈ Dm,n andg1, . . . , gk ∈ Gk such that

ai − di ∈ Um,n and ‖fi − gi‖ϕ ≤ 1/k for i = 1, . . . , n.

Now, for i = 1, . . . , n and x ∈ A,

ϕ(x)[fi(x)ai − gi(x)di] =ϕ(x)[fi(x)− gi(x)]ai + ϕ(x)fi(x)(ai − di)

+ ϕ(x)[gi(x)− fi(x)](ai − di); (*)

hence (using the fact that Um,n is balanced)

ϕ(x)[fi(x)ai − gi(x)di] ∈ Um,n +mUm,n + Um,n = Vm,n.

In consequence, setting h =∑n

i=1 gi ⊗ di we have h ∈ H and for everyx ∈ A,

ϕ(x)[f(x)− h(x)] =n∑

i=1

ϕ(x)[fi(x)ai − gi(x)di] ∈ W


so that f − h ∈ N(ϕ,W ).

Remark. A somewhat more transparent variant of the above proofthat (c) implies (a) can be based on Theorem 6.2.3. Thus, one has to showthat for any ϕ ∈ Bo(X), 0 ≤ ϕ ≤ 1, and any continuous F -seminorm qon E, the space (Cb(X)⊗ E, pϕ) is separable, where

pϕ(f) = supx∈X

q(ϕ(x)f(x)), f ∈ Cb(X,E)

Now, let G be a countable subset dense in (Cb(X), ‖·‖ϕ), and D a count-able set dense in (E, q). Take any f =

∑ni=1 fi ⊗ ai in Cb(X) ⊗ E,

and choose m ∈ N so that ‖fi‖ϕ ≤ m for each i. Given ε > 0, letg =

∑ni=1 gi ⊗ di, where gi ∈ G and di ∈ D. Assume that ‖fi − gi‖ϕ ≤ δ

for all i and some as yet unspecified 0 < δ < 1. Then, making use of (∗),it is easily seen that

pϕ(f − g) ≤n∑

i=1

[q(‖fi − gi‖ϕai) + q(‖fi‖ϕ(ai − di)

+q(‖fi − gi‖ϕ(ai − di))]

≤n∑

i=1

[q(δai) + (m+ 1)q(ai − di)]

and this can be made smaller than ε by taking δ sufficiently small andchoosing the di’s in D sufficiently close to the ai’s. It follows that thecountable set of all g’s of the above form is dense in (Cb(X)⊗E, pϕ).

Next, we obtain:

Theorem 6.2.5. [Kh04] Let E be a non-trivial TVS. Then:(a) (Cb(X)⊗E, u) is trans-separable iff X is a compact metric space

and E is trans-separable.(b) Suppose X is locally compact. Then (Co(X) ⊗ E, u) is trans-

separable iff X is a σ-compact metric space and E is trans-separable.Proof. (a) In this case, (Cb(X) ⊗ E, u) is trans-separable iff it is

separable iff X is a compact metric space. The proof now follows just asin above theorem.

(b) If X is locally compact, then (Co(X), u) is trans-separable iff itis separable iff X is a σ-compact metric space. Again the proof followsjust as in the above theorem.

Again we remark that, if Cb(X) ⊗ E (resp. Co(X) ⊗ E) is u-densein Cb(X,E) (resp. Co(X,E)), the above theorem remains valid withCb(X)⊗ E (resp. Co(X)⊗ E) replaced by Cb(X,E) (resp. Co(X,E)).


Remark.If X has finite covering dimension or E is locally convex,or E has the approximation property or E is complete metrizable witha basis, then Cb(X) ⊗ E is β-dense in Cb(X,E) and that C(X) ⊗ E isk-dense in C(X,E) (see Chapter 4). Hence, under these assumptions,the above two theorems hold with Cb(X)⊗ E, C(X)⊗ E and Co(X)⊗E replaced by Cb(X,E), C(X,E) and Co(X,E), respectively. It is notknown whether or not these ‘density’ results hold for E a locally boundedspace. However, we can prove

Theorem 6.2.6. [Kh08b] Let X be any Hausdorff space and E anylocally bounded space. Then (Cb(X,E), β) is trans-separable iff (C(X ,E), k) is so.

Proof Suppose (C(X,E), k) is trans-separable. Let ϕ ∈ Bo(X) with0 ≤ ϕ ≤ 1, and let W ∈ W. Let V be a balanced bounded neighborhoodof 0 in E, and let S be a closed shrinkable neighborhood of 0 with S ⊆V . The Minkowski functional ρ = ρS of S is continuous and positivehomogeneous and, consequently, for each r > 0, the function hr : E → Edefined by

hr(a) =

a if a ∈ rSr

ρ(a)a if a ∈ E \ rS

is continuous. Further, hr(E) ⊆ rS ⊆ rV, which shows that, for eachf ∈ C(X,E), the function hr f ∈ Cb(X,E). Choose t ≥ 1 such thatV+V ⊆ tS and V+V ⊆ tW . For eachm = 1, 2, ..., there exists a compactset Km ⊆ X such that ϕ(x) < 1/tm2 for x ∈ X\Km. Corresponding toeach Km, choose fmn : n = 1, 2, ... as a N(Km, V )-dense of C(X,E).We show that hm fmn : m,n = 1, 2, ... is β-dense in Cb(X,E). Letf ∈ Cb(X,E) and 0 ≤ δ ≤ 1. Choose integers M ≥ 1/δ and N ≥ 1 suchthat f(X) ⊆ (Mδ/t)V and (fMN − f)(KM) ⊆ (δ/t)V . Let y ∈ X . Ify ∈ KM , then

fMN(y) ∈ f(y) + (δ/t)V ⊆ (Mδ/t)V + (Mδ/t)V ⊆MS,

and so

ϕ(y)[hM fMN(y)− f(y)] = ϕ(y)[fMN(y)− f(y)] ∈ ϕ(y)(δ/t)V ⊆ δW.

If y ∈ X\KM , then, since hM (fMN(y)) ∈ hM (E) ⊆MS,

ϕ(y)[hM fMN (y)− f(y)] ∈ ϕ(y)[MS −Mδ

tV ]

⊆M

tM2[V +

δ

tV ] ⊆

δ

t[V + V ] ⊆ δW.

Thus hM fMN − f ∈ δN(ϕ,W ).


The converse follows from the fact that Cb(X,E) is dense in (C(X,E),k). Indeed, let f ∈ C(X,E), K a compact subset of X and W ∈ W. LetS and V be as above with S ⊆ V . Choose r ≥ 1 with f(K) ⊆ rS. Then,as in the above proof, we get a function hr f ∈ Cb(X,E) such that

hr f(x)− f(x) = f(x)− f(x) = 0 ∈ W for all x ∈ K.

Similar characterizations of trans-separability for the σ and σo topolo-gies are stated without proof since they can be obtained by applyingTheorem 6.2.2 in place of Theorem 6.2.1, with slight modifications.

Theorem 6.2.7. [Kh04] Let E be any non-trivial TVS. Then thefollowing statements are equivalent:

(a) (Cb(X)⊗ E, σ) is trans-separable.(b) (Cb(X)⊗ E, σo) is trans-separable.(c) The closure in βX of each σ-compact subset of X is metrizable

in βX and E is trans-separable.

Note If Cb(X) ⊗ E is u-dense in Crc(X,E), then clearly Theorem7.2.7 holds with Cb(X)⊗ E replaced by Crc(X,E). We do not know ofany general assumption on X other than its compactness ensuring thatCb(X)⊗E is σ- or σo-dense in Cb(X,E). However, without this densityassumption, the following holds.

Theorem 6.2.8. [Kh04] (Cb(X,E), σ) is trans-separable iff (Cb(X ,E), σo) is so.

Proof. (⇒) This follows from the fact that σo 6 σ.(⇐) Suppose (Cb(X,E), σ) is not separable. We show that (Cb(X,E),

σo) is also not trans-separable, i.e., if H = fn is any countable setin Cb(X,E), then there exist a countable set B = xn ⊆ X and aW ∈ W such that Cb(X,E) 6= H +N(B,W ). Since (Cb(X,E), σ) is notseparable, there exists a σ-compact set A ⊆ X and a W ∈ W such thatCb(X,E) 6= H +N(B,W ). Then there exists an f ∈ Cb(X,E) such thatf − fn /∈ N(A,W ) for all n ≥ 1. So, for each n ≥ 1, there exists anxn ∈ A such that

f(xn)− fn(xn) /∈ W.

Then, if B = xn, f − fn /∈ N(B,W ) for all n ≥ 1. Hence Cb(X,E) 6=H + N(B,W ), and so (Cb(X,E), σo) is not trans-separable, as desired.

Next, using Theorem 6.1.1 and the method of proof of Theorem 6.2.5,we obtain:

Theorem 6.2.9. [Kh08b] Let E be a TVS. Then:


(a) (Cb(X)⊗E, u) is trans-separable iff X is a compact metric spaceand E is trans-separable.

(b) Suppose X is locally compact. Then (Co(X) ⊗ E, u) is trans-separable iff X is a σ-compact metric space and E is trans-separable.

Again we remark that, if Cb(X)⊗E (resp. Co(X)⊗E) is u-dense inCb(X,E) (resp. Co(X,E)), Theorem 6.2.9 remains valid with Cb(X)⊗E(resp. Co(X)⊗E) replaced by Cb(X,E) (resp. Co(X,E)).

Finally, we give some examples concerning the above results and re-marks.

Examples 6.2.10. Using the notations of Example 6.1.13, we have1. (Cb(Yo, E), β) and (C(Yo, E), k) are trans-separable for E = ℓp but

not for E = hp. Here every compact subset of Yo is metrizable althoughYo is not.

2. (Cb(Z, ℓp), u) and (Co(Zo, ℓp), u) are (trans-) separable (since Zis compact and metrizable and Zo is σ-compact and metrizable) but(Cb(Y, ℓp), u) and (Co(Yo, ℓp), u) are not (since Y is compact but notmetrizable and Yo is not σ-compact and also not metrizable).

3. (Cb(Y ) ⊗ ℓp, u) and (Cb(Z) ⊗ hp, u) are not trans-separable sinceY is not metrizable and hp is not trans-separable.

4. If ωo the first countable infinite ordinal and

T = [0, ω]× [0, ωo]\(ω, ωo),

the deleted Tychonoff plank, with the product topology [StSe78; Will70],then(Cb(T )⊗ ℓp, σ) is not trans-separable since, if A = ∪∞i=1[0, ω]× n,A is σ-compact and AβT = βT = [0, ω]× [0, ωo] which is not metrizable:in fact the point (ω, ωo) is not even a Gδ in T (see [GuSc72], p. 259).



Section 6.1. The fundamental result on the characterization of sep-arability of (Cb(X), u) was obtained by M. Krein and S. Krein [KK40]in 1940. Later, similar results were obtained by Warner [War58] for(C(X), k) and by Gulick and Schmets [GuSc72] and, independently,by Summers [SumW72] for (Cb(X), β). The notions of submetrizableand separably submetrizable spaces are given in [War58, SumW72]. Inthe case of vector-valued functions, Todd [Tod65] and Choo [Cho79]have considered the separability of (Cb(X,E), β) for E a locally convexTVS and by Katsaras [Kat83], Khan [Kh85, Kh86] and Khan-Rowlands[KR91] for E a TVS.

Section 6.2. As stated in [RobN91], Trans-separability appearsunder many different names in the literature. While workers in thefield of uniform spaces tend to speak simply of separable space (see[Hag74, Isa64]), those who study locally convex spaces favour the phrase”seminorm separable”. Pfister [Pf76] investigated TVSs which were of”countable type”, whereas Drewnowski [Dre75] coined the word ”trans-separable” while working with topological abelian groups and this alsoseems to be suitable in the case of TVSs. This notion has proved to beuseful in the work [Pf76, CO87, RobN91] while studying the metrizabilityof precompact sets in locally convex spaces.

Characterizations of seminorm separability for (Cb(X), u), (C(X), k)and (Cb(X), β) were first given by Gulick and Schmets [GuSc72] Thissection contains results from [Kh99, Kh04, Kh08b] which generalize theresults of [GuSc72] to vector-valued functions by considering the notionof neighborhood separability, or equivalently, trans-separability in thenon-locally convex setting.

CHAPTER 7

Weak Approximation in Function Spaces

In this chapter we are mainly concerned with the weak approximationtype results for Cb(X,E) and CVo(X,E). Throughout this chapter weshall use the notations of Chapters 1 and 4 concerning the strict topologyβ and the weighted topology ωV . As seen in Chapter 4, Cb(X) ⊗ E isβ-dense in Cb(X,E) and CVo(X) ⊗ E is ωV -dense in CVo(X,E) for alocally convex E, and also for some concrete classes of not necessarilylocally convex spaces E (such as admissible spaces and spaces having theapproximation property); the general case of density problem remainsopen.

The main results in this section include a “convexified” version ofthe approximation problem. It states that, if X is locally compact, thenCVo(X)⊗ E is always weakly dense in CVo(X,E). This implies thatCVo(X) ⊗ E is dense in CVo(X,E) equipped with the locally convextopology ωcV associated to ωV (i.e., the strongest locally convex topologyon CVo(X,E) which is weaker than ωV ), where E

c is the locally convexTVS associated to E. These results are used later in Chapter 8 to obtaina representation of β- and ωV -continuous linear functionals on Cb(X,E)and CVo(X,E), respectively.

115

116 7. WEAK APPROXIMATION IN FUNCTION SPACES

1. Weak Approximation in (Cb(X,E), β)

In this section, we shall present some results regarding the weak ap-proximation in the space (Cb(X,E), β), as given in [Naw85]. We shall usehere the measure theoretic terminology of section A.7. Let k(X), z(X)and cz(X) denote the families of all compact, zero and cozero subsets ofX , respectively, let Ba(X) denote the σ-algebra of subsets of X gener-ated by z(X). The family of all positive, tight Baire measures on X willbe denoted by M+

t (Ba(X)). Let C(X, I) denote the subset of Cb(X) ofall functions having values in I = [0, 1]. If f is a function on X into E orR and U ∈ cz(X), then f ≺ U means supp(f) ⊆ Z for some Z ∈ z(X)with Z ⊆ U . We recall that if A ∈ k(X) or A ∈ z(X) and B ∈ cz(X)with A ⊆ B, then:

(a) there are sets F ∈ k(X), Z ∈ z(X) such that A ⊆ F ⊆ Z ⊆ B;(b) there is a function θ ∈ C(X, I) such that θ ≺ B and θ(A) = 1.If E is a TVS, then its topology may be generated by some family

of F -seminorms (§ A.2). This implies that if W is a base of balancedneighborhoods of 0 in E, then W ∩ Ba(E) is also a base of 0 consistingof balanced and zero or cozero sets.

Definition. We define the mt-topology on Cb(X,E) as the lineartopology which has a base at 0 the family of all sets of the form

G = G(µ,W, ε) = f ∈ Cb(X,E) : µ(x : f(x) /∈ W) ≤ ε, (∗)

where µ ∈ M+t (Ba(X)),W ∈ W ∩ Ba(E), and ε > 0. Let J be the

family of all linear topologies τ on Cb(X,E) satisfying τ |H 6 mt|H forany u-bounded subset H of Cb(X,E). We define the γ-topology as supJ .

Lemma 7.1.1. [Naw85] If T is a linear functional on Cb(X,E),then the following statements are equivalent:

(a) T ∈ (Cb(X,E), γ)∗.

(b) T (fα) → 0 for every net fα ⊆ Cb(X,E) which is u-boundedand mt-convergent to 0.

Proof. The implication (a) ⇒ (b) is obvious. (b) implies that theweak topology σ(T ) induced on Cb(X,E) by T belongs to J . Thus σ(T )6 γ, and so T is γ-continuous.

Lemma 7.1.2. [Naw85] The space Cb(X)⊗ E is γ-dense in Cb(X,E).Proof. Fix f ∈ Cb(X,E). Let Sf ⊆ Cb(X) ⊗ E be the family of

all functions of the form∑n

i=1 θi ⊗ f(xi), where θi ∈, C(X, I), xi ∈ X ,θi(xi) = 1 and supp(θi) ∩ supp(θj) = ∅ if i 6= j, i, j = 1, ..., n, n ∈ N. Itis easy to see that Sf is u-bounded. Let G be a mt-neighborhood of 0of the form G(µ,W, ε) given by (∗). We can assume that W is balanced.

1. WEAK APPROXIMATION IN (Cb(X,E), β) 117

Choose a balanced W1 ∈ W such that W1 +W1 ⊆ W and W1 ∈ cz(E).By the tightness of µ, we can find K ∈ k(X) such that

µ(B) ≤ ε/2 for every B ∈ Ba(X), B ∩K = ∅.

The set f(K) is compact, so f(K) ⊆ F + W1 for some finite subset Fof E. It follows that there are sets B1, ..., Bp ∈ Ba(X) such that K ⊆p∪i=1Bi, Bi ∩ Bj = ∅ if i 6= j and

f(x)− f(y) ∈ W for any x, y ∈ Bi, i = 1, ...., p.

Let Zi ∈ z(X), Ui ∈ cz(X) be such that

Zi ⊆ Bi ⊆ Ui and µ(Ui\Zi) ≤ ε/(2p).

We can find sets Oi ∈ cz(X) such that Zi ⊆ Oi ⊆ Ui and Oi ∩Oj = ∅ ifi 6= j, i = 1, ..., p. There exist functions θi ∈ C(X, I) such that

θi(Zi) = 1, supp(θi) = Oi, i = 1, ..., p.

Choose xi ∈ Zi, i = 1, ..., p. Then the function h =∑p

i=1 θi⊗ f(xi)belongs to Sf and

µ(x : (f − h)(x) /∈ W) ≤

p∑

i=1

µ(Ui\Zi) + µ(X\p∪i=1

Ui) ≤pε

2p+ ε/2 = ε.

Therefore f − g ∈ G(µ,W ,ε), so Cb(X)⊗E is γ-dense in Cb(X,E).

Theorem 7.1.3. [Naw85] Every β-continuous linear functional onCb(X,E) is γ-continuous.

Proof. Let T ∈ (Cb(X,E), β)∗. There exist ϕ ∈ Bo(X) and W1 ∈ W

such that

|T (f)| ≤ 1 for all f ∈ N(ϕ,W1). (1)

We may assume thatW1 is balanced and belongs to cz(E). Since β 6

u, we can find a balanced W2 ∈ W such that

N(1,W2) ⊆ N(ϕ,W1), W2 ⊆W1,W2 ∈ z(E). (2)

We observe that:For every ε > 0, there is a Kε ∈ K(X) such that

|T (f)| ≤ ε for all f ∈ N(1,W2) with f(Kε) = 0. (3)

[Indeed, let Kε be a compact subset of X such that x : |ϕ(x)| ≥ ε ⊆Kε. For any f ∈ N(1,W2), f(X) ⊆ W1 by (2). If, moreover, f(Kε) =0, then ϕf(X) ⊆ εW1. Therefore by (1), |T (f)| ≤ ε.]


We define F : cz(X)→ R by

F (U) = sup|T (f)| : f ∈ N(1,W2), f ≺ U for any U ∈ cz(X).

Obviously F is positive, finite and if U1,U2 ∈ cz(X), U1 ⊆ U2, then F (U1)≤ F (U2). We will show that

F (U1 ∪ U2) ≤ F (U1) + F (U2) for any U1, U2 ∈ cz(X). (4)

(4) F (U1 ∪ U2) ≤ F (U1) + F (U2) for any U1,U2 ∈ cz(X).[Let U1, U2 ∈ cz(X), f ∈ N(1,W2) and f ≺ U1 ∪ U2. There is a

Z ∈ z(X) such that supp(f) ⊆ Z ⊆ U1 ∪ U2. Fix ε > 0. Let Kε be as in(3). Put K = Z ∩Kε, H1 = K ∩ U1, H2 = K\H1. For any x ∈ Hi thereare sets Ux ∈ cz(X), Zx ∈ z(X) such that

x ∈ Ux ⊆ Zx ⊆ Ui, i = 1, 2.

The family Ux : x ∈ Kis an open cover of K. By the compactness ofK we can find a finite subset S of K such that K ⊆ ∪Ux : x ∈ S.Let Z ′i = ∪Zx : x ∈ S ∩ Hi, i = 1, 2. Then Z ′i ∈ z(X), Z ′i ⊆ Ui,i = 1, 2. There are sets Zi ∈ z(X) and Oi ∈ cz(X) such that Z ′i ⊆ Oi ⊆Zi ⊆ Ui, i = 1, 2. Let θo, θ1, θ2 ∈ C(X, I) be functions with supportsU1 ∪ U2\(Z ′1 ∪ Z

′2), O1, O2 respectively. We define functions

ϕi = θi(θo + θ1 + θ2)−1, i = 0, 1, 2.

Let fi = ϕif , i = 0, 1.2. Then

f1 ≺ U1, f2 ≺ U2 and fo(Kε) = 0.

Moreover, fi ∈ N(1,W2), i = 1, 2, and f = fo + f1 + f2, so that

|T (f)| ≤ |T (fo)|+ |T (f1)|+ |T (f2)| ≤ ε+ F (U1) + F (U2).

This implies that F (U1 ∩ U2) ≤ F (U1) + F (U2).]Suppose additionally that U1 ∩ U2 = ∅. Fix ε > 0. Let f1, f2 ∈

N(1,W2) be such that Tfi ≥ F (U1) + F (U2)− ε. Thus

F (U1 ∪ U2) = F (U1) + F (U2) for any U1, U2 ∈ cz(X), U1 ∩ U2 = ∅. (5)

From (3) it immediately follows that, for every ε > 0, there exists aKε ∈ k(X) such that

F (U) ≤ εifU ∈ cz(X) and U ∩Kε = ∅. (6)

Moreover, for every ε > 0 and U ∈ cz(X), there is a Z ∈ z(X),Z ⊆ U , such that

F (U\Z) ≤ ε. (7)

1. WEAK APPROXIMATION IN (Cb(X,E), β) 119

[Indeed, if this statement fails to be true for some ε > 0, then byinduction we can find a sequence fn ⊆ N(1,W2) such that

supp(fi) ∩ supp(fj) = ∅ for i 6= j and T (fj) > ε, i, j = 1, 2, ....

But gn = f1+ ..+ fn belongs to N(1,W2) for every n ∈ N and T (gn > nε.This contradicts (1).]

We define µ(B) = infF (U) : U ∈ cz(X), U ⊇ B for B ⊆ X . Itis easy to see that the family B of all subsets B of X such that for anygiven ε > 0 there are Z ∈ z(X), U ∈ cz(X), Z ⊆ B ⊆ U satisfyingµ(U\Z) ≤ ε is an algebra. By (7), cz(X) ⊆ B and so Ba(X) ⊆ B. Thefunction F restricted to Ba(X) is a positive Baire measure on X . From(6), it immediately follows that µ is tight.

We will now show that T is γ-continuous. [Let fαα∈A be a u-bounded net in Cb(X,E) which is mt-convergent to 0. There is a δ > 1such that fα ⊆ δN(1,W2). Fix ε > 0. Let

Zα = x ∈ X : fα(x) /∈ εW1,

Uα = x ∈ X : fα(x) /∈ εW2.

Then Zα ∈ z(X), Uα ∈ cz(X) and Zα ⊆ Uα. We can find functions θα ∈C(X, I) such that

θα ≺ Uα and θα(Zα) = 1, α ∈ A.

Let hα = θαfα and kα = (1 − θα)fα. Then kα ≺ Uα and δ−1hα ∈N(1,W2), so that |T (hα)| ≤ δµ(Uα). Moreover, kα(X) ⊆ εW1, andso kα ∈ sεN(1,W1) where s = sup|ϕ(x)| : x ∈ X. Thus

|T (fα| ≤ |T (hα|+ |T (kα| ≤ δµ(Uα) + sε.

This implies that limα T (fα = 0 and so, by Lemma 7.1.2, T is γ-continuous.]

Theorem 7.1.4. The space Cb(X)⊗E is σ(Y, Y∗

)-dense in Cb(X,E),where Y = (Cb(X,E), β).

Proof. By Theorem 7.1.3, σ(Y, Y∗

) 6 γ. Now, the statement imme-diately follows from Lemma 7.1.2.


2. Weak Approximation in CVo(X,E)

We now present a “convexified” version of weighted approximation,as given in [Naw89].

Notation. Let SM(X) be the family of all set functions µ : cz(X) −→[0,∞], usually called the submeasures, satisfying:

(S1) µ(∅) = 0,(S2) µ(U1) ≤ µ(U2) if U1 ⊆ U2,(S3) µ(U1 ∪ U2) ≤ µ(U1) + (U2),(S4) for every U ∈ cz(X) and an ε > 0 there is a Z ∈ z(X) such

that Z ⊆ U and µ(U\Z) ≤ ε.It easily follows from (S1) - (S4) that the family of all subsets B of

X satisfying(∗∗) “for every ε > 0 there are U ∈ cz(X) and Z ∈ z(X) such that

Z ⊆ B ⊆ U and µ(U\Z) ≤ ε ”is an algebra containing z(X). Therefore, (∗∗) holds for any set B be-longing to Ba(X).

The family SM(X) appears naturally while studying continuous lin-ear functionals on CVo(X,E). Indeed, the following result holds.

Lemma 7.2.1. [Naw89] Let N(v,W ) be an ωV -neighborhood of 0in CVo(X,E), where W is balanced. If T is a linear functional onCVo(X,E) which is bounded on N(v,W ), then the function µ definedon cz(X) by

µ(U) = sup|T (f)| : f ∈ N(v,W ), f ≺ U

belongs to SM(X).Proof. Properties (S1) and (S2) are obvious. We shall show that µ

satisfies (S3). Fix U1, U2 ∈ cz(X), ε > 0 and f ∈ N(v,W ) such thatsupp(f) ⊆ Z ⊆ U1 ∪ U2 for some Z ∈ z(X). vf vanishes at infinityso there is a compact subset Kε of X such that vf(X\Kε) ⊆ εW. LetK = Z ∩Kε. Ui is a cozero set, so we can find a function θi ∈ C(X, I)such that supp(θi) = Ui, i = 1, 2. K is compact, so

inf(θ1 ∨ θ2)(x) : x ∈ K = a > 0.

Let

θ′i := (θi ∨1

3aχX)−

1

3aχX , i = 1, 2, and θo :=

2

3aχX − (

2

3aχX ∧ θ1 ∧ θ2).

Moreover, let ϕi := θ′i(θo + θ′1 + θ′2)−1 for i = 0, 1, 2. We define fi :=

ϕif for i = 0, 1, 2. Then f = fo + f1 + f2, fi ∈ N(v,W ), fi ≺ Ui fori = 1, 2 and fo ∈ εN(v,W ). Therefore

2. WEAK APPROXIMATION IN CVo(X,E) 121

|T (f)| ≤ |T (fo)|+ |T (f1)|+ |T (f2)| ≤ εδ + µ(U1) + µ(U2),

where δ = sup|T (f)| : f ∈ N(v,W ). This implies (S3).If (S4) fails to be true, then for some ε > 0 we can find, by induction,

a sequence fn ⊆ N(v,W ) such that

supp(fn) ∩ supp(fm) = ∅ for n 6= m and T (fn) > ε, n,m = 1, 2, ....

However the functions gn := f1 + ..+ fn belong to N(v,W ) and T (gn >εn for every n ∈ N. This means that T is unbounded on N(v,W ), acontradiction.

Notation. If µ ∈ SM(X), we denote by µ∗ the outer measuredefined on X by µ; i.e.

µ∗(A) = infµ(U) : U ∈ cz(X), χA ≺ U for A ⊆ X.

Definition. Let V be a fixed Nachbin family of weights on X . Wedefine mV -topology as the vector topology on CVo(X,E) which has as abase of neighborhoods of 0 the family of all sets of the form

M(µ, v,W, ε) = f ∈ CVo(X,E) : µ∗(x ∈ X : vf(x) /∈ W) ≤ ε,

where µ ∈ SM(X), v ∈ V, W ∈ W and ε > 0. Let γV be thestrongest vector topology on CVo(X,E) which is weaker than mV onall ωV -bounded subsets of CVo(X,E).

Lemma 7.2.2. [Naw89] Let T be such a linear functional on CVo(X,E)that limα T (fα = 0 for every net fα ⊆ CVo(X,E) which is ωV -boundedand σV -convergent to 0. Then T is γV -continuous.

Proof. It is easily seen that the weakest vector topology on CVo(X,E)for which T is continuous is weaker than mV on every ωV -bounded set.

Theorem 7.2.3. [Naw89] For every Nachbin family V , the topologyγV is stronger than the weak topology w = w(Y, Y

∗

) of the space Y =(CVo(X,E), ωV ).

Proof. Let us fix an ωV -continuous linear functional T on CVo(X,E).We can find an ωV -neighborhood N(v,W ) of 0 such that W is balancedand

|T (f)| ≤ 1 for all f ∈ N(v,W )

Using T and N(v,W ), we define µ ∈ SM(X) as in Lemma 7.2.1; that is

µ(U) = sup|T (f)| : f ∈ N(v,W ), f ≺ U, U ∈ cz(X).


Theorem 7.2.3 will be proved if we show that T is γV -continuous.Let fα be a ωV -bounded net and mV -convergent to 0 in CVo(X,E)

(cf. Lemma 7.2.3). Fix ε > 0. For every α, there are Uα ∈ cz(X) andZα ∈ z(X) such that

Aα := x : vf(x) /∈ εW ⊆ Uα and µ(Uα) ≤ µ∗(Aα) + ε.

There is θα ∈ C(X, I) such that θα ≺ Uα and θα(Zα) = 1. Let hα := θαfαand kα := (1− θα)fα. The net fα is bounded, so there is a δ > 1 suchthat fα ⊆ δN(v,W ). Thus, δ−1hα ∈ N(v,W ) and hα ≺ Uα, so that

|T (hα)| ≤ δµ(Uα) ≤ δ(µ∗(Aα) + ε).

Moreover, kα ∈ εN(v,W ), and so

|T (hα)| ≤ δ(µ∗(Aα) + ε) + ε.

This implies that limα T (fα) = 0.

Notation. In the sequel we will denote by CVo(X)⊗d E the subsetof CVo(X,E) consisting of all functions of the form

∑ni=1 θi ⊗ ai, where

θi ∈ CVo(X), supp(θi) ∩ supp(θj) = ∅ if i 6= j, ai ∈ E, i, j = 1, ..., n, n∈ N.

Theorem 7.2.4. [Naw89] Let V be a Nachbin family of weights ona locally compact Hausdorff space X. For every balanced neighborhoodW of 0 in E and v ∈ V , the γV -closure of N(v, 2W ) ∩ [CVo(X) ⊗dE] contains N(v,W ).

In particular, CVo(X) ⊗ E is γV and weakly dense in CVo(X,E).Proof. Let us fix v,W and f ∈ N(v,W ). The set

Af := θf ∈ C(X, I), supp(θ) is compact

is ωV -bounded and contained in N(v,W ). Moreover, f can be ωV -approximated by elements of Af (see Section 5.2). The topology σVis weaker than ωV , and so γV is weaker than ωV on Af . This impliesthat f belongs to the γV -closure of Af . Therefore, for the proof of thetheorem, we may assume that K = supp(f) is compact.

Let us fix some relatively compact neighborhood U of K, and letD ⊆ C(X, I)⊗E be the set of all functions of the form

∑ni=1 θi ⊗ f(xi),

where θi ∈ C(X, I), supp(θi) ⊆ U , supp(θi) ∩ supp(θj) = ∅, if i 6=j, xi ∈ U , i, j = 1, ..., n, n ∈ N. The set f(U) is bounded in E and eachfunction v ∈ V, being upper semicontinuous, is bounded on U , so D isωV -bounded. We shall show that f can bemV -approximated by elementsof D ∩N(v, 2W ).


[Fix any γV -neighborhoodM(µ, v′,W ′, ε) of 0, where W ′ is balanced.We may assume that supv′(x) : x ∈ U =: a > 0. The set f(K) iscompact, so there exists a finite family B1, ..., Bn of pairwise disjointsubsets of X such that K ⊆ ∪n

i=1Bi ⊆ U, Bi ∈ B(X), and

f(x)− f(y) ∈1

aW ′ for every x, y ∈ Bi, i = 1, ..., n.

The submeasure µ satisfies (∗∗), so we can choose cozero sets U1, ..., Unand zero sets Z1, ..., Zn such that

Zi ⊆ Bi ⊆ Ui ⊆ U and µ(Ui\Zi) ≤ ε/n, i = 1, ..., n.

Clearly each Zi is compact. Using the upper semicontinuity of v, we canfind pairwise disjoint cozero sets Oi, ..., On such that Zi ⊆ Oi ⊆ Ui and

supv(x) : x ∈ Oi < 2 supv(x) : x ∈ Zi for i = 1, ..., n.

Choose xi ∈ Zi satisfying

supv(x) : x ∈ Oi < 2v(xi)

and θi ∈ C(X, I) such that

supp(θi) = Oi and θi(Zi) = 1, i = 1, ..., n.

We define g =∑n

i=1 θi ⊗ f(xi). Then g ∈ D and v(x)θi(x) < 2v(xi) forevery x ∈ Oi, i = 1, ..., n. Thus g ∈ D ∩N(v, 2W ). Moreover,

x ∈ X : v′(f − g)(x) /∈ W ′ ⊆n⋃

i=1

x ∈ Ui : v′(f − g)(x) /∈ W ′

⊆n⋃

i=1

x ∈ Ui : (f − g)(x) /∈ a−1W ′ ⊆

n⋃

i=1

(Ui\Zi).

Therefore,

µ∗(x ∈ X : v′(f − g) /∈ W ′ ≤n∑

i=1

µ(Ui\Zi) ≤ ε,

and so f − g ∈M(µ, v′,W ′, ε), as required]. The second assertion imme-diately follows from the first one and Theorem 7.2.3.

Open Problem [Naw89]. It is not known whether Theorem 7.2.4holds if X is non-locally compact.

Theorem 7.2.5. [Naw89] Let V be Nachbin family on a locallycompact Hausdorff space X. For every Hausdorff TVS E, the locallyconvex topology associated with the weighted topology ωV = ωV (X,E)coincides with the topology induced on CVo(X,E) by the weighted topology


ωV (X,Ec) of the space CVo(X,E

c), where Ec = (E, τ c) is the locallyconvex space associated with E = (E, τ).

Proof. The topology τ c is weaker that τ , so that CVo(X,E) ⊆ CVo(X ,Ec) and the inclusion mapping is continuous. Therefore,

ωcV (X,E) ≥ ωV (X,Ec)|CVo(X,E).

For the proof of the converse inequality let us fix a convex weakly closedωcV (X,E)-neighborhood G of 0 in CVo(X,E). We can find v ∈ V and abalanced neighborhood W of 0 in E such that conv N(v,W ) ⊆ G. Theset

N := N(v, (1/2) convW ) = f ∈ CVo(X,E) : vf(X) ⊆ (1/2) convW

is an ωV (X,Ec)|CVo(X,E)-neighborhood of 0, which is ωV (X,E)-closed.

Therefore, N is closed in the weak topology w of (CVo(X,E), ωV ). Let

Nd := N(v, convW ) ∩ [CVo(X)⊗d E]

andNd,∞ := f ∈ CVo(X)⊗d E : vf(X) ⊆ W∞

where W∞ :=∞⋃n=1

Wn and Wn := 2−n∑2n

i=1W . Since w 6 γV , Theorem

7.2.4 shows thatN = N

w⊆ N

w

d . (1)

W∞ is τ -dense in conv W , and so Nd,∞ is ωV -dense in Nd. Consequently,

Nd,∞ is weakly dense in Nd. (2)

Let us observe now that

Nd,∞ ⊆ conv N(v,W ). (3)

[Indeed, if f ∈ Nd,∞ then we can find θ1, ..., θk ∈ CVo(X) and a1, ..., ak ∈W∞ such that

f =

k∑

i=1

θi ⊗ ai, supp(θi) ∩ supp(θj) = ∅ if i 6= j,

supvθj(x) : x ∈ X ≤ 1, i, j = 1, ..., k.

There are ai,j ∈ W , i = 1, ..., k, j = 1, ..., m,m ∈ N, such that

ai =1

m

m∑

j=1

ai,,j, i = 1, ..., k.

Put fj :=∑k

i=1 θi⊗ai,,j, j = 1, ..., m. Then it is easily seen that vfj(X) ⊆W and f = 1

m

∑mj=1 fj. Therefore, f ∈ conv N(v,W ).]


Combining (1), (2) and (3), we have

N ⊆ Nw

d ⊆ Nw

d,∞ ⊆ convwN(v,W ) ⊆ G.

Finally, G is an ωV (X,Ec)|CVo(X,E) neighborhood of 0.

Remark 7.2.6. [Naw89] It follows from the above results thatωV (X,E) and ωV (X,E

c)|CVo(X,E) produce the same spaces of continu-ous linear functionals on CVo(X,E). Moreover, CVo(X,E) is dense inCVo(X,E

c). Therefore, we can identify the dual spaces of CVo(X,E)and CVo(X,E

c). Indeed, the restriction mapping

CVo(X,Ec)∗ ∋ T −→ T |CVo(X,E) ∈ CVo(X,E)

∗

is an algebraic isomorphism.



Section 7.1. This section contains some results regarding weak ap-proximation in the space (Cb(X,E), β), as given in [Naw85]. This re-quires the measure theoretic terminology of Section A.7. As seen inSection 4.1, Cb(X) ⊗ E is β-dense in Cb(X,E) for a locally convex E,and also for some concrete classes of not necessarily locally convex spacesE (such as admissible spaces and spaces having the approximation prop-erty); the general case of density problem remains open. In connectionwith the Riesz-type representations of functionals on Cb(X,E), this hadbeen obstructing research in the sense that the representation of func-tionals was limited to the subspace Cb(X) ⊗ E (see, e.g., [Shu72, KR81,Kat83]). However, using some idea of “submeasure convergence”, Kalton[Kal83] realized that at least if X is compact (and then β is the uniformtopology), the “representation theory” can avoid the “density problem”.Later Nawrocki [Naw85] proved that a version of this remains valid alsoif X is an arbitrary completely regular space. This is used to obtaindensity results in the weak topology, (i.e. the weakest one defined by theduals of (Cb(X,E), β)). This result is used later in Chapter 8 to obtaina representation of β-continuous linear functionals on Cb(X,E).

Section 7.2. This section contains some results regarding weak ap-proximation in the space (CVo(X,E), ωV ), as given in [Naw89]. The mainresult in [Naw89] states that, if X is locally compact, then CVo(X)⊗ Eis always weakly dense in CVo(X,E). This implies that CVo(X) ⊗ E isdense in CVo(X,E) equipped with the locally convex topology ωcV asso-ciated to ωV (i.e., the strongest locally convex topology on CVo(X,E)which is weaker than ωV ), where E

c is the locally convex space associ-ated to E. Again, this result is required later in Chapter 8 to obtain arepresentation of ωV -continuous linear functionals on CVo(X,E).

CHAPTER 8

The Riesz Representation type Theorems

In this chapter, we first define and establish existence of the integralof functions in Cb(X,E) with respect to certain E∗-valued measures,and then obtain Riesz representation type theorems for the characteriza-tion of dual spaces (Cb(X,E), u)

∗, (Cpc(X,E), β)∗, (Crc(X,E), σ)

∗ and(CVo(X,E), ωV )

∗.

127

128 8. THE RIESZ REPRESENTATION TYPE THEOREMS

1. Vector-valued Measures and Integration

We shall use here the measure theoretic terminology of Section A.7.Let X be a completely regular Hausdorff space and E a real HausdorffTVS with non-trivial dual E∗ and having a base W of closed balancedshrinkable neighborhoods of 0 in E. Let z(X) (resp. c(X), k(X)) be thecollection of all zero (resp. closed, compact) subsets of X . Let B(X) de-note the algebra generated by z(X) and let Ba(X) (resp. Bo(X)) denotethe σ-algebra generated by z(X)(resp. c(X)). The space of all Bairemeasures is denoted by M(Ba(X)), respectively. Let Mσ(Ba(X)) (resp.Mτ (Ba(X)), Mt(Bo(X))) denote the space of all bounded real-valued σ-additive (resp. τ -additive, tight) measures on Ba(X) (resp. Bo(X), Bo(X)).Restricting the above measures to Ba(X), we clearly have Mt(Ba(X)) ⊆Mτ (Ba(X)) ⊆ Mσ(Ba(X)). Note that Mt(Bo(X)) ⊆ Mσ(Ba(X)) neednot hold in general as an m ∈ Mτ (Bo(X)) is not necessarily z(X) regu-lar. Every tight m ∈ M(B(X)) is τ -additive and hence it has a uniqueextension m ∈ Mτ (Bo(X)).

We shall require the following classical versions of the Riesz represen-tation theorem.

Theorem 8.1.1. (a) (Riesz-Alexandroff) (Cb(X), u)∗ = M(B(X))via the linear isomorphism L → µ, where L(g) =

∫xgdµ for all g ∈

Cb(X) and ||L|| = |µ|(X).(b) (Riesz-Buck) (Cb(X), β)∗ = Mt(Bo(X)) via the linear isomor-

phism L→ µ, where L(g) =∫xgdµ for all g ∈ Cb(X) and ||L|| = |µ|(X).

Proof. (a) See ([Var61] or [Whe83]).(b) See ([Gil71] or [Sen72]).

Definition. For each W ∈ W, let MW (B(X), E∗) denote the set ofall finitely-additive E∗-valued set functions m on B(X) such that

(i) for each a 6= 0 in E, ma(F ) = m(F )(a) (F ∈ B(X)) determinesan element ma of M(B(X));

(ii) there exists a constant λ > 0 such that |m|W (X) ≤ λ, where, foreach F ∈ B(X), we define |m|W by

|m|W (F ) = sup |∑

i

m(Fi)(ai)|,

(it would be the same for a balanced W ) the supremum being taken overall finite partitions Fi of F into sets in B(X) (henceforth referred toas a B(X)-partition) and all finite collections ai ⊆W.

1. VECTOR-VALUED MEASURES AND INTEGRATION 129

Let M(B(X), E∗) = ∪W∈W

MW (B(X), E∗). Similarly, we may define

the spaces Mσ(Ba(X), E∗), Mτ (Bo(X), E∗) and Mt(Bo(X), E∗) by re-placing M(B(X)) by Mσ(Ba(X)), Mτ (Bo(X)) and Mt(Bo(X)), respec-tively, in the above definition.

Lemma 8.1.2. [KR81, Kat83] Let W ∈ W. If m ∈MW (B(X), E∗)(resp. Mσ,W (Ba(X), E∗), Mτ,W (Bo(X), E∗), Mt,W (Bo(X), E∗)), then|m|W ∈M(B(X)) (resp. Mσ(Ba(X)), (Mt(Bo(X)), (Mt(Bo(X))).

Proof. First, letm ∈ MW (B(X), E∗). It follows immediately from thedefinition that |m|W is a bounded non-negative set function on X , andit is straightforward to show that |m|W is finitely additive. To show that|m|W is regular, let F ∈ B(X) and ε > 0. There exist a B(X)-partitionFi (1 ≤ i ≤ n) of F and a collection ai(1 ≤ i ≤ n) of points in Wsuch that

|m|W (F ) ≤n∑

i=1

|mai | (Fj) + ε

Since each mai is regular, there exist zero sets Zi(i = 1, ..., n) such thatZi ⊆ Fi and

|mai | (Fj) < |mai | (Zj) + ε/2i.

Let Z = ∪ni=1Zi. Then Z ⊆ F and

|m|W (F ) ≤n∑

i=1

|mai | (Zi) + 2ε ≤ |m|W (F ) + 2ε.

Thus |m|W ∈M(X).Now, let m ∈ Mt,W (Bo(X), E∗). Then, by the above argument, it

easily follows that |m|W is a bounded, non-negative, finitely-additive setfunction on Bo(X). We show that |m|W is countably additive, as follows.Let Ak : k = 1, 2, .... be a sequence of disjoint sets in Bo(X) and

suppose that∞∪k=1

Ak = A. For any n ≥ 1,

|m|W (A) ≥ |m|W (n∪k=1

Ak) =n∑

k=1

|m|W (Ak),

and so

|m|W (A) ≥∞∑

k=1

|m|W (Ak).


Let ε > 0. Then there exist a Bo(X)-partition Fj : j = 1, ..., p of Aand a collection of points aj : j = 1, ..., p ⊆ W such that

|m|W (A) ≤

p∑

j=1

maj (Fj) + ε.

Since each maj is countably additive and Fj ∩ Ak : k = 1, 2, ... is apartition of Fj , we have maj (Fj) =

∑∞k=1maj (Fj ∩ Ak). Hence

|m|W (A) ≤

∣∣∣∣∣

p∑

j=1

∞∑

k=1

maj (Fj ∩ Ak)

∣∣∣∣∣+ ε ≤∞∑

k=1

|m|W (A) + ε.

Since ε is arbitrary, it follows from the above inequalities that |m|W iscountably additive. Next, using the argument of the first part, it easilyfollows that |m|W is k(W )-regular. Thus |m|W ∈ Mt(Bo(X)). The prooffor other parts is similar to the above one.

The advantage of taking the base W of ‘shrinkable’ closed balancedneighborhoods is that the Minkowski functional ρW of each W ∈ W iscontinuous and absolutely homogeneous, and further that W = x ∈X : ρW (x) ≤ 1 (see Section A.4). For this reason, the following ap-proach of defining the integral seems to be simpler than those given in[KR81, Kat83] where the notions of F -seminorms and bipolars are used,respectively.

Definition. [KR81, Kh95] If f ∈ Cb(X,E) and W ∈ W, we write

‖f‖W = ‖ρW f‖ = supx∈X|ρW (f(x))| .

Let m ∈MW (B(X), E∗), (W ∈ W), and f ∈ Cb(X,E). Let D be thecollection of all α = F1, ..., Fn; x1, ..., xn, where Fi (1 ≤ i ≤ n) is aB(X)-partition of X and xi ∈ Fi. If α1, α2 ∈ D, define α1 ≥ α2 iff eachset which appears in α1 is contained in some set in α2. In this way, Dbecomes as indexing set. Let Sα =

∑ni=1m(Fi)(f(xi)). We now define

the integral of f with respect to m over X by∫

X

dmf = limα∈D

Sα.

Regarding the conditions under which this integral exists, we obtain thefollowing lemma.

Lemma 8.1.3. [KR81, Kh95] The integral∫Xdmf , defined above,

exists in each of the following cases:(a) f ∈ Cpc(X,E);


(b) f ∈ Cb(X,E), and |m|W is τ -additive (in particular, |m|W istight);

(c) f ∈ Cb(X,E), and |m|W is σ-additive, and the range of f ismeasure compact (i.e. Mσ(f(X)) =Mτ (f(X))).

Proof. We need to show that, in each case, Sα : α ∈ D is a Cauchynet in R.

(a) Let ε > 0. Choose a balanced shrinkable V ∈ W such thatV + V ⊆ εW . Since f(X) is precompact, there exist Y1, ..., Yn ∈ X such

that f(X) ⊆n⋃i=1

(f(yi) + V ). Let Vi = x ∈ X : ρV (f(x) − f(yi)) ≤ 1.

Then each Vi ∈ B(X). Let G1 = V1 and Gi = Vi\i−1∪j=1

Vj (2 ≤ i ≤ n). By

keeping those Gi’s which are non-empty, we get, G1, ..., Gk say, a B(X)-partition ofX . Choose xi ∈ Gi and let αo = G1, ..., Gk; x1, ..., xk. Notethat if x, y are in the same Gi,

f(x)− f(y) = [f(x)− f(yi)] + [f(yi)− f(x)] ∈ V + V ⊆ εW.

Then, for α1, α2 ≥ αo, we have

|Sα1− Sα2

| ≤ |Sα1− Sαo |+ |Sαo − Sα2

| .

Suppose α1 = F1, ..., Fq; z1, ..., zq, where each Fj is contained in someGi and zj ∈ Fj . Now

|Sα1− Sαo | =

∣∣∣∣∣

q∑

j=1

m(Fj)(f(zj))−k∑

i=1

m(Gi)(f(xi))

∣∣∣∣∣

=

∣∣∣∣∣∣

q∑

j=1

m(Fj)(f(zj))−k∑

i=1

∑

j,Fj⊆Gi

m(Fj)(f(xi))

∣∣∣∣∣∣

= ε

∣∣∣∣∣∣

k∑

i=1

∑

j,Fj⊆Gi

m(Fj)[ε−1(f(zj)− f(xi))]

∣∣∣∣∣∣≤ ε |m|W (X).

Similarly, we can prove that |Sα2− Sαo | ≤ ε |m|W (X). Thus

|Sα2− Sα1

| ≤ 2ε |m|W (X),

and Sα : α ∈ D is a Cauchy net in R.(b) Let ε > 0, and choose an open balanced shrinkable set V ∈ W

with V +V ⊆ εW . The collection V = f−1(f(y)+V ) : y ∈ X is a coverof X consisting of cozero sets. Labelling V as Vλ : λ ∈ I, we make Iinto a directed set by saying that λ ≥ γ ⇔ Vλ ⊆ Vγ. By the τ -additivity


of |m|W , there exist y1, ..., yk ∈ X such that |m|W (X\k∪i=1Vλ1) < ε, where

Vλ1 = f−1(f(yi) + V ) = x ∈ X : ρV (f(x)− f(yi)) < 1.

Define G1 = Vλ1 , Gi = (Vλi\i−1∪j=1

Vλj (2 ≤ i ≤ k), and Gk+1 = X\k∪i=1

Vλi. Assuming that Gi’s are non-empty, choose xi ∈ Gi and let αo =G1, ..., Gk+1; x1, ..., xk+1.

Let α1, α2 ≥ αo. Suppose α1 = F1, ..., Fq; z1, ..., zq, where each Fjis contained in some Gi and zj ∈ Fj . Then

|Sα1− Sαo | ≤

∣∣∣∣∣∣

k∑

i=1

∑

j,Fj⊆Gi

m(Fj)(f(zj)− f(xi))

∣∣∣∣∣∣+

+

∣∣∣∣∣∣

∑

j,Fj⊆Gk+1

m(Fj)(f(zj))

∣∣∣∣∣∣+

∣∣∣∣∣∣

∑

j,Fj⊆Gk+1

m(Fj)(f(xk+1))

∣∣∣∣∣∣≤ ε(|m|W (X) + 2 ‖f‖W ).

Similarly, we obtain |Sα2− Sαo | ≤ ε(|m|W (X)+ 2 ‖f‖W ). Consequently,

Sα : α ∈ D is a Cauchy net in R.(c) Suppose |m|W is σ-additive, and let µ(A) = |m|W (f−1(A)) for

every Baire set A of f(X). Then µ is also σ-additive. Now taking V asin part (b) and using the fact that µ is also τ -additive (since f(X) ismeasure compact), we can complete the proof by the argument of part(b).

Remark. Part (b) and (c) were proved in ([Fon74], Lemma 3.11)assuming E a normed space.

Lemma 8.1.4. [KR81] Let m ∈ MW (X,E∗) (W ∈ W) and f ∈Cb(X,E). If the integral

∫Xdmf exists, then

∣∣∣∣∫

X

dmf

∣∣∣∣ ≤∫

X

(ρW f)d |m|W ≤ ‖f‖W |m|W (X).

Proof. For any ε > 0, There exist a B(X)-partition, Fi : 1 ≤ i ≤ nsay, of X , and points xi ∈ Fi such that

∣∣∣∣∫

X

dmf

∣∣∣∣ ≤∣∣∣∣∣

n∑

i=1

m(Fi)(f(xi))

∣∣∣∣∣ + ε

and ∣∣∣∣∣

n∑

i=1

(ρW f)(xi) |m|W (Fi)

∣∣∣∣∣ ≤∫

X

(ρW f)d |m|W + ε.


Let H1 (resp. H2) be the set of i ∈ 1, ..., n such that ρW (f(xi)) 6=0 (resp. ρW (f(xi)) = 0). We note that, if j ∈ H2, then ρW (tf(xi)) = 0for all t > 0. Then∣∣∣∣

∫

X

dmf

∣∣∣∣ ≤∑

i∈H1

(ρW f)(xi)

∣∣∣∣m(Fi)

(f(xi)

(ρW f)(xi)

)∣∣∣∣

+∑

i∈H2

ε

|m|W (X)

∣∣∣∣m(Fi)

(|m|W (X)f(xi)

ε

)∣∣∣∣+ ε

≤∑

i∈H1

(ρW f)(xi)

∣∣∣∣m(Fi)

(f(xi)

(ρW f)(xi)

)∣∣∣∣+ 2ε.

We note that, if |m|W (X) = 0, then the inequality we are seeking toestablish holds trivially. It follows that∣∣∣∣

∫

X

dmf

∣∣∣∣ ≤∑

i∈H1

(ρW f)(xi) |m|W (Fi) + 2ε

≤

∫

X

(ρW f)d |m|W + 2ε,

and so, since ε is arbitrary,∣∣∣∣∫

X

dmf

∣∣∣∣ ≤∫

X

(ρW f)d |m|W .

The other inequality is straightforward to prove.

Note. It is easy to verify that, if m ∈ M(B(X), E∗), g ∈ Cb(X),and a ∈ E, then ∫

X

dm(g ⊗ a) =

∫

X

gdma.

Remark. An other equivalent approach to define |m|W , due to Kat-saras [Kat83], is as follows: For p a continuous seminorm on E, we defineMp(B(X), E∗) to be the set MW (B(X), E∗) where W = Wp = a ∈ E :p(a) ≦ 1. For an m ∈ Mp(B(X), E∗), we define mp to be the measure|m|W .

Let now W ∈ W be balanced, and let W 00 be the bipolar of W withrespect to the pair 〈E,E∗〉. Let q = qW be the Minkowski functional ofthe absolutely convex set W 00. We have the following:

Lemma 8.1.5. [Kat83] A set function m : B(X) → E∗ belongs toMW (B(X), E∗) iff m ∈Mq(B(X), E∗). Moreover, we have mq = |m|W .

Proof. Since W ⊆ a ∈ E : q(a) ≦ 1, it follows that m ∈MW (B(X), E∗), whenever m ∈ Mq(B(X), E∗), and that |m|W ≦ mq.


On the other hand, let m ∈MW (B(X), E∗) and let A ∈ B(X). For eachϕ ∈ E∗, we define |ϕ|W = sup|ϕ(a)| : a ∈ W. Then W 0 = ϕ ∈ E∗ :|ϕ|W ≦ 1. It is easy to see that |ϕ|W = sup|ϕ(a)| : q(a) ≦ 1. Fromthe definition

|m|W (A) = supn∑

i=1

|m(Ai)|W : Aini=1 a B(X)-partition of A,

it follows now easily that mq(A) ≦ |m|W (A).

2. INTEGRAL REPRESENTATION THEOREMS 135

2. Integral Representation Theorems

We first characterize the u-dual of Cpc(X,E) via the integral repre-sentation. Recall that Cb(X)⊗ E is weakly dense in (Cpc(X,E), u) (seeChapter 7).

Theorem 8.2.1. [Kat83, Kh95] (Cpc(X,E), u)∗ =M(B(X), E∗) via

the linear isomorphism L→ m given by

L(f) =

∫

X

dmf (f ∈ Cpc(X,E)). (1)

Furthermore, if L is represented as in (1) with

m ∈MW (B(X), E∗)(W ∈ W),

then ‖L‖W = |m|W (X), where

‖L‖W = sup|L(f)| : f ∈ Cpc(X,E), ‖f‖W ≤ 1.

Proof. Let m ∈MW (B(X), E∗) (W ∈ W), and suppose that L is thelinear functional on Cpc(X,E) defined by (1). Then, by Lemma 8.1.4,

|L(f)| ≤ ‖f‖W |m|W (X) ≤ |m|W (X),

whenever f ∈ Cpc(X,E) with f(X) ⊆ W . Hence L ∈ (Cpc(X,E), u)∗.

We now show that ‖L‖W = |m|W (X). Clearly, ‖L‖W ≤ |m|W (X). Forany ε > 0, there exist a B(X)-partition F1, ..., Fn of X and a collectiona1, ..., an ⊆W such that

|m|W (X) ≤

∣∣∣∣∣

n∑

i=1

m(Fi)(ai)

∣∣∣∣∣ + ε.

There exist zero sets Zi ⊆ Fi such that |mai | (Zi\Fi) <εn. Since Cb(X)

separates the zero sets in X , the closures Z1, ..., Zn in βX are pairwisedisjoint. Since the cozero sets in βX form a base for open sets, thereexist pairwise disjoint cozero sets U1, ..., U2 in βX with Z i ⊆ Ui. EachUi ∩ X is a cozero set in X containing Zi; there is a cozero set Vi in Xwith

Zi ⊆ Vi ⊆ Ui ∩X and |mai | (Vi\Zi) <ε

n.

Choose hi ∈ Cb(X) with 0 ≤ hi ≤ 1, hi(Zi) = 1, and hi = 0 on X\Vi.Let h =

∑ni=1 hi ⊗ ai. Then h ∈ Cpc(X,E) and ‖h‖W ≤ 1. Now

L(h) =n∑

i=1

L(hi ⊗ ai) =n∑

i=1

∫

Zi

dmai +n∑

i=1

∫

Vi\Zi

hidmai ;


hence ∣∣∣∣∣

n∑

i=1

mai(Fi)

∣∣∣∣∣ ≤∣∣∣∣∣

n∑

i=1

mai(Fi\Zi)

∣∣∣∣∣ +∣∣∣∣∣

n∑

i=1

mai(Zi)

∣∣∣∣∣

≤ ε+ |L(h)|+n∑

i=1

|mai | (Vi\Zi)

≤ ‖L‖W + 2ε.

Thus |m|W (X) ≤ ‖L‖W .Conversely, suppose that L ∈ (Cpc(X,E), u)

∗. Then there exist aW ∈ W and r > 0 such that

|L(f)| ≤ r ‖f‖W (f ∈ Cpc(X,E)). (2)

For each a 6= 0 in E, let La(g) = L(g ⊗ Cb(X)). By (2), we have|La(g)| ≤ r||g||ρW (a) for all g ∈ Cb(X), and so La ∈ (Cb(X), u)∗. Hence,by Theorem 8.1.1(a), there exists an ma ∈ M(B(X)) such that La(g) =∫Xgdma (g ∈ Cb(X)) and ||La|| = |ma|(X).For each F ∈ B(X), define m(F )(a) = ma(F ) (a ∈ E). Then

|m(F )(a)| ≤ |ma|(X) ≤ rρW (a), and consequently m is a finitely ad-ditive E∗-valued set function on B(X). Next |m|W (X) ≤ r, as follows.Let F1, ..., Fp be a B(X)-partition of X and a1, ..., ap ⊆ W , and letε > 0. By using the argument as the one used earlier, there exists anh ∈ Cpc(X,E) with ‖h‖W ≤ 1 such that

∣∣∣∣∣

p∑

i=1

m(Fi)(ai)

∣∣∣∣∣ ≤ |L(h)|+ 2ε ≤ r + 2ε.

So |m|W (X) ≤ r and hence m ∈MW (B(X), E∗).

Now for any f =∑k

i=1 fi ⊗ ai (fi ∈ Cb(X), ai ∈ E) in Cb(X)⊗ E,

L(F ) =

k∑

i=1

L(fi ⊗ ai) =k∑

i=1

∫

X

fidmai =

k∑

i=1

∫

X

dm(fi ⊗ ai) =

∫

X

dmf.

Since∫xCb(X)⊗E is weakly dense in (Cpc(X,E), u), the above holds for

all f ∈ Cpc(X,E).Finally, m is unique, as follows. Suppose there is a µ ∈M(B(X), E∗)

such that L(f) =∫Xdµf for all f ∈ Cpc(X,E). In particular, for any

g ∈ Cb(X) and a ∈ E,∫Xdm(g ⊗ a) =

∫Xdµ(g ⊗ a). Hence

∫Xgdma =∫

Xgdµa for all g ∈ Cb(X), and so by Theorem 8.1.1 (a), ma = µa. Thus,

for any F ∈ B(X) and any a ∈ E, m(F )(a) = ma(F ) = µa(F ). It followsthat m(F ) = µ(F ), and so m = µ.


The following representation theorem gives a characterization for thedual of (Cb(X,E), β).

Theorem 8.2.2. [KR81, Kat83] (Cb(X,E), β)∗ = Mt(Bo(X), E∗)

via the linear isomorphism L→ m given by

L(f) =

∫

X

dmf (f ∈ Cb(X,E)). (3)

Further, if L is represented as in (3) with m ∈Mt,W (Bo(X), E∗) (W ∈W), then

‖L‖W = |m|W (X).

Proof. Let m ∈ Mt,W (Bo(X), E∗) (W ∈ W), and suppose that L isdefined by (3). Then, by Lemma 8.1.2, |m|W ∈ Mt(Bo(X)). It followsfrom Theorem 8.1.1 that the equation

LW (g) =

∫

X

gd |m|W (g ∈ Cb(X)). (4)

defines a β-continuous linear functional LW on Cb(X). Hence there existsa ϕ ∈ Bo(X), 0 ≤ ϕ ≤ 1, such that |LW (g)| ≤ 1 whenever g ∈ Cb(X)with ‖ϕg‖ ≤ 1. For any f ∈ N(ϕ,W ), ||ϕ(ρW f)|| = ||ρW (ϕf)|| ≤ 1and so, by (4) and Lemma 8.1.4,

|L(f)| ≤

∫

X

(ρW f)d |m|W = LW (ρW f) ≤ 1.

Thus L is β-continuous.Conversely, let L ∈ (Cb(X,E), β). Then there exists a θ ∈ Bo(X)

and W ∈ W such that

|L(f)| ≤ 1 for all f ∈ N(θ,W ). (5)

For each a 6= 0 in E, let La(g) = L(g ⊗ a) (g ∈ Cb(X)). It easilyfollows from (5) that La ∈ (Cb(X), β)∗ and so by Theorem 8.1.1, thereexists a unique ma ∈Mt(X) such that

La(g) =

∫

X

gdma (g ∈ Cb(X)).

For each F ∈ Bo(X), m(F ) (a) = ma(F ) (a ∈ E). Since β ≤ u, Lis u-continuous and so there exists a W ∈ W such that |L(f)| ≤ 1whenever f ∈ N(1,W ). Consider g ∈ Cb(X) with ||g|| ≤ 1. ThenρW (g(x)a) ≤ ρW (a) for all x ∈ X , and so g ⊗ a ∈ N(1,W ) wheneverρW (g(x)a) ≤ ρW (a) for all a ∈ X. Thus ||La|| ≤ ρW (a), and so from theinequalities

|m(F )(a)| ≤ |ma(F )| ≤ ||ma|| = ||La|| ≤ ρW (a),


the continuity of m(F ) follows. Consequently m is a finitely additiveE∗-valued set function on Bo(X) such that, for each a 6= 0 in E, ma ∈Mt(X).

Next |m|W (X) < ∞, as follows. Let Fj : j = 1, ..., p be a Bo(X)-partition of X and a1, ..., ap ∈ W , and let ε > 0. Each ma is regular andso there exist compact sets Kj ⊆ Fj with |maj |(Fj\Kj) < ε/2p, and opensets Vj ⊇ Kj with |maj (Fj\Kj)| < ε/2p for j = 1, ..., p; since Kj’s aredisjoint compact sets andX is completely regular, the Vj’s may be chosenpairwise disjoint. Choose functions gj(1 ≤ j ≤ p) in Cb(X), 0 ≤ gj ≤ 1,such that gi = 1 on Kj and supp(gj) ⊆ Vj. Let h =

∑pi=1 gj ⊗ aj . Then

h ∈ Cpc(X,E) and ‖h‖W ≤ 1, and so |L(h)| ≤ 1. By using the aboveinequalities as in the proof of Theorem 8.2.1, we have |m|W (X) ≤ ‖L‖W .Thus m ∈Mt(Bo(X), E∗).

Consider any f =∑k

i=1 fi ⊗ ai (fi ∈ Cb(X), ai ∈ E) in Cb(X) ⊗ E.Then

L(f) =

k∑

i=1

L(fi ⊗ ai) =k∑

i=1

fidmai =

k∑

i=1

∫

X

dm(fi ⊗ ai) =

∫

X

dmf.

Since Cb(X)⊗E is weakly dense in (Cb(X,E), β), the above holds for allf ∈ Cb(X,E). The proof for uniqueness of m and for ‖L‖W = |m|W (X)are the same as in Theorem 8.2.1 with slight modifications.

We next characterize the dual of (Crc(X,E), σ), σ being the σ-compact-open topology. Recall that Crc(X,E) denotes the subspace of Cb(X,E)which consists of those functions f such that f(X) is relatively compact.

Each f in Cb(X) or Crc(X,E) has a continuous extension f to βX , theStone-Cech compactification of X. Let S denote the set of all σ-compactsubsets of X and, for any A ∈ S, let A denote the βX-closure of A. LetMt,W (Bo(A), E∗) (W ∈ W) denote the set of all m ∈Mt,W (Bo(βX), E∗)such that the support of |m|W is contained in some A; let

Mt(Bo(A), E∗) = ∪

W∈WMt,W (Bo(A), E∗).

Theorem 8.2.3. [KR91] (Crc(X,E), σ)∗ = ∪

A∈SMt,W (Bo(A), E∗) via

the linear isomorphism L→ m given by

L(f) =

∫

βX

dmf (f ∈ Crc(X,E)).

Moreover, for some W ∈ W, ‖L‖W = |m|W (βX), where ‖L‖W =|L(f)| : f ∈ Crc(X,E), ‖ρW f‖X ≤ 1.


Proof. Using the Gulick’s version [Gu72] of the Riesz representationtheorem that (Cb(X), σ)∗ = ∪

A∈SMt(Bo(A)), the proof follows as in the

above theorem with appropriate modifications. The details may be foundin ([KR91], Theorem 4.4).

Finally, we state a characterize the dual of the weighted space (CVo(X ,E), ωV ). Here we follow the argument due to Nawrocki [Naw89]. We re-call from Remark 7.2.6 that the topologies ωV (X,E) and ωV (X,E

c)|CVo(X,E)

produce the same spaces of continuous linear functionals on CVo(X,E).Moreover, CVo(X,E) is dense in CVo(X,E

c). Therefore, we can identifythe dual spaces of CVo(X,E) and CVo(X,E

c). Indeed, the restrictionmapping CVo(X,E

c)∗ T −→ T |CVo(X,E) ∈ CVo(X,E)∗ is an algebraic

isomorphism. Therefore, using Theorem 7.2.5 and the known represen-tation of the dual space of CVo(X,E), where E is locally convex (see[Pro71b] or ([Pro77], Theorem 5.42)), we obtain the following result.

Proposition 8.2.4. [Naw89] Let X be a locally compact space, Vany Nachbin family on X and E a TVS with E∗ separates the points ofE. The dual space of (CVo(X,E), ωV ) is isomorphic to

VMt(Bo(X), E∗) ≡ vm : v ∈ V, m ∈Mt(Bo(X), E∗),

the space of all (E∗, σ(E∗, E))-valued bounded tight measures on X.



Section 8.1. Various classes of E∗-valued measures on a topologicalspace X are considered. The existence of the integral of functions inCb(X,E) with respect to these measures is also given. These results aretaken from the papers [Shu72b, KR81, Kat83, Kh95]. Earlier results inthe locally convex setting were obtained in [Kat74, Fon74].

Section 8.2. The Riesz representation type theorems for the charac-terization of dual spaces (Cb(X,E), u)

∗, (Cpc(X,E), β)∗, (Crc(X,E), σ)

∗

and (CVo(X,E), ωV )∗ are obtained via the integral representations. In

fact, each of these dual spaces is identified with a suitable class of E∗-valued measures on X . These results are also taken from the papers[Shu72b, KR81, Kat83, Kh95]. Theorem 8.2.2 has been obtained, in-dependently, in [KR81, Kat83]. For earlier work in the locally convexsetting, several useful references can be found in [Shu72b]. Theorem8.2.4 was obtained in [SumW70] for E a scalar field, in [Pro71b] for E alocally convex space and in [Naw89] for E a general TVS.

There is an extensive work (not considered here) on Bochner andBartle–Dunford–Schwartz integration processes for vector-valued func-tions and also on the Riesz representation of operators T : Co(X,E)→ Efor E a Banach or locally convex space (see [Mez02, Mez08, Khu08]).More recent developments in this area have been made in the papers[DL13, Thom12]. These latter papers deal with the theory of integrationof scalar functions with respect to a measure with values in a TVS E, notnecessarily locally convex. They focus on the extension of such integralsfrom bounded measurable functions to the class of integrable functions,proving adequate convergence theorems, and establishing usable integra-bility criteria.

CHAPTER 9

Weighted Composition Operators

In this chapter, we give some characterizations of multiplication andcomposition operators on the weighted function spaces CVo(X,E) andCVb(X,E) induced by scalar-, vector-, and operator-valued maps. Wealso include characterizations of, compact, equicontinuous, precompactand bounded multiplication/composition operators.

As before, the topology of bounded convergence tu (resp. the topologyof pointwise convergence tp) on CL(E) is the linear topology which hasa base of neighborhood of 0 consisting of all sets of the form

U(D,G) = T ∈ CL(E) : T (D) ⊆ G

= T ∈ CL(E) : ‖T‖D,G ≤ 1,

where D is a bounded (respectively, finite) subset of E, and G ∈ WE isshrinkable. (Here ‖T‖D,G = supρG(T (a)) : a ∈ D.)

141

142 9. WEIGHTED COMPOSITION OPERATORS

1. Multiplication Operators on CVo(X,E)

Let X be a completely regular Hausdorff space, E a Hausdorff TVS,and V a Nachbin family of weights on X . It is assumed throughout thatthe Nachbin family V on X satisfies the following conditions:

(∗) V > 0;(∗∗) CVo(X) does not vanish on X, i.e., corresponding to each x ∈ X ,

there exists an hx ∈ CVo(X) such that hx(x) 6= 0.The above conditions primarily serve to exclude trivial cases along

with certain unnecessary pathological situations. If V > 0, then condi-tion (∗∗) holds, in particular, when each v ∈ V vanishes at infinity orX is locally compact. [First suppose that V ⊆ S+

o (X), and let x ∈ X .Choose v ∈ V such that v(x) 6= 0. Since X is completely regular, thereexists an f ∈ Cb(X) such that f(x) = 1. Since v ∈ S+

o (X), it easilyfollows that vf ∈ CVo(X) and v(x)f(x) 6= 0. Next, suppose that X islocally compact, and let x ∈ X . There exists an f ∈ Coo(X) ⊆ Co(X)such that f(x) = 1. Since V > 0, choose v ∈ V such that v(x) 6= 0. Thenclearly vf ∈ CVo(X) and v(x)f(x) 6= 0.]

We further mention a useful fact that if CVo(X) does not vanish onX, then, for any x ∈ X and any open neighborhood U of x in X , we canchoose a f ∈ CVo(X) with

0 ≤ f ≤ 1, f(X\U) = 0 and f(x) = 1.

This follows from Lemma 4.1.2, due to Nachbin [Nac67], by takingM =CVo(X) ⊆ C(X), K = x, and U = Ui for all i = 1, ..., n.

If CVo(X) does not vanish on X and E is a non-trivial TVS, thenCVo(X)⊗E and hence CVb(X,E) and CVb(X,E) also do not vanish onX . [In fact, for any x ∈ X , choose ϕ in CVo(X) with ϕ(x) 6= 0. Then,if a ( 6= 0) in E, the function ϕ ⊗ a belongs to CVo(X) ⊗ E and clearly(f ⊗ a)(x) = ϕ(x)a 6= 0.]

Note. Earlier in Section 1.2, we have seen by an example thatCVb(X,E) and hence CVo(X,E) may be trivial for relatively nice X(e.g. X = Q). This situation justifies to assume the above conditions(∗) and (∗∗) throughout.

Definition. Let F (X,E) be the vector space of all functions fromX into E. If θ : X → C and ψ : X → E, E a topological algebra, aregiven maps, the scalar multiplication on E and the multiplication on Egive rise to linear mappings Mθ and Mψ from CVo(X,E) into F (X,E)defined by Mθ(f) = θf and Mψ(f) = ψf , both pointwise; that is,

(θf)(x) = θ(x)f(x) and (ψf)(x) = ψ(x)f(x), x ∈ X.

1. MULTIPLICATION OPERATORS ON CVo(X,E) 143

Also if π : X → CL(E), E a TVS, we may define Mπ from CVo(X,E)into F (X,E) by Mπ(f) = πf , where

(πf)(x) = πx(f(x)) := π(x)(f(x)), x ∈ X.

If Mθ, Mψ and Mπ map CVb(X,E) (resp. CVo(X,E)) into itself and arecontinuous, they are called multiplication operators on CVb(X,E)(resp.CVo(X,E)) induced by θ, ψ and π, respectively.

We first give necessary and sufficient conditions forMθ, θ : X → C, tobe the multiplication operator on the weighted space CVo(X,E). Theseresults hold also for the space CVb(X,E) with slight modification in theproofs and are therefore omitted.

For any θ : X → C, we let V |θ| = v|θ| : v ∈ V .

Theorem 9.1.1. [SM91, KT97] For a mapping θ : X → C, thefollowing are equivalent:

(a) Mθ is a multiplication operator on CVo(X,E);(b) (i) θ is continuous and (ii) V |θ| ≤ V.Proof. Let W be a base of closed, balanced and shrinkable neighbor-

hoods of 0 in E.(a)⇒ (b) To prove (i), let xα be a net in X with xα → x ∈ X . By

assumption (∗∗), there exists an h ∈ CVo(X) such that h(x) 6= 0. SinceMθ is a self-map on CVo(X,E), it follows that the function θh from Xinto C is continuous. Hence θ(xα)h(xα) → θ(x)h(x) and consequentlyθ(xα)→ θ(x).

(ii) To show that V |θ| ≤ V , let v ∈ V . By continuity of Mθ, givenG ∈ W, there exist u ∈ V and H ∈ W such that

Mθ(N(u,H)) ⊆ N(v,G). (1)

Without loss of generality, we may assume thatG∪H is a proper subset of

E. Choose a ∈ E\(G∪H), and put t = ρH (a)ρG(a)

. We claim that v|θ| ≤ 2tu.

Fix xo ∈ X . We shall consider two cases: u(xo) 6= 0 and u(xo) = 0.Suppose that u(xo) 6= 0, and let ε = u(xo). Then D = x ∈ X :

u(x) < 2ε is an open neighborhood of xo. Taking M = CVo(X) withassumption (∗∗) in Lemma 4.1.2, there is an h ∈ CVo(X) with

0 ≤ h ≤ 1, h(xo) = 1 and h(X\D) = 0.

Define f = 12ερH(a)

(h⊗ a). Since ρH is homogeneous, for any x ∈ X ,

ρH [u(x)f(x)] = ρH [u(x)1

2ερH(a)h(x)a)] =

u(x)h(x)

2ε.


If x ∈ D, u(x) < 2ε and if x ∈ X\D, h(x) = 0; hence

ρH [u(x)f(x)] <2εh(x)

2ε≤ 1 if x ∈ D,

ρH [u(x)f(x)] = 0 < 1 if x ∈ X\D.

Since H = b ∈ E : ρH(b) ≤ 1, we have f ∈ N(u,H). Hence, by (1),θf ∈ N(v,G). This implies that, for any x ∈ X ,

ρG[1

2ερH(a)θ(x)v(x)h(x)a] ≤ 1,

or v(x)h(x)|θ(x)| ≤2ερH(a)

ρG(a)= 2tε = 2tu(xo),

In particular, v(xo) |θ(xo)| ≤ 2tu(xo).Now suppose that u(xo) = 0 but v(xo)|θ(xo)| > 0. Put

ε =1

2tv(xo)|θ(xo)|.

Let D = x ∈ X : u(x) < ε, and choose an h ∈ CVo(X) with 0 ≤ h ≤1, h(xo) = 1 and h(X\D) = 0. Define g = 1

ερH(a)(h ⊗ a). Since ρH is

homogeneous, for any x ∈ X ,

ρH [u(x)g(x)] = ρH [u(x)1

ερH(a)h(x)a)] =

u(x)h(x)

ε.

If x ∈ D, u(x) < ε and if x ∈ X\D, h(x) = 0; hence

ρH [u(x)f(x)] <εh(x)

ε≤ 1 if x ∈ D

ρH [u(x)f(x)] = 0 < 1 if x ∈ X\D.

Hence g ∈ N(u,H) and so, by (1), θg ∈ N(v,G). This implies that, forany x ∈ X ,

ρG[1

ερH(a)θ(x)v(x)h(x)a] ≤ 1,

or v(x)h(x)|θ(x)| ≤ερH(a)

ρG(a)= tε =

1

2v(xo)|θ(xo)|.

In particular, v(xo)|θ(xo)| ≤12v(xo)|θ(xo)|, which is impossible unless

v(xo)|θ(xo)| = 0.(b) ⇒ (a) We first show that Mθ maps CVo(X,E) into itself. Let

f ∈ CVo(X,E), and let v ∈ V and G ∈ W. Choose u ∈ V such that


v|θ| ≤ u. There exists a compact set K ⊆ X such that u(x)f(x) ∈ G forall x ∈ X\K. Then, since G is balanced,

v(x)Mθ(f)(x) = v(x)θ(x)f(x) ∈ G

for all x ∈ X\K. Hence vMθ(f) vanishes at infinity; further, since θis continuous, Mθ(f) ∈ CVo(X,E). To prove the continuity of Mθ, letfα be a net in CVo(X,E) with fα → 0. Let v,G and u be chosen asabove. Choose an index αo such that fα ∈ N(u,G) for all α ≥ αo. Thenit follows that θfα ∈ N(v,G) for all α ≥ αo. Thus Mθ(fα) → 0. So Mθ

is continuous at 0 and hence, by linearity, on CVo(X,E).

We next consider the case of the operatorMψ, where ψ : X → E andE a topological algebra. Recall that a topological algebra E is calledlocally idempotent if it has a base W of neighborhoods of 0 consistingof idempotent sets. If G ∈ W is idempotent and shrinkable, then itsMinkowski functional ρG is continuous and submultiplicative: ρG(ab) ≤ρG(a)ρG(b) for all a, b ∈ E; further, if E has an identity e, ρG(e) ≥ 1.

Theorem 9.1.2. [SM91, KT97] Let E be a locally idempotent topo-logical algebra with identity e and W a base of neighborhoods of 0. Then,for a mapping ψ : X → E, the following are equivalent:

(a) Mψ is a multiplication operator on CVo(X,E).(b) (i) ψ is continuous and (ii) V ρG ψ ≤ V for every G ∈ W.Proof. We may assume thatW consists of closed, balanced, shrinkable

and idempotent sets.(a) ⇒ (b) (i) Let xα be a net in X with xα → x ∈ X . Choose

an h ∈ CVo(X) with h(x) 6= 0. Since Mψ is a self-map on CVo(X,E),it follows that, for any a ∈ E, the function ψ(h ⊗ a) from X into E iscontinuous. Hence h(xα)ψ(xα)→ h(x)ψ(x) consequently ψ(xα)→ ψ(x).This proves the continuity of ψ.

(ii) Let v ∈ V and G ∈ W. There exist u ∈ V and H ∈ W such that

Mψ(N(u,H)) ⊆ N(v,G). (2)

Without loss of generality, we may assume that H is a proper subset ofE. We claim that vρG ψ ≤ 2ρH(e)u.

Fix xo ∈ X . First suppose that u(xo) 6= 0, and let ε = u(xo). ThenD = x ∈ X : u(x) < 2ε is an open neighborhood of xo and so thereexists an h ∈ CVo(X) such that 0 ≤ h ≤ 1, h(xo) = 1, and h(X\D) = 0.Define f = 1

2ερH(e)(h⊗ e). Then, for any x ∈ X,

ρH(u(x)f(x)) = ρH(u(x)h(x)e/2ερH(e)) ≤ 1;


that is, f ∈ N(u,H). Hence, by (2), ψf ∈ N(v,G). This implies that,for any x ∈ X ,

v(x)h(x)ρG(ψ(x)) ≤ 2ερH(e).

In particular, v(xo)ρG(ψ(xo)) ≤ 2ρH(e)u(xo).Next suppose that u(xo) = 0, but v(xo)ρG(ψ(xo)) > 0. Put ε =

v(xo)ρG(ψ(xo))2ρH (e)

. Let D = x ∈ X : u(x) < ε, and choose an h ∈ CVo(X)

as above. Define g = 1ερH (e)

(h ⊗ e). Then g ∈ N(u,H) and so by (2),

φg ∈ N(v,G). From this we obtain

v(xo)ρG(ψ(xo)) ≤ ρH(e)ε = v(xo)ρG(ψ(xo))/2,

which is impossible unless v(xo)ρG(ψ(xo)) = 0.(b) ⇒ (a) We first show that Mψ maps CVo(X,E) into itself. Let

f ∈ CVo(X,E), and let v ∈ V and G ∈ W. Choose u ∈ V such thatvρG ψ ≤ u. There exists a compact set K ⊆ X such that u(x)f(x) ∈ Gfor all x ∈ X\K. Since ρG is submultiplicative, for any x ∈ X\K,

ρG(v(x)ψ(x)f(x)) ≤ v(x)ρG(ψ(x))ρG(f(x)) ≤ u(x)ρG(f(x)) ≤ 1;

hence Mψ(f) ∈ CVo(X,E). Using again the submultiplicity of ρG, thecontinuity of Mψ follows in the same way as in the proof of Theorem9.1.1.

Wemention that Theorem 9.1.2 also remains true if we replace CVo(X ,E) by CVb(X,E).

We now apply the above results to the cases: V = S+oo(X) and V =

S+o (X).

Theorem 9.1.3. [SM91, KT97] (i) If θ : X → C is a continuousmapping, then Mθ is a multiplication operator on (C(X,E), k).

(ii) If E is a locally idempotent topological algebra with identity eand ψ : X → E a continuous mapping, then Mψ is a multiplicationoperator on (C(X,E), k).

Proof. (i) In view of Theorem 9.1.1, we only need to verify thatV |θ| ≤ V , where V = S+

oo(X). Let v ∈ V . Choose a compact setK ⊆ X with v(x) = 0 for all x ∈ X\K. Let s = sup|θ(x)| : x ∈K, t = supv(x) : x ∈ K and u = stχK . Then u ∈ V and clearlyv(x) |θ(x)| ≤ u(x) for all x ∈ X.

(ii) Let W be a base of neighborhoods of 0 in E consisting ofclosed, balanced, shrinkable and idempotent sets. In view of Theorem9.1.2, we only need to verify that V ρG ψ ≤ V for every G ∈ W,where V = S+

oo(X). Let v ∈ V and G ∈ W. Choose a compact setK ⊆ X with v(x) = 0 for all x ∈ X\K. Let s = supρG(ψ(x)) : x ∈


K, t = supv(x) : x ∈ K and u = stχK . Then u ∈ V and clearlyv(x)ρG(ψ(x)) ≤ u(x) for all x ∈ X .

The following example shows that the above result need not holdwhen (C(X,E), k) is replaced by (Cb(X,E), β).

Example 9.1.4. Let X = R+, E = C, and V = S+o (X). Let

θ = ψ : X → C be given by θ(x) = x2(x ∈ X), and let v ∈ V be givenby v(x) = 1

x(x ∈ X). Then v(x) |θ(x)| = x for all x ∈ X . Since each

u ∈ V is a bounded function, v |θ| ≤ u is not true for every u ∈ V ;i.e.V |θ| ≤ V does not hold and so, by Theorem 9.1.1, Mθ is not amultiplication operator on (Cb(X), β). The same is also true for the space(Cb(X), u), where u is the uniform topology, since β ≤ u. However, ifθ and ψ are “bounded” continuous functions, then it is easily seen thatMθ and Mψ are always multiplication operators on CVo(X,E) for anyNachbin family V .

We now consider multiplication operators induced by operator-valuedmappings and give necessary and sufficient conditions for Mπ, π : X →CLu(E), to be the multiplication operator on the weighted space CVb(X,E).We then use this characterization to show that, in the case of E a lo-cally bounded space, Mπ is a multiplication operator if π is a boundedmapping or V = S+

oo(X).

Theorem 9.1.5. [SM91, KT97] Let X be a k-space and E a TVS.Then, for any continuous mapping π : X → CLu(E), the following areequivalent:

(a) Mπ is a multiplication operator on CVb(X,E).(b) For every v ∈ V and G ∈ W, there exist u ∈ V and H ∈ W such

thatv(x)ρG(πx(y)) ≤ u(x)ρH(y) for all x ∈ X and y ∈ E.

Proof.Wemay assume thatW consists of closed, balanced and shrink-able sets.

(a) ⇒ (b) Let v ∈ V and G ∈ W. By hypothesis, there exist u ∈ Vand H ∈ W such that

Mπ(N(u,H) ⊆ N(v,G). (3)

We claim thatv(x)ρG(πx(y) ≤ 2u(x)ρH(y)

for all x ∈ X and y ∈ E. Let xo ∈ X and yo ∈ E. Then we consider fourcases:

(I) u(xo)ρH(yo) 6= 0.

(II) u(xo) = 0, ρH(yo) 6= 0.


(III) u(xo) 6= 0, ρH(yo) = 0.

(IV) u(xo) = 0, ρH(yo) = 0.

Case I. Suppose u(xo)ρH(yo) 6= 0, and let ε = u(xo)ρH(yo). Thenthe set D = x ∈ X : u(x)ρH(yo) < 2ε is an open neighborhood of xo,and so there exists an h ∈ CVb(X) such that 0 ≤ h ≤ 1, h(xo) = 1, andh(X\D) = 0. Define g = 1

2ε(h⊗ yo). Then, for any x ∈ X,

ρH(u(x)g(x)) =1

2εu(x)h(x)ρH(yo) ≤ 1;

that is, g ∈ N(u,H). Hence, by (3), Mπ(g) ∈ N(v,G). This impliesthat, for any x ∈ X,

v(x)h(x)ρG(πx(yo)) ≤ 2u(xo)ρH(yo).

In particular, by taking x = xo, we have

v(x)ρG(πxo(yo)) ≤ 2u(xo)ρH(yo).

Case II. Suppose u(xo) = 0, ρH(yo) 6= 0, but v(xo)ρG(πxo(yo)) > 0.Put ε = v(xo)ρG(πxo(yo)/2. Then the set D = x ∈ X : u(x)ρH(yo) < εis an open neighborhood of xo, and so there exists an h ∈ CVb(X) suchthat 0 ≤ h ≤ 1, h(xo) = 1, and h(X\D) = 0. Define g = 1

ε(h ⊗ yo).

Then, for any x ∈ X,

ρH(u(x)g(x)) =1

εu(x)h(x)ρH(yo) ≤ 1;

that is, g ∈ N(u,H). Hence, by (3), Mπ(g) ∈ N(v,G). This impliesthat, for any x ∈ X,

v(x)h(x)ρG(πx(yo)) ≤ ε =1

2v(x)ρG(πxo(yo)).

In particular, by taking x = xo, we have

v(x)ρG(πxo(yo)) ≤1

2v(x)ρG(πxo(yo)),

which is impossible unless v(xo)ρG(πxo(yo)) = 0.

(III). Suppose u(xo) 6= 0, ρH(yo) = 0, but v(xo)ρG(πxo(yo)) > 0. Putε = 1

2v(xo)ρG(πxo(yo)). Then the set D = x ∈ X : u(x) < ε + u(xo)

is an open neighborhood of xo, and so there exists an h ∈ CVb(X) such


that 0 ≤ h ≤ 1, h(xo) = 1, and h(X\D) = 0. Define g = h⊗ yo Clearlyg ∈ CVb(X,E) and

0 ≤ ρG g ≤ ρG(yo), ρG g(xo) = ρG(yo), and ρG g(X\D) = 0.

Consider g1 =gε. Then, for any x ∈ X,

ρH(g1(x)) = ρH(1

εh(x)yo) =

1

εh(x)ρH(yo) = 0 ∈ H

Hence g1 ∈ N(u,H) and so, by (3), Mπ(g1) ∈ N(v,G). Therefore, forany x ∈ X

v(x)ρG(πx(g1(x))) ≤ 1, or v(x)ρG(πx(h(x)yo)) ≤ ε =1

2v(xo)ρG(πxo(yo)).

In particular, since h(xo) = 1, v(xo)ρG(πxo(yo)) ≤12v(xo)ρG(πxo(yo)), a

contradiction.

(IV) Suppose u(xo) = 0, ρH(yo) = 0, but v(xo)ρG(πxo(yo)) > 0. Putε = 1

2v(xo)ρG(πxo(yo)). Then the set D = x ∈ X : u(x) < ε is an

open neighborhood of xo, and so there exists an h ∈ CVb(X) such that0 ≤ h ≤ 1, h(xo) = 1, and h(X\D) = 0. Define g = h ⊗ yo. Clearlyg ∈ CVb(X,E) and, for any x ∈ X,

0 ≤ ρG(g(x)) = h(x)ρG(yo) ≤ ρG(yo),

ρG(g(xo)) = ρG(yo), and ρG(g(x)) = 0 if x ∈ X\D;

or 0 ≤ ρG g ≤ ρG(yo), ρG g(xo) = ρG(yo), and ρG g(X\D) = 0.

Consider g1 =gε. Then, for any x ∈ X,

ρH(g1(x)) = ρH(1

εh(x)yo) =

1

εh(x)ρH(yo) = 0 ∈ H

Hence g1 ∈ N(u,H) and so, by (4), Mπ(g1) ∈ N(v,G). Therefore, forany x ∈ X

v(x)ρG(πx(g1(x))) ≤ 1, or v(x)ρG(πx(h(x)yo)) ≤ ε =1

2v(xo)ρG(πxo(yo)).

In particular, since h(xo) = 1, v(xo)ρG(πxo(yo)) ≤12v(xo)ρG(πxo(yo)), a

contradiction.(b) ⇒ (a) We first show that Mπ maps CVb(X,E) into itself. Let

f ∈ CVb(X,E). Since X is a k-space, it is enough to show that Mπ(f)is continuous on any given compact subset K of X . Let xα : α ∈ Ibe a net in K with xα → x ∈ K. Let G ∈ W, and let G1 ∈ W withG1+ G1 ⊆ G. Now f(xα), being convergent, is bounded and so, bycontinuity of π, there exists an α1 ∈ I such that

πxβ(f(xα))− πx(f(xα)) ∈ G1, (4)


for all α ∈ I and all β ≥ α1. Since πx : E → E is a continuous linearmapping, there exists an H ∈ W such that πx(y) ∈ G1 for all y ∈ H.Choose an α2 ∈ I such that f(x) ∈ H for all α ≥ α2; consequently,

πx(f(xα)− f(x)) ∈ G1 (5)

for α ≥ α2. Choose αo ∈ I with αo ≥ α1, α2. Then, by (4) and (5)

πxβ(f(xβ))− πx(f(x)) = πxβ(f(xβ))− πx(f(xβ) + πx(f(xβ)) + πx(f(x)

∈ G1 +G1 ⊆ G

for all β ≥ αo. Hence Mπ(f) ∈ C(X,E). To show that Mπ(f) ∈CVb(X,E), let v ∈ V and G ∈ W. By hypothesis, there exist u ∈ V andH ∈ W such that

v(x)ρG(πx(y)) ≤ ρH(u(x)(y) (6)

for all x ∈ X and y ∈ E. Choose λ > 0 such that u(x)f(x) ∈ λH for allx ∈ X . Then, it follows from (6) that v(x)πx(f(x)) ∈ λG for all x ∈ X .This proves that Mπ(f) ∈ CVb(X,E).

To establish the continuity of Mπ, let fα : α ∈ I be a net inCVb(X,E) with fα → 0, and let v ∈ V and G ∈ W. Choose u ∈ V andH ∈ W as above and which satisfy (6). There exists an αo ∈ I such thatfα ∈ N(u,H) for all α ≥ αo. Then, for any x ∈ X and α ≥ αo

v(x)ρG(πx(fα(x))) ≤ ρH(fα(x)) ≤ 1

or equivalent v(x)πx(fα(x)) ∈ G. So Mπ is continuous at 0 and hence,by its linearity, on CVb(X,E).

We now deduce that, under same additional conditions on π, Mπ isautomatically a multiplication operator.

Corollary 9.1.6. [SM91, KT97] Let π : X → CLu(E) be a constantmapping. Then Mπ is a multiplication operator on CVb(X,E).

Proof. We need to verify that condition (b) of Theorem 9.1.5 holdsin this case. Let v ∈ V and G ∈ W. Choose T ∈ CL(E) such thatπ(x) = T for all x ∈ X . Since T is continuous and linear, there exist anm > 0 and a closed shrinkable H ∈ W such that

v(x)ρG(T (y)) ≤ mρH(y) for all y ∈ E.

Let u = mv. Then, for any x ∈ X and y ∈ E,

v(x)ρG(π(x)(y)) = v(x)ρG(T (y)) ≤ u(x)ρH(y).


Corollary 9.1.7. Suppose E is a locally bounded TVS and π :X → CLu(E) is a continuous and bounded mapping. Then Mπ is amultiplication operator on CVb(X,E).

Proof. Let v ∈ V and G ∈ W. We may assume that G is a boundedneighborhood of 0. Choose a closed and shrinkable H ∈ W with H ⊆ G.Since π(X) is bounded in CLu(E), there exists an m > 0 such thatπ(X) ⊆ mU(H,H); i.e.,

ρH(π(x)(y)) ≤ mρH(y) for all x ∈ X, y ∈ H.

Let u = mv. Then, for any x ∈ X and y ∈ E,

v(x)ρG(π(x)(y)) ≤ v(x)ρH(π(x)(y)) ≤ u(x)ρH(y).

We now apply Theorem 9.1.5 to the cases V = S+oo(X) and V =

S+o (X).

Theorem 9.1.8. [SM91, KT97] Suppose E is a locally bounded TVSand π : X → CLu(E) is a continuous mapping. Then Mπ is a multipli-cation operator on (C(X,E), k).

Proof. We need to verify that condition (a) of Theorem 9.1.5 holdsfor V = S+

oo(X). Let v ∈ V and G ∈ W. Choose a compact K ⊆ X suchthat v(x) = 0 for all x ∈ X\K. We may assume that G is a boundedneighborhood of 0. Choose a closed and shrinkable H ∈ W with H ⊆ G.Since π : X → CLu(E) is continuous, π(K) is compact in CLu(E) andso there exists an m > 0 such that π(K) ⊆ mU(H,H); i.e.,

ρH(π(x)(y)) ≤ mρH(y) for x ∈ K and y ∈ H.

Let t = supv(x) : x ∈ K, and put u = mtχK . Then u ∈ V . Let x ∈ Xand y ∈ E. If x ∈ K, then

v(x)ρG(π(x)(y)) ≤ tρH(π(x)(y)) ≤ u(x)ρH(y).

If x ∈ X\K, the above holds trivially (since v(x) = 0).

Remark. If E = C, then CLu(E) = C and so Example 9.1.4 alsoshows that the above result need not hold when (C(X,E), k) is replacedby (Cb(X,E), β).


2. Weighted Composition Operators on CVo(X,E)

In this section, we consider the more general notion of weighted com-position operators which include the multiplication and composition op-erators.

Definition. Let F (X,E) be the vector space of all functions from atopological space X to a TVS E, and let S(X,E) be a vector subspaceof F (X,E). Let L(E) be the vector space of all linear maps from E intoitself. Then the operator-valued mapping π : X → L(E) and the self-map ϕ : X → X give rise to a linear mappingWπ,ϕ : S(X,E)→ F (X,E)defined as

Wπ,ϕ(f) = π · f ϕ, f ∈ S(X,E);

that is,

Wπ,ϕ(f)(x) = π(x)(f(ϕ(x))) = πx(f(ϕ(x))), f ∈ S(X,E), x ∈ X.

(i) In case S(X,E) is equipped with a linear topology,Wπ,ϕ takes S(X,E)into itself and Wπ,ϕ is continuous, we call Wπ,ϕ the weighted compositionoperator on S(X,E) induced by the mappings π and ϕ.

(ii) If ϕ : X → X is the identity map, the corresponding opera-tor Wπ,ϕ is the multiplication operator Mπ on S(X,E) induced by theoperator-valued mapping π.

(iii) Also, if π(x) = I, the identity operator on E, for every x ∈X, then Wπ,ϕ is the composition operator Cϕ on S(X,E) induced by theself-map ϕ of X, where Cϕ(f) = f ϕ, f ∈ S(X,E).

These operators have been the subject matter of extensive study forthe last several decades on different functions spaces; especially on Lp-spaces, Bergman spaces, and spaces of continuous functions.

As before, it is assumed that the Nachbin family V satisfies (∗) V > 0and (∗∗) CVo(X) does not vanish on X.

Theorem 9.2.1. [MS98] Let X be a completely regular kR-space andE a TVS. Let π : X → CLu(E) and ϕ : X → X be continuous mappings.Then the following conditions are equivalent:

(a) Wπ,ϕ is a weighted composition operator on CVo(X,E).(b) (i) for any v ∈ V and G ∈ W, there exist u ∈ V and H ∈ N

such that

u(ϕ(x))y ∈ H implies that v(x)πx(y) ∈ G for all x ∈ X, y ∈ E;

(ii) for each v ∈ V and G ∈ W and compact set K ⊆ X, the set

ϕ−1(K) ∩ x ∈ X : v(x)πx(y) /∈ G is compact for each 0 6= y ∈ E.

2. WEIGHTED COMPOSITION OPERATORS ON CVo(X,E) 153

Proof. (a) ⇒ (b) Suppose that Wπ,ϕ is a weighted composition op-erator on CVo(X,E). To establish condition (b)(i), we fix v ∈ V andG ∈ W. By the continuity of Wπ,ϕ, there exist u ∈ V and H ∈ W suchthat

Wπ,ϕ(N(u,H) ⊆ N(v,G).

Let ω = 2u. Then we claim that

ω(ϕ(x))y ∈ H implies that v(x)πx(y) ∈ G for all x ∈ X, y ∈ E.

[Assume that for some xo ∈ X and yo ∈ E, we have ω(ϕ(xo))yo ∈ H butv(xo)πxo(yo) /∈ G. Let D = x ∈ X : ω(x)yo ∈ H. Then D is an openneighborhood of ϕ(xo) and therefore there exists f ∈ CVo(X) such that0 ≤ f ≤ 1, f(ϕ(xo)) = 1 and f(X\D) = 0. Define g = 2f ⊗ yo. Now, itis easy to see that g ∈ N(v,H) and therefore we have Wπ,ϕg ∈ N(v,G).Thus it follows that v(xo)πxo(yo) ∈

12G ⊆ G, which is a contradiction.

This proves our claim.]To prove (b)(ii), let v ∈ V and G ∈ W and compact set K ⊆ X .

There exists f ∈ CVo(X) such that f(K) = 1. Fix 0 6= y ∈ E anddefineg = f ⊗ y. Then clearly Wπ,ϕg ∈ CVo(X,E). Now if we put

D = x ∈ X : v(x)πx(g(ϕ(x))) /∈ G

and

S = ϕ−1(K) ∩ x ∈ X : v(x)πx(y) /∈ G,

then clearly D is a compact subset of X such that S ⊆ D. Thus S, beinga closed subset of D, is compact.

(b)⇒ (a) Suppose that conditions (b)(i) and (b)(ii) hold. First of allwe show that Wπ,ϕ is an into map. [Let f ∈ CVo(X,E) and let K ⊆ Xbe a compact set. Let xα : α ∈ I be a net in K such that xα → x. FixG ∈ W. Choose balanced H ∈ W such that H +H ⊆ G. Let

B = f(ϕ(xα)) : α ∈ I,

a bounded set in E. Since π : X → CLu(E) is continuous and U(B,H)is a neighborhood of 0 in CLu(E), there exists αo ∈ I such that

(πxα − πx)(f(ϕ(xα))) ∈ H for all α ≥ αo.

Again, since πx(f(ϕ(x))) → πx(f(ϕ(x))) in E, there exists αo ∈ I suchthat

π(x)(f(ϕ(xα)))− πx(f(ϕ(x))) ∈ H for all α ≥ αo.

Let α1 ∈ I be such that α1 ≥ αo and α1 ≥ αo. Then, for every α ≥ α1,we have

πxα(f(ϕ(xo)))− πx(f(ϕ(x)))


= πxα(f(ϕ(xo)))− πx(f(ϕ(x))) + πx(f(ϕ(xα))− πx(f(ϕ(x)))

∈ H +H ⊆ G.

This proves that Wπ,ϕf ∈ C(X,E).To show that Wπ,ϕf ∈ CVo(X,E), fix v ∈ V and G ∈ W. We need to

prove that the set S = x ∈ X : v(x)πx(f(x))) /∈ G is compact. Choosea balanced G1 ∈ W with G1 + G1 ⊆ G. According to condition (b)(i),there exist u ∈ V and H ∈ W such that

u(ϕ(x))y ∈ H implies that v(x)πx(y) ∈ G1 for allx ∈ X, y ∈ E.

Now, if we putK = x ∈ X : v(x)f(x) /∈ H, then K is a compact subsetof X and ϕ(S) ⊆ K. Let m = supu(x) : x ∈ K. Obviously m > 0.Since f(ϕ(S)) is precompact in E, there exist a finite set f(ϕ(xi)) : 1 ≤i ≤ nin f(ϕ(S)) such that

f(ϕ(S)) ⊆n⋃

i=1

f(ϕ(xi)) +1

2mH.

Again, condition (b)(ii) implies that for each i ∈ 1, 2, ..., n the set

Ci = ϕ−1(K) ∩ x ∈ X : v(x)πx(f(ϕ(xi))) /∈ G1

is compact. Now S ⊆ ∪ni=1Ci, as follows. [Let xo ∈ S. Then there existsj ∈ 1, 2, ..., n such that

f(ϕ(xo))− f(ϕ(xj)) ∈1

2mH.

That is,

u(ϕ(xo))[f(ϕ(xo))− f(ϕ(xj))] ∈1

2mH.

Further, it implies that

v(xo)πxo [f(ϕ(xo))− f(ϕ(xj))] ∈ G1.

Now, we claim that

v(xo)πxof(ϕ(xj))) /∈ G1.

Assume the opposite. Then

v(xo)πxo(f(ϕ(xo))) = v(xo)πxo [f(ϕ(xo))− f(ϕ(xj))] + v(xo)πxof(ϕ(xj))

∈ G1 +G1 ⊆ G

which is impossible. Thus xo ∈ Cj ⊆ ∪ni=1Ci.] Therefore S, being aclosed subset of the compact set ∪ni=1Ci, is itself compact. This provesthat Wπ,ϕf ∈ CVo(X,E).

To show that Wπ,ϕ is continuous, fix v ∈ V and G ∈ W. Then, bycondition (i), there exist u ∈ V and H ∈ W such that u(ϕ(x))y ∈ H


implies that v(x)πx(y) ∈ G for all x ∈ X and y ∈ E. We claimthat Wπ,ϕ(N(u,H)) ⊆ N(v,G). [Let f ∈ CVo(X,E) be such thatf ∈ N(u,H). Then we have u(ϕ(x))f(ϕ(x)) ∈ H for all x ∈ X.Further, it implies that v(x)πx(f(ϕ(x))) ∈ G for all x ∈ X. That isWπ,ϕf ∈ N(v,G).] This proves that Wπ,ϕ is continuous at the origin inCVo(X,E) and hence continuous on CVo(X,E). ThusWπ,ϕ is a weightedcomposition operator on CVo(X,E).

In the next theorem, we shall characterize the weighted compositionoperators Wπ,ϕ on CVo(X,E) induced by the mappings π : X → E andϕ : X → X , where E is a topological algebra.

Theorem 9.2.2. [MS98] Let X be a completely regular kR-space,and let E be a topological algebra with hypocontinuous multiplication andidentity e. Let ϕ : X → X be a continuous map. Then the followingconditions are equivalent:

(a) Wπ,ϕ is a weighted composition operator on CVo(X,E).(b) (i) π : X → E is continuous;(ii) for any v ∈ V and G ∈ W there exist u ∈ V and H ∈ W such

that

u(ϕ(x))y ∈ H implies that v(x)πx(y) ∈ G for all x ∈ X, y ∈ E;

(iii) for any v ∈ V , G ∈ W and compact set K ⊆ X, the set

ϕ−1(K) ∩ x ∈ X : v(x)πx(y) /∈ G is compact for each y( 6= 0) ∈ E.

Proof. (a) ⇒ (b) If Wπ,ϕ is a weighted composition operator onCVo(X,E), then we need to prove condition (b)(i), whereas the proof ofconditions (b)(ii) and (b)(iii) can be obtained with slight modificationsas in the proof of conditions (b)(i) and (b)(ii) of Theorem 9.2.1. Fixxo ∈ X. Then there exists f ∈ CVo(X,E) such that f(ϕ(xo)) 6= 0. Also,there exists an open neighborhood D of xo in X such that f(ϕ(x)) 6= 0for all x ∈ D. Let h : D → C be defined as h(x) = (f(ϕ(x)))−1 for allx ∈ D. Let g : X → E be defined as g(x) = f(x)e for all x ∈ X . Thenclearly g ∈ CVo(X,E) and therefore Wπ,ϕg ∈ CVo(X,E). Now, since themapping π · g ϕ · h is continuous at xo, it follows that the mapping π iscontinuous at xo. This proves that the mapping π is continuous on X .

(b) ⇒ (a) Suppose that conditions (b)(i) through (b)(iii) hold. Letf ∈ CVo(X,E) and letK ⊆ X be a compact set. Fix xo ∈ K andG ∈ W.Then there exists H ∈ W such that H + H ⊆ G. Let B1 = f(ϕ(K))and B2 = π(xo). Then B1, B2 ∈ b(E). Since the multiplication in Eis hypocontinuous, there exist H1, H2 ∈ W such that H1B1 ⊆ H andH2B2 ⊆ H . Since the map π is continuous at xo, there exists an open


neighborhood D1 of xo in K such that

π(x)− π(xo) ∈ H1 for all x ∈ D1.

Again, since f ϕ is continuous at xo, there exists an open neighborhoodD2 of xo in K such that

f(ϕ(x))− f(ϕ(xo)) ∈ H2 for all x ∈ D2.

f(ϕ(x)) − f(ϕ(xo)) ∈ H2, for all x ∈ D2. Let D = D1 ∩ D2. Then, forany x ∈ D, we have

π(x)f(ϕ))− π(xo)f(ϕ(xo))

= (π(x)− π(xo)f(ϕ(x)) + π(xo)(f(ϕ(x))− f(ϕ(xo)))

∈ H1B1 +H2B2 ⊆ H +H ⊆ G.

This proves that π · f ϕ is continuous at xo in K and hence on X .Thus Wπ,ϕf ∈ C(X,E). Again with slight modifications in the proof ofTheorem 9.2.1, it can be shown that Wπ,ϕf ∈ CVo(X,E) and Wπ,ϕ iscontinuous on CVo(X,E).

Remark 9.2.3. In case X is a completely regular kR-space and E isa topological algebra with hypocontinuous multiplication not necessar-ily containing an identity e, then the continuous mappings π : X → Eand ϕ : X → X induce the weighted composition operator Wπ,ϕ onCVo(X,E) iff conditions (b)(ii) and (b)(iii) of Theorem 9.2.2. are satis-fied.

Theorem 9.2.4. [MS98] Let X be a completely regular space andE be a TVS, and let ϕ : X → X be a mapping. Then the following areequivalent:

(a) ϕ induces the composition operator Cϕ on CVo(X,E);(b) ϕ induces the composition operator Cϕ on CVo(X,E) and CVo(X,E)

is invariant under Cϕ;(c) (i) ϕ is continuous,(ii) for any v ∈ V and G ∈ W there exist u ∈ V and H ∈ W such

that

u(ϕ(x))y ∈ H implies that v(x)y ∈ G for all x ∈ X, y ∈ E,

(iii) for any v ∈ V , G ∈ W and compact set K ⊆ X, the set

ϕ−1(K) ∩ x ∈ X : v(x)y /∈ G is compact for each y( 6= 0) ∈ E.

Proof. (a) ⇒ (c) In view of assumption (∗∗), it follows that if ϕ :X → X is any function for which the corresponding composition mapCϕ induced on CVo(X,E) has its range contained in C(X,E), then ϕ isnecessarily continuous. Hence (c)(i) holds.


To prove condition (c)(ii), we fix v ∈ V and G ∈ W. Since Cϕ iscontinuous on CVo(X,E), there exist u ∈ V and H ∈ W such that

Cϕ(N(u,H)) ⊆ N(v,G).

Let ω = 2u. We claim that ω(ϕ(x))y ∈ H implies that v(x)y ∈ G forall x ∈ X and y ∈ E. Assume that for some xo ∈ X and yo ∈ E, wehave ω(ϕ(xo))yo ∈ H but v(xo)yo /∈ G. Let D = x ∈ X : ω(x)yo ∈ H.Then D is an open neighborhood of ϕ(xo) and therefore there existsf ∈ CVo(X) such that 0 ≤ f ≤ 1, f(ϕ(xo)) = 1 and f(X\D) = 0.Define g = 2f ⊗ yo. Then g ∈ N(u,H) and thereby Cϕg ∈ N(v,G).Further, it implies that v(xo)yo ∈ G, which is impossible. Thus conditionc(ii) is satisfied.

To see condition (c)(iii), let v ∈ V and G ∈ W and compact setK ⊆ X . Then there exists f ∈ CVo(X) such that f(K) = 1. Fixyo( 6= 0) ∈ E. Define g = f ⊗ yo. Let

S = x ∈ X : v(x)g(ϕ(x)) /∈ G

and

M = ϕ−1(K) ∩ x ∈ X : u(x)yo /∈ G.

Then obviously S is a compact subset of X and M ⊆ S. Thus M iscompact, being a closed subset of S. This proves condition (c).

(c) ⇒ (b) To show that Cϕ is a composition operator on CVb(X,E),we fix v ∈ V and G ∈ W. By condition (c)(ii), there exist u ∈ Vand H ∈ W such that u(ϕ(x))y ∈ H implies that v(x)y ∈ G1, for allx ∈ X and y ∈ E. Now we claim that Cϕ(N(u,H)) ⊆ N(v,G). [Letf ∈ N(u,H). Then since u(ϕ(x))f(ϕ(x)) ∈ H for all x ∈ X , it followsthat v(x))f(ϕ(x)) ∈ G for all x ∈ X . That is, Cϕf ∈ N(v,G). Thisproves the continuity of Cϕ at the origin in CVb(X,E) and hence Cϕ iscontinuous on CVb(X,E).

To complete the proof of condition (b), it remains to show thatCVo(X,E) is invariant under Cϕ. [Fix f ∈ CVo(X,E). Let v ∈ Vand G ∈ W. Consider the set S = x ∈ X : v(x)f(ϕ(x)) /∈ G. Choosea balanced G1 ∈ W with G1 + G1 ⊆ G. According to condition (c)(ii),there exist u ∈ V and H ∈ W such that

u(ϕ(x))y ∈ H implies that v(x)y ∈ G1 for all x ∈ X, y ∈ E.

Let K = x ∈ X : u(x)f(x) /∈ H. Then K is a compact subset of Xand ϕ(S) ⊆ K. Let λ = supu(x) : x ∈ K. Then obviously λ > 0.Since f(ϕ(S)) is precompact in E, there exists a finite set f(ϕ(xi))ni=1


in f(ϕ(S)) such that

f(ϕ(S)) ⊆n⋃

i=1

f(ϕ(xi)) +1

2λH.

In view of condition (c)(iii), it follows that for each i ∈ 1, 2, ..., n, theset

Ai = ϕ−1(K) ∩ x ∈ X : v(x)f(ϕ(xi)) /∈ G1

is compact. If we put A = ∪ni=1Ai, then A is a compact subset ofX . Now, to show that S ⊆ A, let y ∈ S. Then there exists j ∈1, 2, ..., n such that f(ϕ(y)) − f(ϕ(xj)) ∈

12λH . Further, it implies

that u(ϕ(y))[f(ϕ(y))− f(ϕ(xi))] ∈12H . By hypothesis, it follows that

v(y)[f(ϕ(y))− f(ϕ(xj))] ∈ G1.

Now, if v(y)(f(ϕ(xi)) ∈ G1,

v(y)f(ϕ(y)) = v(y)[f(ϕ(y))− f(ϕ(xj))] + v(y)(f(ϕ(xi)) ∈ G1 +G1 ⊆ G,

a contradiction (since y ∈ S). Hence v(y)(f(ϕ(xi)) /∈ G1,showing thaty ∈ A. Thus S, being a closed subset of A, is compact. This proves thatCϕf ∈ CVo(X,E).]

(b)⇒ (a). This is clear.

Remark 9.2.5. If ϕ : X → X is a surjective homeomorphism, thencondition (c)(iii) is automatically satisfied and such a function will there-fore induce a composition operator on CVo(X,E) as soon as condition(c)(ii) is satisfied.

3. COMPACT WEIGHTED COMPOSITION OPERATORS 159

3. Compact Weighted Composition Operators

This section contains characterization of compact weighted compo-sition operators and, in particular, of precompact, equicontinuous andbounded multiplication operators on the weighted space CVo(X,E), whereE is a non-trivial TVS.

As before, it is assumed that the Nachbin family V satisfies (∗) V > 0and (∗∗) CVo(X) does not vanish on X.

Theorem 9.3.1. [MS98] Let X be a locally compact space andE a TVS. Let V ⊆ S+

o (X) and π : X → CLu(E) and ϕ : X → Xbe mappings with ϕ continuous, and suppose that Wπ,ϕ is a weightedcomposition operator on CVo(X,E) Then the following conditions areequivalent:

(a)Wπ,ϕ is a compact operator on CVo(X,E).(b)(1) π : X → CLu(E) is continuous.(2) For every x ∈ X, πx := π(x) is a compact operator on E.(3) ϕ is locally constant on X\z(π), where z(π) = x ∈ X : π(x) =

0.Proof. (a)⇒ (b)) Suppose Wπ,ϕ is a compact operator on CVo(X,E).

Then conditions (b)(1)-(3) hold, as follows.(b) (1) Let xo ∈ X, B ∈ b(E), G ∈ W. Since X is locally compact,

choose a neighborhood Ko of ϕ(xo) such that cl-Ko is compact. Choosef ∈ CVo(X) such that f(cl-Ko) = 1. Let Do = ϕ−1(Ko). For eachy ∈ B, define gy = f ⊗ y. Then the set M = gy : y ∈ B is boundedin CVo(X,E). Further, it implies that the set Wπ,ϕ(M) is relativelycompact in CVo(X,E). Now, by the Arzela-Ascoli Theorem 3.1.4, theset Wπ,ϕ(M) is equicontinuous at xo. Thus there exists a neighborhoodD1 of xo such that

πx(gy(ϕ(x)))− πxo(gy(ϕ(xo))) ∈ G for all x ∈ D1, y ∈ B.

Let D = Do ∩D1. Then we have

πx(y)− πxo(y) ∈ G for all x ∈ D, y ∈ B.

That is,

πx − πxo ∈ N(B,G) for all x ∈ D.

This proves that π : X → CLu(E) is continuous at xo and hence on X .(b)(2) Let xo ∈ X, B ∈ b(E). We need to show that πxo(B) is rel-

atively compact in E. Choose f ∈ CVo(X) such that f(ϕ(xo) = 1. Foreach y ∈ B, define gy = f ⊗ y. Let F = gy : y ∈ B. Then it followsthat Wπ,ϕ(F ) is relatively compact in CVo(X,E). So, by Theorem 3.1.4,


the set Wπ,ϕ(F )(xo) is relatively compact in E. But

Wπ,ϕ(F )(xo) = πxo(gy(ϕ(xo))) : y ∈ B =

= πxo(f((ϕ(xo)y) : y ∈ B = πxo(B);

hence πxo(B) is relatively compact in E. This proves that πxo is a com-pact operator on E.

(b)(3) Suppose ϕ is not locally constant on X\z(π). Then there existsxo ∈ X\z(π) such that ϕ is not locally constant on any open neighbor-hood D of xo. Let B be any open base at xo. The set B can be ordered bythe relation D1 ≤ D2 iff D1 ⊆ D2. Thus there exists a net xD : D ∈ Bin X such that xD → xo such that ϕ(xD) 6= ϕ(xo). For each D ∈ B, wechoose fD ∈ Cb(X) such that fD(ϕ(xo) = 1 and fD(ϕ(xD) = 0. Againchoose h ∈ CVo(X) such that h(ϕ(xo) = 1. Let y ∈ E be such thatπxo(y) 6= 0. Then there exists H ∈ W such that πxo(y) /∈ H. For eachD ∈ B, define gD = (hfD)⊗y. Since CVo(X) is a Cb(X)-module, it easilyfollows that the set M = gD : D ∈ B is bounded in CVo(X,E) andconsequently the setWπ,ϕ(M) is relatively compact in CVo(X,E). Again,by Theorem 3.1.4, the set Wπ,ϕ(M) is equicontinuous at xo. Thus, for agiven net xD → xo and H ∈ W, we have

supρH [πxD(gD(ϕ(xD)))− πxo(gD(ϕ(xo)))] : D ∈ B → 0.

That is, ρH(πxo(y)) → 0, which is impossible because πxo(y) /∈ H. Thisproves that ϕ is locally constant on X\z(π).

(b)⇒ (a)

Suppose that the conditions (b)(1)-(3) hold. Let F ⊆ CVo(X,E) bea bounded set. We need to show that Wπ,ϕ(F ) is compact in CVo(X,E).It is enough to show that the set cl-Wπ,ϕ(F ) satisfies conditions (i) -(iii)of the Arzela-Ascoli Theorem 3.1.4.

(i) Clearly, cl-Wπ,ϕ(F ) is β-closed. Since the closure of a boundedset is bounded, it follows that cl-Wπ,ϕ(F ) is β-bounded.

(ii) Let xo ∈ X and B = f(ϕ(xo)) : f ∈ F. Then clearly the set Bis bounded in E. Since πxo is a compact operator, it readily follows that

cl-Wπ,ϕ(F )(xo) = cl-πxo(f(ϕ(xo))) : f ∈ F = cl-πxo(B),

which is relatively compact in E.(iii) To show that cl-Wπ,ϕ(F ) is equicontinuous on each compact sub-

set of X , it suffices that Wπ,ϕ(F ) is equicontinuous on each compactsubset of X . Suppose K is any compact subset of X. Let xo ∈ K andG ∈ W. Consider the set S = f(ϕ(x)) : x ∈ X, f ∈ F. Then Sis bounded in E. In case xo ∈ z(π), it is easy to see that Wπ,ϕ(F ) isequicontinuous at xo. Now suppose xo /∈ z(π). Then, by condition (3),


there exists an open neighborhood D1(xo) of xo such that ϕ is constanton D1(xo). Since π : X → CLu(E) is continuous, there exists an openneighborhood D2(xo) of xo such that

πx − πxo ∈ N(S,G) for all x ∈ D2(xo).

Let D = D1(xo) ∩ D2(xo). Then D is a neighborhood of xo in K suchthat

πx(f(ϕ(x)))− πxo(f(ϕ(xo))) ∈ G for all x ∈ D and f ∈ F.

So Wπ,ϕ(F ) is equicontinuous at xo and hence on K.

Theorem 9.3.2. [MS98] Let X be a locally compact space and Ea Hausdorff topological algebra with hypocontinuous multiplication con-taining an identity e. Let V ⊆ S+

o (X) and π : X → CLu(E) andϕ : X → X be mappings with ϕ continuous, and suppose that Wπ,ϕ

is a weighted composition operator on CVo(X,E). Then the followingconditions are equivalent:

(a) Wπ,ϕ is a compact weighted composition operator on CVo(X,E).(b)(1) π : X → CLu(E) is continuous;(2) For every x ∈ X, πx := π(x) is a compact operator on E;(3) ϕ is locally constant on X\z(π).

Proof. The proof can be obtained by slight modification in the proofof Theorem 9.2.2 and Theorem 9.3.1.

Corollary 9.3.3. [MS98] Let X be a locally compact space and Ea an infinite dimensional TVS, and let V ⊆ S+

o (X). Then there is nocompact composition operator on CVo(X,E).

Proof. We first note if π : X → CLu(E) is the map given by π(x) = Ifor all x ∈ X, then the weighted composition operator Wπ,ϕ becomes thecomposition operator Cϕ. Since E is an infinite dimensional TVS, theidentity operator I on E is not compact. Consequently, the condition(b) of Theorem 9.3.2 cannot hold.

Remarks. In Theorems 9.3.1 and 9.3.2, if we take U as a systemof weights on X such that U ⊆ V, then the results hold for the spaceCUo(X,E).

(ii) Theorems 9.3.1 and 9.3.2 make sure that there are non-zero com-pact weighted composition operator on these spaces of continuous func-tions, where as it is not the case with Lp-spaces. However, there need notbe non-zero compact weighted composition operator on Lp-spaces (withnon-atomic measures) [Tak92].

Example. Let X = N with the discrete topology, and let E = Cb(R),Let π : N → CL(E) be defined by π(n) = 1

nAn for all n ∈ N, where


An : E → E is defined by Anf(t) = f(n) for all f ∈ E and t ∈ R.Clearly, π(n) is a compact operator on R. Let ϕ : N → N be definedby ϕ(n) = n2, n ∈ N. Let ω : N → R+ be defined by ω(n) = 1

n, n ∈ N.

Consider V = λω : λ ≥ 0. Now, it is easy to see that the mappings πand ϕ satisfy all the conditions of Theorem 9.3.1 and therefore Wπ,ϕ is acompact weighted composition operator on CVo(X,E).

We next consider the particular case of multiplication operators butcharacterize more general notions of equicontinuous, precompact andbounded multiplication operator Mπ on CVo(X,E) and CVb(X,E). Re-call that a linear map T : CVb(X,E) −→ CVb(X,E) is said to becompact (resp. precompact, equicontinuous, bounded) if it maps some0neighborhood in CVb(X,E) into a compact (resp. precompact, equicon-tinuous, bounded) subset of CVb(X,E).

Lemma 9.3.4. [Oub02, AAK09] For any v ∈ V , G ∈ W and x ∈ X,

1

v(x)= supρG(f(x)) : f ∈ N(v,G)

=supρG(f(x)) : f ∈ CVb(X,E) with ‖f‖v,G ≤ 1. (1)

Proof. Let x ∈ X , v ∈ V , G ∈ W. There exist f ∈ CVb(X,E) andH ∈ W such that (ρH f)(x) = 1. Choose a ∈ E with ρG(a) = 1.

Case I. Suppose v(x) = 0. For each n ≥ 1, set

Un := y ∈ X : v(y) <1

nand 1−

1

n< ρH(f(y)) < 1 +

1

n;

and consider hn ∈ Cb(X) with

0 ≤ hn ≤ n, hn(x) = n, and supp(hn) ⊆ Un.

Clearly, the function gn := nn+1

hn ρH f ⊗ a ∈ CVb(X,E) and

‖gn‖v,G = supv(y)n

n+ 1hn(y)ρH(f(y))ρG(a) : y ∈ X

= supv(y)n

n+ 1hn(y)ρH(f(y))ρG(a) : y ∈ Un

<1

n·

n

n+ 1· n · (1 +

1

n) · 1 = 1,

hence,

supρG(f(x)) : f ∈ CVb(X,E), ‖f‖v,G ≤ 1 ≥ supρG(gn(x)) : n ∈ N

= supn

n+ 1hn(x)ρH(f(x))ρG(a) : n ∈ N

= supn

n+ 1· n · 1 · 1 : n ∈ N =∞ =

1

v(x).


Case II. Suppose v(x) 6= 0. For n > 1v(x)

, let

Fn : = y ∈ X :v(x)

v(x) + 12n

≤ ρH(f(y)) ≤v(x)

v(x)− 12n

,

Un : = y ∈ X :v(x)

v(x) + 1n

< ρH(f(y)) <v(x)

v(x)− 1n

.

Then choose hn, kn ∈ Cb(X) with

0 ≤ hn ≤1

v(x) + 1n

, hn(x) =1

v(x) + 1n

on Fn, and supp(hn) ⊆ Un,

0 ≤ kn ≤ 1, kn(x) = 1, and

supp(kn) ⊆ Jn := Fn ∩ y ∈ X : v(y) < v(x) +1

n.

The function gn :=v(x)− 1

2n

v(x)hnknρH f ⊗ a ∈ CVb(X,E) and

‖gn‖v,G = supv(y)ρG(gn(y)) : y ∈ X

= supv(y)v(x)− 1

2n

v(x)hn(y)kn(y)ρH(f(y))ρG(a) : y ∈ X

= supv(y)v(x)− 1

2n

v(x)hn(y)kn(y)ρH(f(y))ρG(a) : y ∈ Jn

< (v(x) +1

2n)v(x)− 1

2n

v(x)·

1

v(x) + 12n

· 1v(x)

v(x)− 12n

= 1;

hence

supρG(f(x)) : f ∈ CVb(X,E), ‖f‖v,G ≤ 1

≥ supρG(gn(x)) : n ∈ N

= supv(x)− 1

2n

v(x)·

1

v(x)− 12n

· 1 · 1 · 1 : n ∈ N =1

v(x).

On the other hand,

supρG(f(x)) : f ∈ CVb(X,E), ‖f‖v,G ≤ 1

=1

v(x)supv(x)ρG(f(x)) : f ∈ CVb(X,E), ‖f‖v,G ≤ 1

≤1

v(x)supsup

y∈Xv(y)ρG(f(y)) : f ∈ CVb(X,E), ‖f‖v,G ≤ 1


=1

v(x)sup‖f‖v,G : f ∈ CVb(X,E), ‖f‖v,G ≤ 1 =

1

v(x).

Thus, in each case, (1) holds.

Recall that if A ⊆ X, then a point x ∈ A is called an isolated pointof A if x is not a limit point of A, i.e. if there exists a neighborhood U ofx in X such that U ∩A = x. Consequently, if X has no isolated point,then, for each x ∈ X,

Theorem 9.3.5. [Oub02, AAK09] Let π : X −→ CL(E) be amap such that Mπ(CVb(X,E)) ⊆ C(X,E) and supposeX has no isolatedpoints. If Mπ is equicontinuous on CVb(X,E), then Mπ = 0.

Proof. Suppose Mπ is equicontinuous on CVb(X,E) but Mπ(fo) 6= 0for some fo ∈ CVb(X,E). Then there exists xo ∈ X with πxo(fo(xo)) 6= 0.SinceMπ is equicontinuous, there exist some v ∈ V and balanced G ∈ Wsuch that Mπ(N(v,G)) is equicontinuous on X and in particular at xo.We may assume that fo ∈ N(v,G) (since N(v,G) is absorbing). Hence,for every balanced H ∈ W, there exists a neighborhood D of xo in Xsuch that

πy(f(y))− πxo(f(xo)) ∈ H for all y ∈ D and f ∈ N(v,G).

Since xo is not isolated, there exists some y ∈ D with y 6= xo. Choosethen gy ∈ Cb(X) with 0 ≤ gy ≤ 1, gy(y) = 0, and gy(xo) = 1. Thengyfo ∈ N(v,G) and so

πy(gy(y)fo(y))− πxo(gy(xo)fo(xo)) ∈ H ;

that is, πxo(fo(xo)) ∈ H . Since H ∈ W is arbitrary and E is Haus-dorff (i.e. ∩

U∈WU = 0), we have πxo(fo(xo)) = 0. This is the desired

contradiction.

Corollary 9.3.6. [Oub02, AAK09] Let π : X −→ CL(E) be amap such that Mπ(CVb(X,E)) ⊆ C(X,E). If X is a VR-space withoutisolated points, then Mπ is precompact iff Mπ = 0.

Proof. Suppose Mπ is precompact. Then, by Theorem 3.2.1, Mπ isequicontinuous. Hence, by Theorem 9.3.5, Mπ = 0. The converse istrivial.

Theorem 9.3.7. [Oub02, AAK09] Let π : X −→ CL(E) be suchthat Mπ(CVb(X,E)) ⊆ C(X,E). Then Mπ is a bounded multiplicationoperator on CVb(X,E) iff there exist v ∈ V and G ∈ W such that forany u ∈ V, H ∈ W, there exists λ > 0 such that

λu(x)a ∈ G implies that u(x)πx(a) ∈ H, x ∈ X, a ∈ E,


or equivalently

u(x)ρH(πx(a)) ≤ λv(x)ρG(a), x ∈ X, a ∈ E. (A)

Proof. Similar to that of Theorem 9.1.1.



Section 9.1. Singh and Summers [SS88] have studied the notion ofcomposition operators on CVo(X,C). Later, Singh and Manhas [SM91,SM92] made an analogous study of multiplication operatorsMθ, Mψ andMπ from CVb(X,E) (resp. CVo(X,E)) into itself, assuming E a locallyconvex space or a locally m-convex algebra. These operators are inducedby certain maps θ : X → C and ψ : X → E and π : X → CL(E).Thissection contains results from the papers [KT97, KT02, MS98] which gen-eralize the results of above authors to the case when E is a TVS ortopological algebra which is not necessarily locally convex. (See [SK97,Oub05]).

Section 9.2. This section contains characterizations of the more gen-eral notion of weighted composition operatorsWπ,ϕ from CVb(X,E) (resp.CVo(X,E)) into itself, due to Manhas and Singh [MS98]. Here ϕ : X →X and π : X → CL(E).

Section 9.3. A systematic study of compact weighted compositionoperators on spaces of continuous functions was initiated by Kamowitz[Kam81] on the Banach algebra C(X) for a compact Hausdorff spaceX . This study has been further extended to function algebra settingand Banach lattices by Feldman [Fel90], Singh and Summers [SS87] andTakagi [Tak88]. Jamison and Rajagopalan [JR88] and Takagi [Tak91]have further generalized these results to the spaces C(X,E). Finally, in[MS97] Manhas and Singh have obtained a study of compact and weaklycompact weighted composition operators on the weighted locally convexspaces CVo(X,E), where V ⊆ S+

o (X) is a Nachbin family on X and Eis a quasi-complete locally convex space. This section contains charac-terization of compact Weighted Composition Operators Wπ,ϕ and also ofprecompact, equicontinuos and bounded multiplication operators on thenon-locally convex weighted spaces CVo(X,E), as given in [MS98, Oub02,AAK09] for E a non-trivial TVS. More general results for operatorsWπ,ϕ

from CVb(X,E) into CUb(Y,E) can be found in [Oub05]; here X and Yare completely regular spaces and ϕ : Y → X and π : Y → CL(E) arethe corresponding maps maps.

CHAPTER 10

The General Strict Topology

The general strict topology was defined by Busby in [Bus68] in thecase of Y a C∗-algebra and A a closed two-sided ideal in Y as the onegiven by the seminorms x → max‖ax‖, ‖xa‖ for a ∈ A, x ∈ Y. In-dependently, this topology was studied in greater detail and in a moregeneral setting of Y a Banach left A-module by Sentilles and Taylor in aninteresting paper [ST69] as the one given by the seminorms x → ‖a.x‖for a ∈ A, x ∈ Y . More recently, Khan-Mohammad-Thaheem [KMT05]have considered the case of Y a locally idempotent F -algebra which con-tains A as a closed two-sided ideal, while Shantha [Shan04, Shan05] hastaken Y a locally convex module over a locally convex algebra A.

In this chapter, we begin with the study of general strict and re-lated topologies on a topological left A-module Y, where Y is HausdorffTVS and A a Hausdorff topological algebra having a two-sided boundedapproximate identity. These results were originally given in the papers[ST69, Shan04, KMT05, SumW72] in particular cases and extended re-cently in [Kh07, Kh08]. We shall also consider essentiality of topologi-cal left A-modules (Cb(S,E), u) and (C(S,E), k). Finally, we study theproperties of the topological module HomA(A, Y ) of continuous homo-morphisms.

167

168 10. THE GENERAL STRICT TOPOLOGY

1. Strict Topology on Topological Modules

Recall that if Y is a TVS and A be a topological algebra, both overthe same field K (= R or C), then Y is called a topological left A-moduleif it is a left A-module and the module multiplication (a, x)→ a.x fromA × Y into Y is separately continuous. If b(A) (resp. b(Y )) denote thecollection of all bounded sets in A (resp. Y ), then module multiplication(a, x) → ax is called b(A)- (resp. b(Y )-) hypocontinuous if, given anyneighborhood G of 0 in Y and any D ∈ b(A) (resp. B ∈ b(Y )), thereexists a neighborhood H of 0 in Y (resp. V of 0 in A) such that D.H ⊆ G(resp. V.B ⊆ G).Wemention that if (Y, τ) is an ultrabarrelled topologicalleft A-module, then the module multiplication is b(A)-hypocontinuous.(See section A.6).

Definition. Let A be a topological algebra and (Y, τ) a TVS whichis a topological left A-module, and let W be a base of τneighborhoodsof 0 in Y . For any bounded set D ⊆ A and G ∈ W, we set

M(D,G) = x ∈ Y : D.x ⊆ G.

The uniform topology u = uA (resp. strict topology β = βA) on Y isdefined as the linear topology which has a base of neighborhoods of 0consisting of all sets of the form M(D,G), where D is a bounded (resp.finite) subset of A and G ∈ W. If A has a two-sided approximate identityeλ : λ ∈ I, the compact-open topology k on Y is defined just as abovewith D being a finite subset of eλ : λ ∈ I.

As mentioned earlier, β was introduced by Sentilles and Taylor ([ST69],p.145) in the case of Y a left Banach A-module as the one given by theseminorms x→ ‖a.x‖ for a ∈ A, x ∈ Y .

As a classical example, if A = Co(S) and Y = Cb(S,E) with S alocally compact space and E a TVS and Y endowed with τ = u, theuniform topology and A endowed with the sup-norm topology and hasan approximate identity. In this particular case, β is the strict topologyand k is the topology of uniform convergence on compact subsets of S.Another instance of the above is that A is a topological algebra andY = CMd(A), the set of all double multipliers (S, T ), where A is atopological algebra and S, T : A → A are continuous linear mappingssatisfying aS(b) = T (a)b for all a, b ∈ A [Jo64, KMT99]. We shallconsider more cases of essential topological left A-modules (Cb(S,E), u)and (C(S,E), k) in the later part of this section.

In the sequel, we shall assume, unless stated otherwise, that Y is atopological left A-module, where Y = (Y, τ) is a Hausdorff TVS with

1. STRICT TOPOLOGY ON TOPOLOGICAL MODULES 169

W a base of neighborhoods of 0 and A a Hausdorff topological algebrawith eλ : λ ∈ I a bounded two-sided approximate identity. We shallalso assume that A is faithful in Y in the sense that, for any x 6= 0 inY , a.x = 0 for all a ∈ A implies that x = 0 (cf. [Jo64, ST69]). Insome results, the assumption of ultrabarrelledness on A or Y is requiredessentially to apply the uniform boundedness principle.

Theorem 10.1.1. [Kh07] Let (Y, τ) be a topological left A-modulewith A having a two-sided approximate identity eλ : λ ∈ I.Then:

(a) k 6 β 6 u.(b) k, β, and u are Hausdorff.

(c) For each x ∈ Y , eλ.xβ−→ x.

(d) ∪λ∈Ieλ.Y is β-dense in Y ; in particular, Ye is β-dense in Y.

If, in addition the module multiplication is b(A)-hypocontinuous, then:(e) u 6 τ.(f) (Y, k), (Y, β) and (Y, u) are topological A-modules.

Proof. (a) Since eλ : λ ∈ I ⊆ A, it follows that k 6 β. Since everyfinite set in A is bounded, we have β 6 u.

(b) We only need to show that k is Hausdorff. Let x ∈ ∩λ∈J

∩G∈W

M(eλ, G). Since τ is Hausdorff, ∩G∈W

G = 0 and so eλ.x = 0 for all

λ ∈ I. Then, for any a ∈ A, aeλ → a in A and so, by separate continuityof module multiplication,

a.x = limλ(aeλ).x = lim

λa.(eλ.x) = 0.

Since A is faithful in Y , we have x = 0. Thus (Y, k) is Hausdorff.(c) Let x ∈ Y , and let D be a finite set in A and G ∈ W. Since

aeλ → a in A for all a ∈ D and D is finite, by separate continuity ofmodule multiplication, there exists λo ∈ I such that for all λ ≥ λo anda ∈ D

(aeλ − a).x ∈ G, or a.(eλ.x− x) ∈ G.

Hence eλ.x− x ∈M(D,G) for all λ ≥ λo. So eλ.xβ−→ x.

(d) Let x ∈ Y . Since eλ.x : λ ∈ I ⊆ ∪λ∈Ieλ.Y and, by (c), eλ.x

β−→ x,

we have x ∈ β-cl (∪λ∈I

eλY ). Thus ∪λ∈Ieλ.Y is β-dense in Y .

(e) We need to verify that, for any net xα : α ∈ J in Y , xατ−→ 0

implies that xαu−→ 0. Let D be a bounded set in A and G ∈ W. By

b(A)-hypocontinuity, choose balanced H ∈ W such that D.H ⊆ G. Since


D is bounded, there exists r > 1 such that D ⊆ rV . Since xατ−→ 0,

there exists αo ∈ J such that

xα ∈ H for all α ≥ αo.

Then, for any a ∈ D and α ≥ αo,

a.xα ∈ D.H ⊆ G;

that is, xα ∈M(D,G) for all α ≥ αo. Hence xαu−→ 0, and so u 6 τ .

(f) We prove the result only for (Y, u) as the proofs for (Y, β) and(Y, k) are similar to it. First, let bα : α ∈ J ⊆ A with bα → 0 and fix

x( 6= 0) in Y . To show that bαxu−→ 0 in Y , let D be a bounded subset of

A and G ∈ W. By b(A)-hypocontinuity, choose balanced H ∈ W suchthat D.H ⊆ G. Since bα → 0 in A, by separate continuity of modulemultiplication, bα.x

τ−→ 0 in Y and so there exists αo ∈ J such that

bα.x ∈ H for all α ≥ αo.

Hence, for any a ∈ D and α ≥ αo, abα.x ∈ D.H ⊆ G; that is, bα.x ∈M(D,G), for all α ≥ αo. Thus bα.x

u−→ 0.

Next, let xα : α ∈ J ⊆ Y with xαu−→ 0 and fix a( 6= 0) in A. Let

D be a bounded set in A and G ∈ W. Since the map Ra : A→ A givenby Ra(b) = ba, b ∈ A, is linear and continuous (by separate continuity of

multiplication in A), it follows that Da is bounded in A. Since xαu−→ 0,

there exists αo ∈ I such that for all α ≥ αo

xα ∈M(Da,G), or a.xα ∈ M(D,G).

Hence a.xαu−→ 0 in Y . Consequently, (Y, u) is a left semitopological

A-module.

Theorem 10.1.2. [Kh07] Let (Y, τ) be a topological left A-modulewith b(Y )-hypocontinuous module multiplication and A having a two-sided approximate identity eλ : λ ∈ I.Then:

(p) k = β on τ -bounded sets.(q) If a sequence xn ⊆ Y is τ -bounded and k-convergent, then it is

β-convergent.

Proof. (p) Let S be a τ -bounded in Y , and let xα : α ∈ J be a net

in S with xαk−→ x ∈ S. Let D be a finite set in A and G ∈ W. Choose

balanced H ∈ W with H +H +H ⊆ G. By b(Y )-hypocontinuity, thereexists a neighborhood V of 0 in A such that V.S ⊆ H. Since D is finite,there exists λo ∈ I such that


aeλo − a ∈ V for all a ∈ D.

Since xαk→ xo, eλo .xα

τ→ eλo .xo and so, for each a ∈ D, a.(eλo .xα)

τ→

a.(eλo .xo). Since D is finite, choose αo ∈ J such that

a.(eλo .xα − eλo .xo) ∈ H for all α ≥ αo and a ∈ D.

Then, for any a ∈ D and α ≥ αo,

a.(xα − x) = (a− aeλo).xα + aeλo .(xα − x)] + (aeλo − a).x

∈ V.S +H + V.S ⊆ H +H +H ⊆ G;

that is, xα − x ∈M(D,G) for all α ≥ αo. Hence xαβ−→ x, and so β 6 k

on S. Thus k = β on S.

(q) Suppose that xn is τ -bounded and xnk−→ x ∈ Y . By (1), k = β

on the τ -bounded set x, xn : n ≥ 1; hence xnβ−→ x.

Lemma 10.1.3. [Kh07] Let Y be a topological left A-module with Aultrabarralled. Let S be a β-bounded set in Y . Then, for any boundedset D ⊆ A, D.S is τ -bounded in Y .

Proof. For any x ∈ Y , define Rx : A → Y by Rx(a) = a.x, a ∈ A.Clearly, each Rx is linear and also continuous (by separate continuity ofmodule multiplication). We next show that Rx : x ∈ S is pointwisebounded in CL(A, Y ). Let a ∈ A and G ∈ W. Since S in β-bounded,there exists r > 0 such that S ⊆ rM(a,G). So

Rx(a) : x ∈ S = a.x : x ∈ S ⊆ rG,

showing that Rx : x ∈ S is pointwise bounded. Since A is ultra-barralled, by the principle of uniform boundedness, Rx : x ∈ S isequicontinuous and hence uniformly bounded in CL(A, Y ). Since D isbounded in A, D.S = ∪

x∈SRx(D) is τ -bounded in Y .

Theorem 10.1.4. [Kh07] Let (Y, τ) be a topological left A-modulewith A ultrabarralled. Then:

(i) β and u have the same bounded sets in Y .(ii) If (Y, β) is ultrabornological, then β = u on Y .If, in addition, A has a two-sided approximate identity eλ : λ ∈ I,

then:(iii) k = β on u-bounded sets.(iv) A sequence xn ⊆ Y is β-convergent iff it is u-bounded and

k-convergent.


Proof. (i) Since β 6 u, every u-bounded set is β-bounded. Now, letS be any β-bounded set in Y . By Lemma 10.1.3, for any bounded setD ⊆ A, D.S is τ -bounded in Y or, equivalently, S is u-bounded.

(ii) By (i), the identity map i : (Y, β) → (Y, u) takes bounded setsinto bounded sets. Since Y is ultrabornological, i is continuous. Henceu 6 β.

(iii) Let S be a u-bounded set in Y , and let xα : α ∈ J ⊆ S with

xαk→ xo ∈ S. Let D be a finite set in A and G ∈ W. Choose a balanced

H ∈ W with H + H + H ⊆ G. As in the proof of Lemma 10.1.3, bythe principle of uniform boundedness, Rx : x ∈ S is equicontinuousin CL(A, Y ). Hence there exists a neighborhood V of 0 in A such thatV.S ⊆ H. Choose λo ∈ I such that

aeλo − a ∈ V for all a ∈ D.

Since xαk→ xo, eλo .xα

τ→ eλo .xo and so by separate continuity, for each

a ∈ D,

aeλo .xατ→ aeλo .xo.

Since D is finite, choose αo ∈ J such that

aeλo .(xα − xo) ∈ H for all α ≥ αo and a ∈ D.

Hence, for any a ∈ D and α ≥ αo,

a.(xα − xo) = (a− aeλo).xα + aeλo .(xα − xo) + (aeλo − a).xo

∈ V.S +H + V.S ⊆ H +H +H ⊆ G.

So xαβ→ xo and hence β 6 k on S.

(iv) Let xn ⊆ Y with xnβ−→ x ∈ Y . Then it is β-bounded and

hence, by part (i), u-bounded. Further, since k 6 β, xnk−→ x. Con-

versely, suppose that xn is u-bounded and xnk−→ x ∈ Y . By (iii),

k = β on the u-bounded set x, xn : n ≥ 1; hence xnβ−→ x.

The next result characterizes the equivalence of bounded sets in thetopologies β, u, and τ on Y .

Theorem 10.1.5. [Kh07] Let (Y, τ) be a topological left A-modulewith A ultrabarralled. Then the following are equivalent:

(a) u and τ have the same bounded sets in Y .(b) β and τ have the same bounded sets in Y .If the module multiplication is either b(A)-hypocontinuous or b(Y )

hypocontinuous, then each of (a) and (b) is equivalent to:(c) The β-closure of a τ -bounded set is τ -bounded.


Proof. (a) ⇔ (b) since β and u have the same bounded sets byTheorem 10.1.4(i).

(b)⇒ (c). Let S be a τ -bounded set in Y , and let D = eλ : λ ∈ I.Then D.S is τ -bounded in Y (cf. [Mal86], p. 28). [Indeed, let G ∈ W.By b(Y )-hypocontinuity, there exists a neighborhood V of 0 in A suchthat V.S ⊆ G. Since D is a bounded set in A, there exists r > 0 such thatD ⊆ rV. Hence D.S ⊆ rV.S ⊆ rG.] Further, by b(A)-hypocontinuity andTheorem 10.1.1((a), (e)), β 6 τ and so D.S and hence β-cl(D.S) is β-

bounded. By (b), β-cl(D.S) is τ -bounded. Now, for any x ∈ S, eλ.xβ→ x

(by Theorem 10.1.1(c)); since each eλ.x ∈ D.S, we have x ∈ β-cl(D.S).Hence S ⊆ β-cl(D.S) and so β-cl(S) is τ -bounded.

(c) ⇒ (a). Since the module multiplication is b(A)-hypocontinuous,by Theorem 10.1.1(e), u 6 τ and so every τ -bounded set is u-bounded.Now, let S be a u-bounded set in Y . Then, if D = eλ : λ ∈ I,D.S is τ -bounded. [Indeed, let G ∈ W. Since D is a bounded set in A,M(D,G) is a uneighborhood of 0 in Y and so there exists r > 0 such thatS ⊆ rM(D,G) = M(D, rG); that is, D.S ⊆ rG.] By (c), β-cl(D.S) isτ -bounded. But, again by Theorem 10.1.1(c), S ⊆ β-cl(D.S); hence S isτ -bounded.

We now consider a characterization of the equivalence of the topolo-gies β, u, and τ on Y .

Theorem 10.1.6. [Kh07] Let (Y, τ) be a topological left A-modulewith Y an F -space and A an F -algebra. Consider the following condi-tions:

(a) There exists a ∈ A such that the map x→ a.x is an isomorphismof Y onto Y .

(b) β = τ on Y .(c) (Y, β) is metrizable.(d) (Y, β) is ultrabornological.(e) β = u on Y .

(f) For each x ∈ Y, eλ.xu→ x.

Then (a)⇒ (b)⇒ (c)⇒ (d)⇒ (e); if A has a two-sided approximateidentity, (e) ⇒ (f). If (Y, τ) is a Banach algebra and A has a boundedtwo-sided approximate identity, then (f) ⇒ (a); that is, (a) − (f) areequivalent:

Proof. (a) ⇒ (b). Suppose (a) holds, and let xαβ→ x in Y . Now

φ : x → a.x is a continuous, linear and one-one map from (Y, τ) onto(Y, τ). As a consequence of the open mapping theorem, the inverse map


φ−1 : a.x → x is continuous. Since a.xατ→ a.x, we have φ−1(a.xα)

τ→

φ−1(a.x); that is, xατ→ x. So τ 6 β.

(b) ⇒ (c). This is obvious since (Y, τ) is metrizable.(c) ⇒ (d) since every metrizable TVS is ultrabornological.(d) ⇒ (e). This follows from Theorem 10.1.4(ii).(e) ⇒ (f). This follows from Theorem 10.1.1(c).Now suppose that (Y, τ) is a Banach algebra and A has a bounded

two-sided approximate identity. Then (f) ⇒ (a) follows from ([ST69],Theorem 2.4).

Remark. We can apply above results to extend an observation oninner derivation, due to Phillips [Ph95]. Let (Y, τ) be a topological al-gebra and A a closed two-sided ideal in Y, with Y not necessarily anF -algebra. A linear map D : A→ A is called a derivation on A if

D(ab) = D(a)b+ aD(b), a, b ∈ A.

For any x ∈ Y, the map δx : A → A defined by δx(a) = [x, a] = xa −ax, a ∈ A, is a derivation on A, called the inner derivation on A. Aderivation D : A→ A is called approximately inner if there exists a netaα : α ∈ J in A such that D(a) = limα δaα(a) for all a ∈ A. We claimthat, for any x ∈ Y, δx is approximately inner. Indeed, by Theorem 10.1.1(d), A is β-dense in Y and so there exists a net aα : α ∈ J in A such

that aαβ→ x. Then, for any a ∈ A,

δx(a) = limα(aαa− aaα) = lim

αδaα(a).

This was observed earlier in ([Ph95], p. 244) in the case of Y =Md(A).

We now consider the notion of essential modules, due to Rieffel [Rie67].

Definition. Let (Y, τ) be a topological left A-module with A havinga left approximate identity eλ. The essential part Ye of Y is defined as

Ye = x ∈ Y : eλ.xτ−→ x. We say that Y is essential if Y = Ye.

Clearly, Ye is a topological left A-submodule of Y. Note that, for anya ∈ A and x ∈ Y, eλ.(a.x) = (eλa).x

τ−→ a.x (by separate continuity of

module multiplication) and so a.x ∈ Ye; hence [A.Y ] ⊆ Ye, where [A.Y ] isthe vector subspace of Y spanned by A.Y = a.x :∈ A, x ∈ Y . Further,

for any x ∈ Ye, clearly eλ.x : λ ∈ I ⊆ A.Y and eλ.xτ−→ x and so

x ∈ τ -cl[A.Y ]. Thus [A.Y ] ⊆ Ye ⊆ τ -cl[A.Y ].


We can now state an analogue of factorization theorem 10.1.1 fortopological modules, again due to Ansari-Piri [AP90]. A topological leftA-module Y is called A-factorable if, for each x ∈ Y, there exist a ∈ Aand y ∈ Y such that x = a.y.

Theorem 10.1.7. [AP90] Let A be a fundamental F -algebra with auniformly bounded left approximate identity. If Y is an F -space whichis an essential topological left A-module, then Y is A-factorable.

We mention that if Y is A-factorable, then Y is essential since Y =A.Y ⊆ Ye ⊆ Y, or that Y = Ye. Note that the above is a so-called left-hand version of the factorization theorem. Clearly its right-hand versionalso holds.

We next consider some properties of the essential part Ye of Y

Theorem 10.1.8. [Kh07] Let (Y, τ) be a topological left A-modulewith Y ultrabarralled and A having a bounded left approximate identityeλ : λ ∈ I. Then:

(i) Ye is τ -closed in Y.(ii) Ye = τ

-cl[A.Y ].

Proof. (i) Let x ∈ τ -cl(Ye). We need to show that eλ.xτ−→ x. Let

G ∈ W. Choose a balanced H ∈ W such that H + H + H ⊆ G. Foreach λ ∈ I, define Lλ : Y → Y by Lλ(y) = Leλ(y) = eλ.y, y ∈ Y .Since D = eλ : λ ∈ I is bounded in A, it follows from the proofof Lemma 10.1.3 that Lλ : λ ∈ I is pointwise bounded and henceequicontinuous in CL(Y, Y ) by the principle of uniform boundedness.There exists H1 ∈ W such that Lλ(H1) ⊆ H for all λ ∈ I. Since x ∈ τ -cl(Ye), we can choose xo ∈ Ye such that

x− xo ∈ H1 ∩H.

Since eλ.xo → xo, choose λo ∈ I such that

eλ.xo − xo ∈ H for all λ ≥ λo.

Hence, for any λ ≥ λo,

eλ.x− x = eλ.(x− xo) + (eλ.xo − xo) + (xo − x)

∈ Lλ(H1 ∩H) +H +H1 ∩H ⊆ H +H +H ⊆ G;

that is, eλ.xτ−→ x and so x ∈ Ye. Hence Ye is τ -closed.

(ii) As seen above, [A.Y ] ⊆ Ye ⊆ τ -cl[A.Y ]. Hence, by (i),

Ye = τ


-cl[A.Y ].

Corollary 10.1.9. Let Y and A be as above. Then Y is essentialiff Y = [A.Y ]

τ.

Proof. By definition, Y is essential if Y = x ∈ Y : eλ.xτ−→ x.

Hence, by above theorem, Y is essential iff Y = [A.Y ]τ.

We mention that if A is a fundamental F -algebra with a uniformlybounded left approximate identity and Y is an F -space which is an essen-tial topological left A-module, then, by Theorem 10.1.7, Y is A-factorableand so Y = A.Y ⊆ Ye ⊆ Y, or thatY = A.Y = Ye.

Following Rieffel ([Rie67], p. 454), we may also consider a version ofthe Theorem 10.1.8 for the u-topology on Y , as follows.

Theorem 10.1.10. [Kh07] Let (Y, τ) be an essential topological leftA-module with b(A)-hypocontinuous module multiplication and A havinga bounded left approximate identity eλ : λ ∈ I. Then u-cl[A.Y ] = x ∈

Y : eλ.xu−→ x.

Proof. Let Y1 = x ∈ Y : eλ.xu−→ x. For any y ∈ Y1, each

eλ.y ∈ A.Y1 ⊆ A.Y and eλ.yu−→ y, and so clearly y ∈ u-cl[A.Y ]; hence

Y1 ⊆ u-cl[A.Y ]. Now, let x ∈ u-cl[A.Y ]. Then there exists a net xα :

α ∈ J in [A.Y ] such that xαu−→ x. To show that eλ.x

u−→ x, let D

be a bounded subset of A and G ∈ W. Choose balanced H ∈ W suchthat H + H + H ⊆ G. By b(A)-hypocontinuity, choose H1 ∈ W withD.H1 ⊆ H . Put D1 = D ∪ eλ : λ ∈ I, a bounded set in A. Since

xαu−→ x, choose αo ∈ J such that

xα − x ∈M(D1, H1 ∩H) for all α ≥ αo.

Since (Y, τ) is essential, eλ.xαo

τ−→ xαo and so there exists λo ∈ I such

thateλ.xαo − xαo ∈ H1 for all λ ≥ λo.

Hence, for any a ∈ D and λ ≥ λo,

a.(eλ.x− x) = aeλ.(x− xαo) + a.(eλ.xαo − xαo) + a.(xαo − x)

∈ a.(H1 ∩H) + a.H1 +H1 ∩H

⊆ H +H +H ⊆ G;

that is, eλ.xu−→ x, and so x ∈ Y1. Thus u-cl[A.Y ] = Y1.

2. ESSENTIALITY OF MODULES OF VECTOR-VALUED FUNCTIONS 177

2. Essentiality of Modules of Vector-valued Functions

We now present two results on essentiality of topological left A-modules of continuous vector-valued functions endowed with the uni-form and compact-open type topologies. Recall that if X be a Hausdorfftopological space and E a non-trivial Hausdorff TVS with WE a base ofneighborhoods of 0 in E, then the uniform topology u on Cb(X,E) is thelinear topology which has a base of neighborhoods of 0 consisting of allsets of the form

N(X,W ) = f ∈ Cb(X,E) : f(X) ⊆W,

where W varies over WE .

Theorem 10.2.1. [Kh07] Suppose A is a topological algebra with abounded approximate identity eλ : λ ∈ I and E a topological left A-modules such that the module multiplication is both b(A)-hypocontinuousand b(E)-hypocontinuous. Then:

(i) (Cb(X,E), u) and (Co(X,E), u) are topological left A-moduleswith respect to the module multiplication (a, f) → a.f as pointwise ac-tion:

(a.f)(x) = a.f(x), a ∈ A, f ∈ Cb(X,E), x ∈ X.

(ii) If X is locally compact, then (Co(X,E), u) is essential.Proof. (i) First, we need to show that if aα : α ∈ J ⊆ A with

aα → 0 and f ( 6= 0) ∈ Cb(X,E), then aα.fu→ 0 in Cb(X,E). Let

G ∈ WE . Since f(X) is bounded in E, by b(E)-hypocontinuity, thereexists a neighborhood V of 0 in A such that V.f(X) ⊆ G. Since aα → 0in A, there exists αo ∈ J such that aα ∈ U for all α ≥ αo. Hence

aα.f(X) ⊆ V.f(X) ⊆ G or aα.f ∈ N(X,G) for all α ≥ αo.

Next, let a ( 6= 0) ∈ A and fα : α ∈ J ⊆ Cb(X,E) with fαu→ 0

in Cb(X,E). Then a.fαu→ 0 in Cb(X,E), as follows. Let G ∈ WE . By

b(A)-hypocontinuity, there exists an H ∈ WE such that aH ⊆ G. Since

fαu→ 0 in Cb(X,E), there exists an αo ∈ J such that fα(X) ⊆ H for all

α ≥ αo. Hence

a.fα ∈ N(X, aH) ⊆ N(X,G) for all α ≥ αo.

(ii) Fix f ∈ Co(X,E). We need to show that eλ.fu→ f. Let G ∈ WE .

Choose a balanced H ∈ WE with H+H+H ⊆ G. Since D = eλ : λ ∈ Iis bounded in A, by b(A)-hypocontinuity, there exists V ∈ WE such that

D.V ⊆ H.


Since f ∈ Co(X,E), there exists a compact set K ⊆ X such that

f(x) ∈ V ∩H for all x ∈ X\K.

Since f(K) is compact, there exists a finite set z1, ..., zn ⊆ K such that

f(K) ⊆ ∪ni=1(f(zi) + V ∩H).

Since E is an essential A-module, there exists λo ∈ I such that

eλf(zi)− f(zi) ∈ H for all λ ≥ λo and i = 1, ..., n.

Now, let x ∈ X and λ ≥ λo. First suppose x ∈ K. Choose i ∈ 1, ..., nsuch that

f(x)− f(zi) ∈ V ∩H.

Then

eλf(x)− f(x) = eλ(f(x)− f(zi)) + (eλf(zi)− f(zi)) + (f(zi)− f(x))

∈ D.(V ∩H) +H − (V ∩H) ⊆ H +H +H ⊆ G.

If x ∈ X\K,

eλf(x)− f(x) ∈ eλ.(V ∩H)− V ∩H ⊆ eλ.V −H ⊆ H +H ⊆ G.

Definition. If X is a completely regular hemicompact space withKn an increasing sequence of compact sets which form a base for com-pact sets on X , we define the compact-open type topology k on C(X,E)as the linear topology which has a base of neighborhoods of 0 consistingof all sets of the form

N(Kn,W ) = f ∈ C(X,E) : f(Kn) ⊆W,

where n ≥ 1and W ∈ WE .

Theorem 10.2.2. [Kh07] Let X,E and k be as in the above def-inition. Suppose A is a topological algebra with a bounded approximateidentity eλ : λ ∈ I and E a topological left A-module such that themodule multiplication is both b(A)- and b(E)- hypocontinuous. Then:

(i) (C(X,E), k) is a topological left A-module with respect to the mod-ule multiplication (a, f)→ a.f.

(ii) (C(X,E), k) is essential.Proof. (i) First, we need to show that if bα : α ∈ J ⊆ A with bα → 0

and f ( 6= 0) ∈ C(X,E), then bα.fk→ 0 in C(X,E). Fix any Km ∈ Kn

andG ∈ WE . By b(E)-hypocontinuity, there exists a neighborhood V of 0in A with V.f(Km) ⊆ G. Since bα → 0 in A, there exists αo ∈ J such that

2. ESSENTIALITY OF MODULES OF VECTOR-VALUED FUNCTIONS 179

bα ∈ V for all α ≥ αo. Hence, for any α ≥ αo, bα.f(Km) ⊆ V.f(Km) ⊆ Gand so

bα.f(Km) ⊆ V.f(Km) ⊆ G and so bα.f ∈ N(Km, G).

Next, let b ( 6= 0) ∈ A and gα : α ∈ J ⊆ C(X,E) with gαk→ 0 in

C(X,E). Then b.gαk→ 0 in C(X,E), as follows. Fix any Km ∈ Kn and

G ∈ WE . By b(A)-hypocontinuity, there exists H ∈ WE with b.H ⊆ G.Choose αo ∈ J such that gα(Km) ⊆ H for all α ≥ αo. Hence, for anyα ≥ αo, b.gα(Km) ⊆ b.H ⊆ G and so

b.gα ∈ N(Km, b.H) ⊆ N(X,G).

(ii) Fix f ∈ C(X,E).We need to show that eλ.fk→ f. LetKm ∈ Kn

andG ∈ WE . Choose a balanced H ∈ WE with H +H +H ⊆ G. SinceD = eλ : λ ∈ I is bounded in A, by b(A)-hypocontinuity, there existsV ∈ WE such that

D.V ⊆ H.

Since f(Km) is compact, there exists a finite set z1, ..., zn ⊆ Km suchthat

f(Km) ⊆ ∪ni=1(f(zi) + V ∩H).

Since E is an essential A-module, there exists λo ∈ I such that

eλf(zi)− f(zi) ∈ H for all λ ≥ λo and i = 1, ..., n.

Now, let x ∈ Km and λ ≥ λo. Choose i ∈ 1, ..., n such that

f(x)− f(zi) ∈ V ∩H.

Then

eλf(x)− f(x) = eλ.(f(x)− f(zi)) + (eλ.f(zi)− f(zi)) + (f(zi)− f(x))

∈ D.(V ∩H) +H + V ∩H ⊆ H +H +H ⊆ G.


3. Topological Modules of Homomorphisms

In this section, we shall study the properties of the topological mod-uleHomA(A, Y ) of continuous homomorphisms. These results were origi-nally given in the papers [Rie67, ST69, Shan04] and in the present generalform in [Kh08].

Definition: Let E and Y be topological left A-modules, where E andY are TVSs and A a topological algebra. Then a mapping T : E → Y iscalled an A-module homomorphism if T (a.x) = a.T (x) for all a ∈ A andx ∈ E ([Rie67], p. 447). (Similarly, if E and Y are right A-modules, thenwe can define an A-module homomorphism as a mapping T : E → Ysatisfying T (x.a) = T (x).a for all a ∈ A and x ∈ E. We will stateresults for left modules over A, similar results holding, of course, forright modules.)

Our main interest is the study of A-module homomorphism from Ainto Y .

Lemma 10.3.1. [Kh08a] Let Y be a left A-module. Suppose thatA is faithful in Y . Then any A-module homomorphism T : A → Y ishomogeneous (that is, T (λa) = λT (a) for all λ ∈ K and a ∈ A).

Proof. Let a ∈ A and λ ∈ K. Then, for any c ∈ A,

c.T (λa) = T (c.(λa)) = T ((λc)a) = (λc).T (a) = c.λT (a)

Since A is faithful in Y, T (λa) = λT (a).

We now establish the linearity and continuity of an A-module homo-morphisms using the factorization theorem. The proof of the followingtheorem is analogous to those given in [Jo66, Cra69, Rie69, SumM72,KMT99].

Theorem 10.3.2. [Kh08a] Let Y be a topological left A-module withY metrizable and A strongly factorable. Then any A-module homomor-phism T : A→ Y is linear and continuous.

Proof. To show that T is linear, let a1, a2 ∈ A and α, β ∈ K. If wetake an = a1, a2, 0, 0, ..., then clearly an → 0; since A is stronglyfactorable, there exist b, c1, c2 ∈ A such that a1 = c1b, a2 = c2b. So

T (αa1 + βa2) = T ((αc1 + βc2)b) = (αc1 + βc2).T (b)

= αT (c1b) + βT (c2b) = αT (a1) + β(a2);

hence T is linear. Since Y is metrizable, to show that T is continuous, itsuffices to show that if an ⊆ A with an → 0, then T (an) → 0. Using

3. TOPOLOGICAL MODULES OF HOMOMORPHISMS 181

again the strong factorability of A, we can write an = cnb, where b ∈ Aand cn ⊆ A with cn → 0. Then

T (an) = T (cnb) = cn.T (b)→ 0.T (b) = 0.

(by the separate continuity of module multiplication). Thus T is contin-uous.

Definition. (Cf. [Rie67], p. 447) Let E and Y be topologicalleft A-modules, where E and Y are TVSs and A a topological algebra.Let HomA(E, Y ) denote the vector space of all continuous linear left A-module homomorphisms of E into Y. If E is an A-bimodule, then defining(a ∗ T )(x) = T (x.a), HomA(E, Y ) becomes a left A-module. In fact, forany b ∈ A, x ∈ E,

(a ∗ T )(b.x) = T ((b.x).a) = T (b.(x.a)) = b.T (x.a) = b.(a ∗ T )(x).

In particular, HomA(A, Y ) is a left A-module. Note that if A is commu-tative, then defining (T ∗ a)(x) = T (a.x), HomA(E, Y ) becomes a rightA-module. In fact, for any b ∈ A, x ∈ E,

(T ∗ a)(b.x) = T (a.(b.x)) = T (b.(a.x)) = b.T (a.x) = b.(T ∗ a)(x).

Lemma 10.3.3. [Kh08a] Let E and Y be topological left A-moduleswith A having an approximate identity eλ : λ ∈ I. If E is anessential A-module, then HomA(E, Y ) = HomA(E, Ye). In particular,HomA(A, Y ) = HomA(A, Ye).

Proof. Since Ye ⊆ Y, clearly HomA(E, Ye) ⊆ HomA(E, Y ). Now letT ∈ HomA(E, Y ). Then, for any x ∈ E, since eλ.x→ x,

limλ

eλ.T (x) = limλ

T (eλ.x) = T (x).

Therefore T (x) ∈ Ye, i.e. T ∈ HomA(E, Ye).We mention that HomA(E, Y ) has been extensively studied in the

case of E and Y as the Banach modules of Banach-valued function spacesL1(G,A) and Co(G,A), where G is a locally compact abelian group andA a commutative Banach algebra (see, e.g., [Wen52, Rie69, HL92]. IfE = Y = A, then HomA(A,A) is the usual multiplier algebra of A, andis denoted by M(A) (see section 10.1).

Definition: Let A be a Hausdorff topological algebra and (Y, τ) aHausdorff TVS which is a topological left A-module and has a baseWY ofneighborhoods of 0 in Y . The topology of bounded convergence tu = tuA(resp. the topology of pointwise convergence tp = tpA) on HomA(A, Y )


is defined as the linear topology which has a base of neighborhood of 0consisting of all sets of the form

U(D,G) = T ∈ HomA(A, Y ) : T (D) ⊆ G,

where D is a bounded (respectively, finite subset) of A and G ∈ WY .Clearly, tp 6 tu. Further, we obtain:

Lemma 10.3.4. [Kh08a] Let (Y, τ) be a topological left A-modulewith b(A)-hypocontinuous module multiplication. Then both (HomA(A,Y ), tu) and (HomA(A, Y ), tp) are topological left A-modules.

Proof. We prove the result only for (HomA(A, Y ), tu). For any a ∈ Aand T ∈ HomA(A, Y ), the map (a, T )→ a ∗ T is separately continuous,as follows. First, let aα : α ∈ J be a net in A with aα → a ∈ A, and letD be a bounded subset of A and G ∈ WY . By b(A)-hypocontinuity, thereexists a balanced H ∈ WY such that D.H ⊆ G. Since T is continuous,there exists αo ∈ J such that

T (aα)− T (a) ∈ H for all α ≥ αo.

Now, for any b ∈ D and α ≥ αo,

(aα ∗ T − a ∗ T )(b) = T (baα)− T (ba) = b.[T (aα)− T (a)] ∈ D.H ⊆ G;

that is, aα ∗ T − a ∗ T ∈ U(D,G) for all α ≥ αo. Hence aα ∗ Ttu→ a ∗ T.

Now, let Tα : α ∈ J be a net in HomA(A, Y ) with Tαtu→ T ∈

HomA(A, Y ), and letD be a bounded subset of A andG ∈ WY . Since themap Ra : A→ A given by Ra(b) = ba, b ∈ A, is linear and continuous (byseparate continuity of multiplication in A), it follows that Ra(D) = Da

is bounded in A. Since Tαtu→ T , there exists αo ∈ J such that

Tα − T ∈ U(Da,G) for all α ≥ αo.

Now, for any b ∈ D and α ≥ αo,

(a ∗ Tα − a ∗ T )(b) = (Tα − T )(ba) ∈ G;

that is, a ∗ Tα− a ∗ T ∈ U(D,G) for all α ≥ αo. Hence a ∗ Tαtu→ a ∗ T.

The following results generalize some results of [Wan61, Bus6, Pom92,KMT99] to modules of continuous homomorphisms.

Theorem 10.3.5. [Kh08a] Let Y be a topological left A-module.Then:

(i) If Y is an F-space and A is strongly factorable, then both (HomA(A,Y ), tu) and (HomA(A, Y ), tp) are complete.


(ii) If Y is complete and A is ultrabarrelled having a bounded ap-proximate identity, then both (HomA(A, Y ), tp) and (HomA(A, Ye), tp)are complete.

Proof. (i) Let Tα : α ∈ J be a tu-Cauchy net in HomA(A, Y ).Since tp 6 tu, Tα : α ∈ J is an tp-Cauchy net in HomA(A, Y ); inparticular, for each a ∈ A, Tα(a) is a Cauchy net in X . Consequently,by completeness of Y , the mappings T : A → Y , given by T (a) =limα Tα(a) (a ∈ A), is well-defined. Further, for any a, b ∈ A,

T (ab) = limαTα(ab) = lim

αa.Tα(b) = a.T (b).

Since Y is metrizable and A is strongly factorable, by Theorem 10.3.2,

T ∈ HomA(A, Y ). We now show that Tαtu−→ T . Let D be a bounded

subset of A and take closed G ∈ WY . There exists an index αo such that

Tα(a)− Tγ(a) ∈ G for all a ∈ D and α, γ ≥ αo.

Since G is closed, fixing α ≥ αo and taking limγ, we have

Tα(a)− T (a) ∈ G for all a ∈ D.

Hence,

Tα − T ∈ U(D,G) for all α ≥ αo.

Thus (HomA(A, Y ), tu) is complete. By a similar argument, (HomA(A,Y ), tp) is also complete.

(ii) We first show that HomA(A, Y ) is tp-complete. [Let Tα : α ∈ Jbe a tp-Cauchy net in HomA(A, Y ). Then, for each a ∈ A, Tα(a) : α ∈J is a Cauchy net in Ye and hence in Y for all a ∈ A. Since (Y, τ) iscomplete, Tα(a) is convergent for all a ∈ A. Define T : A→ Y by

T (a) = limαTα(a), a ∈ A.

Then T is a left A-module homomorphism. Also Tα : α ∈ J is point-wise bounded. Since A is ultrabarrelled, by the principle of uniformboundedness, Tα : α ∈ J is equicontinuous. Therefore, given anyclosed G ∈ WY , there exists a neighborhood V of 0 in A such that

Tα(V ) ⊆ G for all α ∈ J ; that is ∪α∈J Tα(V ) ⊆ G.

If a ∈ V, Tα(a) ∈ Tα(V ) ⊆ G. Therefore, Tα(a) : α ∈ J is a net in Gfor all a ∈ V. Thus

T (a) = limαTα(a) ∈ cl-G = G; that is, T (V ) ⊆ G.


Therefore T : A → Y is continuous, and so T ∈ HomA(A, Y ). ThusHomA(A, Y ) is tp-complete.] Since A has an approximate identity, byLemma 10.3.3,

HomA(A, Y ) = HomA(A, Ye).

Hence HomA(A, Ye) is also tp-complete.

Definition: For any x ∈ Y, consider the map Rx : A→ Y given by

Rx(a) = a.x, a ∈ A.

Clearly, Rx is linear and continuous (by separate continuity of modulemultiplication); further, for any a, b ∈ A,

Rx(ab) = (ab).x = a.(b.x) = a.Rx(b),

so that Rx ∈ HomA(A, Y ). Now define a map µ : Y → HomA(A, Y ) by

µ(x) = Rx, x ∈ Y.

It is easily seen that µ(Y ) = Rx : x ∈ Y is a left A-submodule ofHomA(A, Y ). [In fact, for any a ∈ A and x ∈ Y,

(a ∗Rx)(b) = Rx(ba) = b.(a.x) = Ra.x(b), b ∈ A,

so that a ∗Rx ∈ HomA(A, Y ).]

Theorem 10.3.6. [Kh08a] Let Y be a topological left A-module withA having a two-sided approximate identity eλ : λ ∈ I. Then µ(Ye) istp-dense in HomA(A, Ye); in particular, µ(Y ) is tp-dense inHomA(A, Ye).

Proof. Let T ∈ HomA(A, Ye). For each λ ∈ I, define xλ = T (eλ).Then

limγeγ .xλ = lim

γeγ.T (eλ) = lim

γT (eγeλ) = T (eλ) = xλ,

and so xλ ∈ Ye. Now, for any a ∈ A, a.xλ = a.T (eλ) = T (aeλ) → T (a);hence

µ(xλ)(a) = Rxλ(a) = a.xλ = a.T (eλ) = T (a.eλ)→ T (a)

Therefore µ(xλ)tp→ T. Thus T ∈ tp-cl[µ(Ye)]; that is, µ(Ye) is tp-dense in

HomA(A, Ye).

Remark: Wemention that µ(Y ) need not be tu-closed inHomA(A, Y )even if Y = A is a metrizable locally C∗-algebra or a Banach algebra (seealso ([Lar70], p. 17) and ([Ph88], p.187)). We include the following ex-ample, due to Wang ([Wan61], p. 1138), shows that µd(A) need not beu-closed in CMd(A), in general. (See also ([Lar70], p. 17) and ([Ph88],p.187)).


Example. Let ϕ : R→ R be a function with ϕ(x) ≥ 1 (x ∈ R) and

limx→+∞

ϕ(x) = limx→−∞

ϕ(x) = +∞.

Define

A = Aϕ = f ∈ C(R,C) : ‖f‖ϕ = supx∈R| ϕ(x)f(x) |<∞.

Then A forms a commutative normed algebra under pointwise operationsand the above norm. To see that A is complete, let fn be a Cauchysequence in A. Then fn and fnϕ are Cauchy sequences in the usualsupremum norm ‖.‖∞. Let f and g be, respectively, the uniform limitsof fn and fnϕ. Then

supx∈R| g(x) |<∞ and fϕ = g.

Hence A is a Banach algebra. Further, A is faithful. It is apparent thatA ⊆ Co(R,C) but A 6= Co(R,C). Indeed any real-valued function f ∈Co(R,C) such that f(x)

√ϕ(x) ≥ 1 belongs to Co(R,C)\ A. However, A

with the operator norm is dense in Co(R,C), and so A is not complete inthe operator norm. For instance, let f ∈ Co(R,C) and, since Coo(R,C)is dense in Co(R,C), there exists a sequence fn ⊆ A of functions withcompact support such that

limn→∞‖fn − f‖∞ = 0.

Then

sup‖g‖≤1

‖fng − fg‖ϕ = sup‖g‖≤1

supx∈R| (fn(x)− f(x))g(x)ϕ(x) |

≤ ‖fn − f‖∞ sup‖g‖≤1

supx∈R| g(x)ϕ(x) |

≤ ‖fn − f‖∞,

showing that f ∈ cl-(A). Consequently, A is not closed in Co(R,C) inthe operator norm.

Theorem 10.3.7. Let X be a topological left A-module with A havinga two-sided approximate identity eλ : λ ∈ I. If (X, β) is complete, themap µ : X → HomA(A,Xe) defined by µ(y) = Ry, y ∈ X, is onto.

Proof. By Theorem 10.3.6, µ(X) is tp-dense in HomA(A,Xe). Wenow show that µ(X) is tp-closed in HomA(A,Xe). Let T ∈ tp-cl µ(X).

There exists a net xα : α ∈ J ⊆ X such that Rxα

tp→ T. Then xα :

α ∈ J is β-Cauchy in X . [Let D be a finite subset of A and G ∈ WX .


Choose a balanced H ∈ WX with H + H ⊆ G. Since Rxα

tp→ T, there

exists αo ∈ I such that for all α ≥ αo,

Rxα − T ∈ N(D,H) or Rxα(a)− T (a) ∈ H for all a ∈ D.

Then, for any a ∈ D and α, γ ≥ αo,

a.xα − a.xγ = [Rxα(a)− T (a)] + [T (a)− Rxγ (a)] ∈ H +H ⊆ G.]

Since (X, β) is complete, xαβ→ xo, xo ∈ X. Hence Rxα

tp→ Rxo . By

uniqueness of limit in Hausdorff spaces, T = Rxo ∈ µ(X). Thus µ(X) =HomA(A,Xe).



Section 10.1. In 1968, Busby [Bus68] also defined a general stricttopology on any B∗-algebra Y which contains A as a closed two-sidedideal; this is given by the seminorms x → max||ax||, ||ax|| (a ∈ A).Independently, this topology was studied in greater detail and in a moregeneral setting by Sentilles and Taylor [ST69] where A is a Banach al-gebra having an approximate identity and Y is a left Banach A-module.More recently, Khan-Mohammad-Thaheem [KMT05] have considered thecase of Y a locally idempotent F -algebra which contains A as a closedtwo-sided ideal, while Shantha [Sha04] has taken Y a locally convex mod-ule over a locally convex algebra A. As pointed out in Sentilles and Taylor[ST69], the essential ingredient of arguments used by Buck [Buc58] andlater authors is the presence of an approximate identity in Co(X) or A.

This section contains some results from the paper [Kh07] whish dealwith the study of general strict and related topologies on a topological leftA-module Y, where Y is Hausdorff TVS and A a Hausdorff topologicalalgebra having a two-sided bounded approximate identity. These includegeneralization of several results of Sentilles and Taylor [ST69] from thesetting of Banach left modules to topological left modules obtained byShantha [Shan04, KMT05].

Section 10.2. This section two results from [Kh07] on the essential-ity of topological left A-modules of continuous vector-valued functions,which were originally proved in [SumW72] for a particular case.

Section 10.3. In this section, a study is made of the properties ofthe topological module HomA(A, Y ) of continuous homomorphisms.

In [Rie67, Rie69], Rieffel made an extensive study of the Banach mod-ule HomA(A,X) of continuous homomorphisms. Further results in thisdirection have been obtained in [ST69, Rue77] in connection with thestudy of general strict topology. More recently, Shantha [Sha04, Shan05]has studied homomorphisms in the case of locally convex modules. Thissection contains some results from a recent work [Kh08a] which has in-vestigated the extent to which some of the results of above authors arealso true in the non-locally convex setting of topological modules. Asapplication, an observation on inner derivation, due to Phillips [Ph95], ispresented in a more general setting.

We mention that particular cases of the module HomA(A,X) are thealgebras Mℓ(A),Mr(A),M(A) and Md(A) which consists of all contin-uous left multipliers, right multipliers, multipliers and double multipli-ers. There is an extensive literature on multipliers; see, e.g., [Wan61,


Jo64, Jo66, Tay70, TW70, APT73, Fon75, Lat76, Tom76, Dav77, CL78,Won85, Ph88, Hus89, KMT99, MKT01].

CHAPTER 11

Mean Value Theorem and Almost Periodicity

189

190 11. MEAN VALUE THEOREM AND ALMOST PERIODICITY

1. The Mean Value Theorem in TVSs

We present here the mean value theorem and the mean value inequal-ity for class of Gateaux differentiable functions f : A ⊆ E → F, where Eand F are TVSs (both over the field R of real numbers) and A ⊆ E anopen set. These results are given in [Vain64] for E and F normed spacesand in the present general form in [Kh97].

Definition. A function f : A→ F is said to be Gateaux differentiableat xo ∈ A if there exists a mapping from E into F , denoted by f ′(xo),such that, given any z ∈ E and a balanced neighborhood V of 0 in F ,there exists a δ > 0 satisfying,

f(xo + tz)− f(x)

t− f ′(xo)(z) ∈ V,

whenever 0 < |t| < δ; f ′(xo) is called the Gateaux derivative of f at xoand we briefly write as,

f ′(xo)(z) = limt→0

f(xo + tz)− f(xo)

t, z ∈ E. (1)

Some authors also require f ′(xo) to be linear, but it is not assumedhere. The following example shows that a Gateaux differentiable functionneeds not being continuous.

Example. The function f : R2 → R, given by

f(x) =a2b

a4 + b2if x = (a, b) 6= (0, 0), and f((0, 0)) = 0,

is not continuous at x = (0, 0), although f ′(0) exists.

We first consider the Lagrange form of the mean value theorem forreal-valued functions on E. For any x, y ∈ E, the line segment [x, y]joining x and y is defined as the set

[x, y] = (1− t)x+ ty : 0 ≤ t ≤ 1.

Theorem 11.1.1. [Kh97] Let E be a Hausdorff TVS over R, andlet A ⊆ E be an open set. Let g : A → R be a function continuous andGateaux differentiable at each point of a segment [xo, xo+ h] in A. Thenthere exists a θ ∈ (0, 1) such that

g(xo + h)− g(xo) = g′(xo + θh)(h).

Proof. Define φ : [0, 1]→ R by

φ(t) = g(xo + th), t ∈ [0, 1].

1. THE MEAN VALUE THEOREM IN TVSS 191

Then φ is differentiable on [0, 1]. In fact, using (1), we obtain

[φ′(t) = lim∆t→0

[g(xo + (t +∆t)h)− g(xo + th)]/∆t

= g′(xo + th)(h).

By the classical mean value theorem, there exists a θ ∈ (0, 1) such that

φ(1)− φ(0) = φ′(θ).

Hence

g(xo + h)− g(xo) = φ(1)− φ(0) = g′(xo + θh)(h).

An exact analogue of Theorem 11.1.1 for vector-valued (or even forcomplex-valued) functions need not hold as is shown below:

Example. Consider the function f : [0, 2π]→ C defined by

f(x) = cos x+ i sin x = eix, x ∈ [0, 2π].

Clearly f ′(x) = ieix. Now f(2π)− f(0) = 1− 1 = 0, but

(2π − 0)f ′(c) = 2πieic 6= 0 for all 0 < c < 2π.

Hence

f(2π)− f(0) 6= (2π − 0)f ′(c) for all 0 < c < 2π.

However, f(x) does satisfy the Mean Value Inequality for the valuesxo = 0, xo + h = 2π since

|f(2π)− f(0)| = 0 ≤ sup|2π − 0||ieic| : 0 ≤ c ≤ 2π = 2π.

However, for the class of these functions, the following version of themean value theorem holds.

Theorem 11.1.2. [Kh97] Let E and F be a Hausdorff topologicalvector spaces over R, and let A ⊆ E be an open set. Suppose F hasa non-trivial (real) continuous dual F

∗

, and let f : A → F be a func-tion continuous and Gateaux differentiable at each point of the segment[xo, xo + h] in A. Then, given u ∈ F

∗

, there exists a θ ∈ (0, 1) such that

< f(xo + h)− f(xo), u >=< f ′(xo + θh)(h), u > . (2)

Proof. Define g : [xo, xo + h]→ R by

g(x) =< f(x), u >, x ∈ [xo, xo + h].

Then it follows from the linearity and continuity of u that

g′(x)(h) =< f ′(xo)(h), u > .


By Theorem 11.1.1, there exists a θ ∈ (0, 1) such that

g(xo + h)− g(xo) = g′(xo + θh)(h).

This establishes (2).

The above result clearly reduces to Theorem 11.1.1 when F = R.

Theorem 11.1.3. (Mean Value Inequality) [Kh97] Under the hy-pothesis of Theorem 11.1.2, given a continuous seminorm p on F , thereexists a θ ∈ (0, 1) such that,

p(f(xo + h)− f(xo)) ≤ p(f ′(xo + θh)). (3)

Proof. By the analytic form of the Hahn-Banach theorem ([Ko69],Theorem 8, §.17.3), we can choose a u( 6= 0) ∈ F

∗

such that

< y, u >≤ p(y) for all y ∈ F

and< f(xo + h)− f(xo), u >= p(f(xo + h)− f(xo)).

By Theorem 11.1.2, there exists a θ ∈ (0, 1) such that

< f(xo + h)− f(xo), u >=< f ′(xo + θh)(h), u > .

Consequently, we obtain (3).

Remark. We have seen above that the function f : [0, 2π] → Cdefined by

f(x) = cosx+ i sin x = eix, x ∈ [0, 2π].

does not satisfy Theorem 11.1.1, the Mean Value Theorem, for the valuesxo = 0, xo + h = 2π. However, f(x) does satisfy Theorem 3, the MeanValue Inequality, for the values

xo = 0, xo + h = 2π

since

|f(2π)− f(0)| = 0 ≤ sup|2π − 0||ieic| : 0 ≤ c ≤ 2π = 2π.

As an application of the above theorem, we obtain:

Corollary 11.1.4. [Kh97] Suppose F is locally convex and A ⊆ Ean open connected set, and let f : A → F be continuous and Gateauxdifferentiable at each point of A. If f ′(xo) = 0 for each x ∈ A, then fis constant on A.

Proof. Fix x ∈ A, and let B = x ∈ A : f(x) = f(xo). Clearly, B isnon-empty and closed in A. B is also open in A, as follows. Let x ∈ B.Choose a balanced neighborhood V of 0 in E such that U = x+ V ⊆ A.

1. THE MEAN VALUE THEOREM IN TVSS 193

Then, for each y ∈ U , [x, y] ∈ U . Hence, for each y ∈ U and anycontinuous seminorm p on F , it follows from Theorem 11.1.3 and thehypothesis that

p(f(y)− f(x)) ≤ supz∈[x,y]

(f ′(z)(y − x)) = p(0) = 0.

Since F is locally convex and Hausdorff, f(y) = f(x) = f(xo) for ally ∈ U. Hence U ⊆ B. Since A is connected, B = A. Thus f(x) = f(xo)for all x ∈ A.

Theorem 11.1.5. (Mean Value Inclusion) [AS67, Kh97] Suppose Fis a locally convex space. Then, under the hypothesis of Theorem 2,

f(xo + h)− f(x) ∈ co.(B),

where B = f ′(xo)(h) : x ∈ [xo, xo + h] and co.(B) is the closed convexhull of B.

Proof. Suppose f(xo + h)− f(xo) /∈ co.(B). By ([Ko69], Theorem 2,§. 20.7), there exists a u( 6= 0) ∈ F

∗

and an r ∈ R such that,

< y, u >≤ r < f(xo + h)− f(xo), u > (4)

for all y ∈ co.(B). Define φ : [0, 1] → R by φ(t) =< f(xo + th), u >.Then φ′(t) =< f ′(xo + th)(h), u >. There exists a θ ∈ (0, 1) such thatφ(1)− φ(0) = φ′(θ). Then

< f(xo + h)− f(xo), u >= φ(1)− φ(0) =< f ′(xo + θh)(h), u >

which contradicts (4).

Remark. Theorem 11.1.5 need not hold if F is not locally convex, orif co.(B) is replaced by B or co.(B). This follows from ([AS67], Examples1.23-1.25).


2. Almost Periodic Functions with Values in a TVS

In this section, we consider the concept of almost periodicity of func-tions having values in a topological vector space E. The results are takenmainly from [NGu84, KA11].

Definition: A subset P of R is called relatively dense in R if thereexists a number ℓ > 0 such that every interval of length ℓ in R containsat least one point of P .

Definition: Let (E, τ) be TVS with a base W = WE of balancedneighborhoods of 0. A function f : R → E is called almost periodic if itis continuous and, for each W ∈ W, there exists a number ℓW > 0 suchthat each interval of length ℓW in R contains a point τW such that

f(t + τW )− f(t) ∈ W for all t ∈ R. (∗)

A number τW ∈ R for which (∗) holds is called W -translation numberof f. The above property says that, for each W ∈ W, the function f hasa set of W -translation numbers PW,f which is relatively dense in R.

The set of all almost periodic functions f : R → E is denoted byAP (R, E). For any f : R → E and a fixed h ∈ R, the h-translate of fis defined as the function fh : R→ E defined by

fh(t) = f(t+ h), t ∈ R.

We shall denote H(f) = fh : h ∈ R, the set of all translates of f .

Definition: LetX be a Hausdorff space and (E, d) a metrizable TVS.The uniform topology u on Cb(X,E) is defined as the topology given bythe metric ρ

ρ(f, g) = supx∈X

d(f(x), g(x)), f, g ∈ Cb(X,E).

Theorem 11.2.1. [NGu84, KA11] Let X be a Hausdorff space and(E, d) a metric linear space.

(a) If a sequence fn ⊆ Cb(X,E) is u-convergent to a functionf : X → E, then f ∈ Cb(X,E).

(b) If fn is a u-Cauchy sequence in Cb(X,E) and if there is afunction f : X → E such that fn(x) → f(x) for each x ∈ X, then

fnu−→ f and f ∈ Cb(X,E).(c) If E is complete, then so is (Cb(X,E), u).Proof. (a) f is continuous as follows. Let xo ∈ X and W ∈ W. We

show that there exists a neighborhood G of xo in X such that

f(y) ∈ f(xo) +W for all y ∈ G.

2. ALMOST PERIODIC FUNCTIONS WITH VALUES IN A TVS 195


there exists an integer N such that

fn(x)− f(x) ∈ V for all n ≥ N and all x ∈ X. (A)

Since fN is continuous at xo, there exists a neighborhood G of xo in Xsuch that

fN (y) ∈ fN(xo) for all y ∈ G. (B)

Then, for any y ∈ G, using (A) and (B),

f(y)− f(xo) = (f(y)− fN(y)) + (fN(y)− fN(xo)) + (fN(xo)− f(xo))

∈ V + V + V ⊆W.

Thus f is continuous on X .Next, f is bounded as follows. Suppose f is unbounded. Then there

exists a W ∈ W such that f(X) is not contained in nW for all n ≥ 1.Hence, for each n ≥ 1, there exists xn ∈ X such that f(xn) /∈ nW .

Choose a balanced V ∈ W such that V + V ⊆W . Since fnu−→ f , there

exists an integer K such that

fn(x)− f(x) ∈ V for all n ≥ K and x ∈ X.

In particular,fK(xn)− f(xn) ∈ V for all n ≥ 1.

Since fK is bounded, there exists an integer m > 1 such that fK(X) ⊆mV . Then

f(xm) = [f(xm)− fK(xm)] + fK(xm) ∈ V +mV ⊆ mV +mV ⊆ mW,

a contradiction. Hence f is bounded.(b) Let fn be a u-Cauchy sequence in Cb(X,E). By hypothesis, for

each x ∈ X, f(x) := limn→∞

fn(x) exists in E. Then fnu−→ f , as follows.

Let W ∈ W be closed. Since fn is u-Cauchy, there exists an integer Jsuch that

fn(x)− fm(x) ∈ W for all n,m ≥ J and x ∈ X.

Fix any n ≥ J . Since W is closed and, for each x ∈ X, fm(x) −→ f(x),we have

fn(x)− f(x) ∈ W for x ∈ X.

Thus fn(x)− f(x) ∈ W for all n ≥ J and all x ∈ X ; i.e.

fn − f ∈ N(X,W ) for all n ≥ J.

This proves that fnu−→ f . By (a), f ∈ Cb(X, Y ).

(c) Let fn be a u-Cauchy sequence in Cb(X,E). Since p ≤ u, fnis a p-Cauchy sequence in C(X,E); that is, for each x ∈ X , fn(x) is


a Cauchy sequence in E. Since E is complete, for each x ∈ X, f(x) :=

limn→∞

fn(x) exists in E. Then, by (b), fnu−→ f and that f is continuous

and bounded. Hence (Cb(X,E), u) is complete.

Theorem 11.2.2. [NGu84, KA11] Let E be a TVS. Let f ∈ AP (R, E).Then:

(a) f has precompact range f(R); hence f is bounded.(b) f is uniformly continuous on R.

Proof. (a) LetW ∈ W. Choose a balanced V ∈ W such that V +V ⊆W. Since f is almost periodic, there exists a number ℓ = ℓV > 0 suchthat each interval of length ℓ in R contains a point τV such that

f(t+ τV )− f(t) ∈ V for all t ∈ R. (1)

By continuity of f , the set f [0, ℓ] is compact in E and hence precompact.So there exist x 1, ..., x n ∈ f [0, ℓ] such that

f [0, ℓ] ∈n⋃

i=1

(xi + V ). (2)

We claim that f(R) ⊂n⋃i=1

(xi + W ). [Take an arbitrary t ∈ R. By (1),

there exists τ ∈ [−t,−t+ ℓ] such that f(t+ τ)− f(t) ∈ V. By (2), choosexk ∈ x1, ..., xn such that f(t+ τ)− xk ∈ V. Then

f(t)− xk = [f(t)− f(t+ τ)] + [f(t + τ)− xk] ∈ −V + V ⊆W,

and therefore f(t) ∈ x k +W .] Thus f(R) is precompact in E.(b) LetW ∈ W. Choose a balanced V ∈ W such that V +V +V ⊆W.

There exists a number ℓ = ℓV > 0 such that each interval of length ℓ inR contains a point τV such that

f(t+ τV )− f(t) ∈ V for all t ∈ R. (3)

Now f , being almost periodic, is continuous on R. Then f is uniformlycontinuous on the compact set [−1, 1 + ℓV ], so there exists δ = δV > 0(we may assume 0 < δ < 1 without loss of generality) such that

f(s)− f(t) ∈ V for all s, t ∈ [−1, 1 + ℓV ] with |s− t| < δ. (4)

Let a, b ∈ R with |a − b| < δ and assume a < b. We claim that f(a) −f(b) ∈ W. [We may assume that a < b. Choose a τV ∈ [−a,−a + ℓV ]satisfying (3). Then

a+ τV ∈ a+ [−a,−a + ℓV ] = [0, 0 + ℓV ] ⊆ [−1, 1 + ℓV ];


since 0 < b− a < δ < 1,

b+ τV ∈ b+[−a,−a+ ℓV ] = [b−a, b−a+ ℓV ] = [0, 1+ ℓV ] ⊆ [−1, 1+ ℓV ].

Also, |(a+ τV )− (b+ τV )| = |a− b| < δ, so by (4),

f(a+ τV )− f(b+ τV ) ∈ V. (5)

Therefore, by (3) and (5),

f(a)− f(b) = [f(a)− f(a+ τV )] + [f(a+ τV )− f(b+ τV )]

+[f(b+ τV )− f(b)]

∈ −V + V + V ⊆W.]

Thus f is uniformly continuous on R.

Remark. Clearly, by Theorem 11.2.2(a), AP (R, E) ⊆ Cb(R, E).

Theorem 11.2.3. [NGu84, KA11] Let E be a TVS. If fn is a

sequence in AP (R, E) such that fnu−→ f , then f ∈ AP (R, E).

Proof. Clearly, by Theorem 11.2.2(b), f is continuous on R. LetW ∈ W. Choose a balanced V ∈ W such that V + V + V ⊆ W. Sincefn

u−→ f , there exists an integer N ≥ 1 such that such that

fn(t)− f(t) ∈ V for all t ∈ R and n ≥ N. (6)

Since fN is almost periodic, there exists a number ℓ = ℓV > 0 such thateach interval of length ℓ in R contains a point τ = τV such that

fN(t + τ)− fN(t) ∈ V for all t ∈ R. (7)

Then, by (6) and (7), for any t ∈ R

f(t+ τ)− f(t) = [f(t + τ)− fN(t+ τ)] + [fN(t + τ)− fN(t)]

+[fN (t)− f(t)]

∈ −V + V + V ⊆W.

Since the set PV,fN of almost periods of fN is relatively dense, we takePV,f = PV,fN . Hence f is also almost periodic

Theorem 11.2.4. [NGu84, KA11] Let E be a TVS. If f : R → Eis almost periodic, then the functions (i) λf (λ ∈ K), (ii) f(t) ≡ f(−t)and (iii) fh(t) = f(t+ h) (h ∈ R) are also almost periodic.

Proof. (i) This is trivial if λ = 0. Suppose λ 6= 0. Let W ∈ W bebalanced. Then V = λ−1W ∈ W. Since f is almost periodic, there existsa number ℓ = ℓV > 0 such that each interval of length ℓ in R contains apoint τV such that

f(t+ τV )− f(t) ∈ V for all t ∈ R.


Then

(λf)(t+ τV )− (λf)(t) = λ[f(t+ τV )− f(t)] ∈ λV = W for all t ∈ R.

Hence λf is almost periodic.(ii) Let W ∈ W be balanced. There exists a number ℓ = ℓW > 0 such

that each interval of length ℓ in R contains a point τ such that

f(t+ τ)− f(t) ∈ W for all t ∈ R.

Put s = −t; we get:

f(s− τ)− f(s) = f(−s+ τ)− f(−s) = f(t+ τ)− f(t) ∈ W.

Therefore f(s − τ) − f(s) ∈ W for every s ∈ R. This shows that f isalmost periodic with −τ as a W -translation number.

(iii) Let W ∈ W be balanced, and let h ∈ R. There exists a numberℓ = ℓW > 0 such that each interval of length ℓ in R contains a point τsuch that


Replacing t by t+ h, we have

f(t+ h+ τ)− f(t+ h) ∈ W for all t ∈ R,

or

fh(t+ τ)− fh(t) ∈ W for all t ∈ R.

Theorem 11.2.5. [NGu84, KA11] Let E and F be a TVSs, and

let f ∈ AP (R, E). If g : f(R)→F is any continuous function, then thecomposed function g f ∈ AP (R, F ).

Proof. Let U ∈ WF be balanced. Since f(R) is compact, g is uni-

formly continuous on f(R) and so there exists a W ∈ WE such that

g(x)− g(y) ∈ U for all x, y ∈ f(R) with x− y ∈ W.

Since f is almost periodic, there exists a number ℓ = ℓW > 0 such thateach interval of length ℓ in R contains a point τ such that


Consequently,

g[f(t+ τ)]− g[f(t)] ∈ U for all t ∈ R.

Thus g f ∈ AP (R, F ).

We now proceed to establish the Bochner’s criteria for almost peri-odicity. As a first step, we obtain:


Theorem 11.2.6. [NGu84, KA11] Let E be a TVS and f : R → Ea continuous function. If the set of translates H(f) = fh : h ∈ R isu-sequentially compact in Cb(R, E), then f is almost periodic.

Proof. Suppose f is not almost periodic Then there exists a W ∈ Wsuch that for every ℓ > 0, there exists an interval of length ℓ, [−a,−a+ℓ](say) which contains no W -translation number of f. Consequently, forevery h ∈ [−a,−a + ℓ], there exists th ∈ R such that

f(th + h)− f(th) /∈ W.

Let us consider h1 ∈ R and an interval (a1, b1) with b1−a1 > 2 |h1| whichcontains no W -translation number of f . Now Let h2 =

a1+b12

. Since

−b1 − a1

2< h1 <

b1 − a12

,

h2−h1 ∈ (a1, b1) and therefore h2−h1 cannot be aW -translation numberof f . Let us consider another interval (a2, b2) with b2−a2 > 2(|h1|+ |h2|),which contains no W -translation number of f . Let h3 = a2+b2

2; then

h3 − h1, h3 − h2 ∈ (a2, b2) and therefore h3 − h1and h3 − h2 cannot beW -translation number of f . We proceed and get a sequence hn suchthat no hm − hn (m > n) is a W -translation number of f ; that is, thereexists tmn ∈ R with

f(tmn + hm − hn)− f(tmn) /∈ W. (8)

Put smn = tmn − hn. Then (8) becomes:

f(smn + hm)− f(smn + hn) /∈ W. (9)

Since H(f) is sequentially compact in Cb(R, E), there exists a subse-quence knof hnsuch that ft + kn) is u-convergent on R. Then,for W ∈ W, there exists an N = NW ≥ 1 such that if m, n > N (wemay take m > n), we have:

f(t+ km)− f(t+ kn) ∈ W for every t ∈ R. (10)

Taking t = smn in (10), we get a contradiction to (9). Thus f is almostperiodic.

As a converse of the above theorem, we obtain:

Theorem

11.2.7. [NGu84, KA11] Let E be an F -space and f : R → E analmost periodic function. Then the set of translates H(f) = fh : h ∈ Ris u-compact in Cb(R, E).

Proof. Since E is an F -space, by Theorem 11.2


.1, (Cb(R, E), u) is also an F-space. Therefore, it suffices to show thatany sequence fhn in H(f) has a u-Cauchy subsequence.

Proof. Let S = an be a dense sequence in R. Since f(R) is precom-pact and hence relatively compact in the F-space E, we can extract fromfhn(a1) a convergent subsequence. Let fh1,n be the subsequence offhn which converges at a1. We apply the same argument as above tothe sequence fh1,n to choose a subsequence fh2,n which converges ata2. We continue the process and consider the diagonal sequence fhn,nwhich converges at each an in S. Call this last sequence by fkn. Weclaim that this sequence is u-Cauchy on R.

Proof. Let to ∈ R, and let W ∈ W. Choose a balanced V ∈ Wsuch that V + V + V + V + V ⊂ W . By almost periodicity of f , letℓ = ℓ(V ) > 0 be such that each interval of length ℓ in R contains a pointτ = τV such that

f(t+ τ)− f(t) ∈ V for all t ∈ R. (11)

By uniform continuity of f over R, there exists δ = δV > 0 such that

f(t)− f(t′) ∈ V for every t, t′ ∈ R with |t− t′| < δ. (12)

We divide the interval [0, ℓ] into subintervals I1, ..., Ir, each of lengthsmaller than δ. Since S is dense in R, for each 1 ≤ j ≤ r, Ij containsa point bj of S and set So = b1, ..., br. Since fkn is p-convergent onS, it follows that fkn is u-convergent on the finite set So. Then thereexists an integer N = NV ≥ 1 such that

f(bi + kn)− f(bi + km) ∈ V for every i = 1, ..., J and n,m > N. (13)

Now, choose τo ∈ [−to,−to + ℓ] satisfying (11). Then to + τ ∈ [0, ℓ], sochoose bj ∈ So such that |to + τo − bj | < δ. So, by (12),

f(to + τo + kn)− f(bj + kn) ∈ V for every n ≥ 1. (14)

Therefore, if n,m > N , by applying (11), (13), (14) we get

fkn(to)− fkm(to) = f(to + kn)− f(to + km)

= [f(to + kn)− f(to + kn + τo)]

+[f(to + kn + τo)− f(bj + kn)]

+[f(bj + kn)− f(bj + km)]

+[f(bj + km)− f(to + τo + km)]

+[f(to + km + τo)− f(to + km)]

∈ V + V + V + V + V ⊂W


Therefore the subsequence fkn of fhn is u-Cauchy on R. Conse-quently, fkn is u-convergent on R.

Combining Theorems 11.2.6 and 11.2.7, we have:

Corollary 11.2.8. (Bochner’s Criterion) [NGu84, KA11] Let E bean F -space. Then a continuous function f : R → E is almost periodicfunction iff the set H(f) = fh : h ∈ R is u-compact in Cb(R, E).

Theorem 11.2.9. Let E be an F -space, and let f1, ..., fm ∈ AP (R, E).Define F : R→ Em by

F (t) = (f1(t), ..., , fm(t)), t ∈ R.

Then F ∈ AP (R, Em). In particular, for any W ∈ W, f1, ..., fm havecommon W -translation numbers.

Proof. Let hn be a sequence in R. Consider the sequence f1,hn oftranslates of function f1 corresponding to hn. Since f1 is almost peri-odic, by using Bochner’s criteria, we can extract from f1,hn a uniformlyconvergent subsequence, denoting again by f1,hn. Continuing this pro-cess, we extract from fm,hn a uniformly convergent subsequence, de-noted also by fm,hn. Then the sequence (f1,hn..., fm,hn) has a subse-quence which is easily seen to be u-convergent on R. Hence F is almostperiodic. Next, for any W ∈ W, there exists a number ℓ = ℓW > 0 suchthat each interval of length ℓ in R contains a point τW such that

F (t+ τW )− F (t) ∈ Wm for all t ∈ R.

Consequently,

fi(t + τW )− fi(t) ∈ W for all t ∈ R and i = 1, ..., m.

Theorem 11.2.10. [NGu84, KA11] Let E be a F -space, and letf, g ∈ AP (R, E). Then f + g ∈ AP (R, E).

Proof. In view of the Bochner’s criteria, we need to show that H(f +g) = fh + gh : h ∈ R is u-compact in Cb(R, E). Define F : R→ E2 by

F (t) = (f(t), g(t)), t ∈ R.

By Theorem 11.2.8, F ∈ AP (R, E2) and hence, by Bochner’s criteria,H(F ) = (fh, gh) : h ∈ R is u-compact in Cb(R, E2). Now defineS : Cb(R, E)× Cb(R, E)→ Cb(R, E) by

S(u, v) = u+ v, u, v ∈ Cb(R, E).

This is a continuous function, hence S(H(F )) = H(f + g)) is u-compactin Cb(R, E).

We next consider completeness of the space (AP (R, E), u).


Theorem 11.2.11. [NGu84, KA11] Let E be an F -space. Then thevector space AP (R, E) is u-complete.

Proof. By Theorems 11.2.4. and 11.2.10, AP (R, E) is a vector space.Further, since each f ∈ AP (R, E) is bounded, AP (R, E) is a vectorsubspace of Cb(R, E). Since E is complete, by Theorem 11.2.1, Cb(R, E)is u-complete. So we need only to see that AP (R, E) is u-closed inCb(R, E).

Proof. Let f ∈ Cb(R, E) with f ∈ u-cl[AP (R, E)]. Then there exists

a sequence fn ⊆ AP (R, E) such that fnu−→ f . Since each fn is almost

periodic and fnu−→ f , by Theorem 11.2.3, f is almost periodic. Hence

f ∈ AP (R, E). Thus AP (R, E) is u-closed in Cb(R, E).

Remark. We mention that a continuous functions f : [a; b]→ E neednot be integrable in the Riemann sense, if E is not locally convex ([Rol85],p. 123). However, if E is a locally pseudoconvex F-space, then all an-alytic functions f : [a; b] → E are integrable in the Bochner-Lebesguesense ([Rol85], Theorem 3.5.2). Using a similar approach, almost peri-odicity of functions with values in p-Frechet spaces, 0 < p < 1, has beenconsidered in [GaNGu07]. This paper also contains some applications todifferential equations and to dynamical systems.



Section 11.1. A useful reference for the background of the resultsin this section is [FM91]; see also [Mcl65, Vain64]. These results aregiven in [Vain64] for E and F normed spaces and in [AS67] for E andF locally convex space. In the general case, similar results are given in[Llo73] without proof. The proofs in the present form are taken from[Kh97]. These include a shorter and direct proof of Theorem 11.5 basedon the separation form of the Hahn-Banach theorem (cf. [AS67]). Thereader is referred to [PRS01] for more recent versions of the theorem inthe TVS setting.

We now indicate further possible generalizations of the mean valuetheorem.

(I)

For set-valued mappings. In this case, two useful references arethe papers: (i) P.H. Sach: Differentiation of set-valued maps in Banachspaces, Math. Nachr. 139(1988), 215-235. (ii) P.H. Sach and N.Q. Lan,A mean value theorem for set-valued maps, Rev. Roumaine Math. PuresAppl. 36(1993), 359-368. These papers contain versions of the MeanValue Theorem in the Banach space setting and it may be possible toextend these results to the setting of topological vector spaces.

(II)

For Fuzzy mappings. A useful references in this direction is thepaper: M. Ferraro and D.H. Foster, Differentiation of fuzzy continu-ous mappings on fuzzy topological vector spaces, J. Math. Anal. Appl.,121(1987), 589-601. This paper does not contain any version of the MeanValue Theorem, but it seems likely that such a theorem can be formu-lated.

Section 11.2. The theory of almost periodic functions was mainlycreated and published during 1924-1926 by the Danish mathematicianHarold Bohr; Bohr’s work was preceded by the important investigationsof P. Bohl and E. Esclangon (see references in [Cor89, LeZh82]). The the-ory attracted the interest of a number of researchers because it has beeneffectively applied to solutions of diverse problems, initially in the the-ory of harmonic analysis and differential equations. In 1933, S. Bochner[Bo33] published an important article devoted to the extension of thetheory of almost periodic functions on the real line R with values in aBanach space E. His results were further developed by several mathe-maticians (see the monographs by L. Amerio and G. Prouse [AmPr71],C. Corduneanu [Cor89], B.M. Levitan and V.V. Zhikov [LeZh82], and


S. Zaidman [Za85]). The theory of almost periodic functions takingvalues in a complete metrizable locally convex space E has been initi-ated by G.M. N’Guerekata [NGu84] and further developed in [BuNgu04,GaNGu07, NGu01]. A survey paper by A.I. Shtern [Sh05] also considersalmost periodic functions and representations in locally convex spaces.

In this section, we have considered the concept of almost periodicityof functions having values in a topological vector space E, not neces-sarily locally convex. We are mainly concerned with studying the topo-logical properties of almost periodic functions and demonstrate the va-lidity of several known results, including some from [NGu84, BuNgu04][[GaNGu07] to this general setting, as given in [KA11].

CHAPTER 12

Non-Archimedean Function Spaces

Let X be a Hausdorff topological space. As mentioned earlier, thestrict topology was introduced for the first time by Buck in [4] on thespace of all bounded continuous functions on a locally compact space X .Several other authors have extended Buck’s results by taking as X anarbitrary completely regular space and considering spaces of continuousfunctions on X which are either real-valued or have values in a classicallocally convex space or even in a classical topological vector space. In thenon-Archimedean case some authors studied strict topologies on spaces ofcontinuous functions, on a zero-dimensional topological space, with val-ues either in a non-Archimedean valued field F or in a non-Archimedeanlocally convex space over F. In this chapter we will consider the casewhere the functions take their values in a non-Archimedean TVS.

Throughout this chapter, unless it is stated explicitly otherwise, X willbe a non-empty Hausdorff topological space, F a non-Archimedian non-trivially valued field (with unit element e) and E a non-trivial Haus-dorff TVS over F with a base W of neighborhoods of 0 in E consist-ing of balanced and absorbing sets. As before, we denote by C(X,E)the space of all continuous E-valued functions on X . By Cb(X,E)(resp. Co(X,E), Coo(X,E), Crc(X,E)), we will denote the space of allf ∈ C(X,E) which are bounded (resp. vanish at infinity, vanish outsidesome compact set, have relatively compact range) in E. In case E = F,we will write C(X,F), Cb(X,F), Co(X,F), Coo(X,F) and Crc(X,F), re-spectively.

205

206 12. NON-ARCHIMEDEAN FUNCTION SPACES

1. Compact-open Topology on C(X,E)

In this section, we shall study the τK topology which is the non-Archimedian analogues of the usual compact open topology studied inearlier chapters.

First we recall some terminology (§ A.1, A.5). A topological space(X, τ) is called zero-dimensional if, for each x ∈ X and each neighbor-hood U of x, there exists a clopen (i.e., both closed and open) set V suchthat x ∈ V and V ⊆ U . The family of all clopen subsets of X will bedenoted by Clo(X). A valued field F is said to be non-trivially valuedif there is an α ∈ F such that 0 < |α| 6= 1. By the F-characteristicfunction of a set A ⊆ X , we mean the function χA : X → F defined byχA(x) = e if x ∈ A and χA(x) = 0 if x /∈ X (x ∈ X). Note that, forany ϕ ∈ C(X,F) and ε > 0, the set t ∈ X : |ϕ(t)| ≥ ε is clopen in Xsince it is the inverse image of the clopen set F\B(0, ε) under a continu-ous map. For any non-zero scalar λ ∈ F , we shall denote its inverse byλ−1 (instead of 1

λ). By S(X,F) we will denote the space spanned by the

functions χA, A ∈ Clo(X), while S(X,F)⊗E denotes the space spannedby the functions χA ⊗ a, A ∈ Clo(X) and a ∈ E.

Let (E, τ) be a topological vector space (in short, a TVS) over avalued field (F, | · |). A subset S of E is called a non-Archimedean set ifS + S ⊆ S. The pair (E, τ) is called a non-Archimedean TVS if it has abase of τneighborhoods of 0 consisting of non-Archimedean sets

. Every non-Archimedean neighborhood of 0 in a TVS is a clopen(i.e. both closed and open) set. Every non-Archimedean valued field(F, | · |) is a non-Archimedean TVS over itself. Further, any indiscreteTVS is a non-Archimedean TVS. In particular, the trivial TVS E = 0is a non-Archimedean TVS.

Recall that a subset A of X is said to be bounding if every continuousreal valued function on X is bounded on A. If every f ∈ C(X,F) isbounded on A, then A is said to be F-bounding. It is easy to see thatcontinuous images of bounding (resp. F-bounding ) sets are bounding(resp. F-bounding ). The space X is said to be pseudocompact (resp.F-pseudocompact), if every continuous real function on X (resp. everyf ∈ C(X,F)) is bounded.

Lemma 12.1.1 If X is zero-dimensional, then a subset A of X isbounding iff it is F-bounding.

Proof. Assume that A is not bounding and let g be a continuous realfunction on X which is not bounded on A. Then there exist a sequence(nk) of positive integers and a sequence (xk) in A such that nk < |g(xk)| <

1. COMPACT-OPEN TOPOLOGY ON C(X,E) 207

nk+1 for all k. For each k, there exists a clopen neighborhood Bk of xkcontained in x : nk < |g(x)| < nk+1. The set Bo = X \

⋃∞k=1Bk is open

since Bo =⋃∞k=1Gk, where

Gk = x : |g(x)| < nk+1\k⋃

i=1

Bi.

Hence the sets Bo, B1, . . . form a clopen partition of X . Choose λ ∈ F,|λ| > 1, and let h =

∑∞k=1 λ

kχBk.Then h ∈ C(X,F) and h(xk) = λk, and

so h is not bounded on A, which implies that A is not F-bounding. Nowthe result clearly follows.

Recall that a completely regular Hausdorff space Y is c-complete oruniversally complete iff Y is homeomorphic to a closed subspace of aproduct of metrizable topological spaces. If Y is c-complete, then everybounding subset of Y is relatively compact. The c-completion of Y is thesmallest c-complete subspace of the Stone-Cech compactification βY ofY , which contains Y , and it is denoted by θY .

Definition. A basic sequence in E is a sequence U = (Vn) of balancedabsorbing subsets of E such that

(1) Vn+1 + Vn+1 ⊆ Vn for all n.(2) For each n and each non-zero scalar λ, there exists an m such that

Vm ⊆ λVn.Every basic sequence U = (Vn) induces a pseudometrizable linear

topology τU on E for which the sequence (Vn) is a base at zero. If everyVn ∈ W, then the basic U is called topological. Given any such U , theset kerU = ∩Vn is a subspace of E. Let EU = E/ kerU be the quotientspace and π = πU the quotient map. Then π(U) = (π(Vn)) is a basicsequence in EU .

The following Theorem is contained in [BBNW75].

Theorem 12.1.2. Every Hausdorff topological vector space E overK is topologically isomorphic to a subspace of a product of metrizabletopological vector spaces.

Proof. We will only sketch the proof. Let F be the collection of alltopological basic sequences in E and let G = ΠU∈FEU . Then the mapϕ : E → G given by

ϕ(a)U = πU(a), a ∈ E,

is a linear homeomorphism between E and ϕ(E).

Remark. Using the preceding Theorem we get that E is completelyregular.


If E is the completion of E, then the family of all closures WE

inE, for closed W ∈ W, is a base at 0 in E. Using this fact, we get thefollowing

Lemma 12.1.3. If A ⊆ E, then A is totally bounded in E iff it istotally bounded in E.

Proof. It is clear that if A is totally bounded in E, then it is totallybounded in E. For the converse, suppose that A is totally bounded in Eand let W ∈ W be a closed. Choose another closed W1 ∈ W such thatW1 +W1 ⊆ W . Let V and V1 be the closures of W and W1 in E andlet S = e1, e2 . . . , en) be a finite subset of E such that A ⊆ S + V1. Fork = 1, . . . , n, choose ak ∈ E such that ek − ak ∈ V1. Given x ∈ A, thereexist k and z ∈ V1 such that

x = ek + z ∈ ak + V1 + V1 ⊆ ak + V.

Thus x− ak ∈ V ∩ E = W and so A ⊆ a1, a2 . . . , an+W .

Theorem 12.1.4 (1). If E is complete, then E is c-complete.(2). Every bounding subset of E is totally bounded.(3). If E is complete, then for a subset A of E the following are

equivalent:(a) A is bounding.(b) A is totally bounded.(c) A is relatively compact.Proof. (1). By Theorem 12.1.2, E is topologically isomorphic to a

subspace F of a product Πi∈IEi of metrizable topological vector spaces.The space F is closed in the product since E is complete. It follows thatE is c-complete.

(2). Let A be a bounding subset of E. Then A is a bounding subset of

the completion E of E. Since E is c-complete, the closure AEis compact

and hence totally bounded, which implies that A is totally bounded.(3). (a)⇒ (b) follows from (2).(b)⇒(c) follows since E is complete.(c)⇒(a) Suppose A is relatively compact. Then A is compact and

hence bounding, which implies that A is bounding.

Theorem 12.1.5 Assume that X is zero-dimensional. For a subsetA of X, the following are equivalent:

(1) A is bounding.(2) A is F-bounding.(3) For each f ∈ C(X,E), the set f(A) is totally bounded in E.(4) For each f ∈ C(X,E), the set f(A) is bounded in E.


Proof. (1)⇔(2) follows from Lemma 12.1.1.(1)⇒(3) follows since continuous images of bounding sets are bound-

ing and in view of the preceding Theorem 12.1.4.(3)⇒(4) is clear.(4)⇒(1) Assume that (4) holds and let ϕ ∈ C(X,F). Since E is

Hausdorff and non-trivial, there exists an u ∈ E and a closed balancedW ∈ W not containing u. Since the function f = ϕ ⊗ e is boundedon A, there exists a non-zero scalar λ such that ϕ(A)u ⊆ λW . If nowx ∈ A, then λ−1ϕ(x)u ∈ W and so |λ−1ϕ(x)| < 1 since W is balancedand u /∈ W . It follows that |ϕ(x)| < |λ| for all x ∈ A. Thus A isF-bounding.

Notation. For V ∈ W and A ⊆ X , we denote by

N(A, V ) : = N(χA, V ) = f ∈ C(X,E) : f(A) ⊆ V ,

Nb(A, V ) : = Nb(χA, V ) = N(A, V ) ∩ Cb(X,E).

Throughout, K denotes a family of compact subsets ofX which coversX and it is upwards directed by set inclusion. If A ∈ K is compact, theneach N(A, V ) is absorbing. Further, if V + V ⊆ W , then N(A, V ) +N(A, V ) ⊆ N(A,W ). We get now easily the following

Theorem 12.1.6. The family

ΩK,B = N(A, V ) : A ∈ K, V ∈ W

is a base at 0 in C(X,E) for a Hausdorff linear topology τK. MoreoverτK does not depend on the particular choice of B.

Note. (1) Let Ao ∈ K and Vo ∈ W. If (Vn) is a basic topologicalsequence in E such that V1+V1 ⊆ Vo, then (N(Ao, Vn)) is a basic sequencein C(X,E) and N(Ao, V1) +N(Ao, V1) ⊆ N(Ao, Vo).

(2) If K = K(X) is the family of all finite(resp. compact) subsets ofX , we will denote τK by τk (resp. τρ).

(3) The topology of uniform convergence τu on Cb(X,E) has a baseof neighborhood of 0 consisting of all sets of the form N(X, V ), V ∈ W .

Recall that a sequence (Bn), of subsets of of a TVS G, is called afundamental sequence of bounded sets, if each Bn is bounded and eachbounded subset of G is contained in some Bn.

Lemma 12.1.7 If (C(X,E), τK) has a fundamental sequence (Gn)of bounded sets, then E also has the same property.

Proof. For each A ∈ K and each n the set Gn(A) is bounded in E.In particular, since K covers X , each Gn(x), x ∈ X , is bounded in E.For a ∈ E, denote by a the function which is defined on X by a(x) = a.


Let Bn = a ∈ E : a ∈ Gn. For x ∈ X , we have that Bn ⊆ Gn(x)and so Bn is a bounded subset of E. Let B be any bounded subset ofE and G = a : a ∈ B. Since G(A) ⊆ B for every A ∈ K, it followsthat G is a bounded subset of (C(X,E), τK) and so G ⊆ Gn, for some n,which implies that B ⊆ Bn. Therefore (Bn) is a fundamental sequenceof bounded subsets of E.

Recall that a Hausdorff zero-dimensional topological space X is aP -space if every countable intersection of clopen subsets of X is clopen.

Theorem 12.1.8. If X is a zero-dimensional P -space and E se-quentially complete, then each of the spaces (C(X,E), τρ) and (C(X,E), τk)is sequentially complete.

Proof. (1). Let fn be a τρ-Cauchy sequence. Since E is sequentiallycomplete, the limit f(x) = lim fn(x) exists in E for each x. The functionf is continuous. Indeed, let x ∈ X and let V ∈ W be closed. Let V1 bethe interior of V and let m be such that

fn(x) ∈ f(x) + V1 for all n ≥ m.

For each n ≥ m, there exists a clopen neighborhood An of x such thatfn(An) ⊆ f(x) + V1. The set A =

⋂n≥mAn is clopen and

fn(y) ∈ f(x) + V for all y ∈ A and all n ≥ m.

It follows that f(y) ∈ f(x) + V for all y ∈ A, which proves that f iscontinuous at x.

(2). Suppose that fn is τk-Cauchy. Then fn is τρ-Cauchy andhence fn → f pointwise, for some f ∈ C(X,E). Given a closed V ∈ Wand a compact subset Y of X , there exists no such that

(fn − fm)(Y ) ⊆ V for all n,m ≥ no.

Since V is closed and lim fm(x) = f(y) for all y, it follows that (fn −

f)(Y ) ⊆ V for all n ≥ no, which proves that fnτk→ f .

Theorem 12.1.9. Let K be a compact subset of the zero-dimensionaltopological space X and let g be a continuous E-valued function on K.Then:

(1) Given a V ∈ W, there exists an f ∈ S(X,F)⊗E such that f(X)is a finite subset of g(K) and (g − f)(K) ⊆ V .

(2) If E is complete and metrizable, then there exists a continuousE-valued function f on X which is an extension of g and whose imagein E is relatively compact.


Proof. (1). For each x ∈ K, there exists a clopen neighborhood Axof x in K such that

g(y)− g(x) ∈ V for all y ∈ Ax.

By the compactness of K, there are x1, x2, . . . , xn in K such that K =∪nk=1Axk . Let

B1 = Ax1, B2 = Ax2 \B1, . . . , Bn = Axn \ ∪n−1k=1Bk.

For each k there exists a clopen subset Zk of X such that Zk ∩K = Bk.Replacing Zk by Zk \ ∪j 6=kZj, we may assume that the sets Z1, . . . , Znare pairwise disjoint. Now it suffices to take f =

∑nk=1 χZk

g(xk).(2). Assume that E is metrizable and complete and let λ ∈ F, 0 <

|λ| < 1. There exists a base (Vn) of closed balanced neighborhoods of 0in E such that Vn+1+ Vn+1 ⊆ Vn and Vn+1 ⊆ λVn for all n. By (1), thereexists a function f1 ∈ S(X,F)⊗ E such that

f1(X) ⊆ g(K) and (g − g1)(K) ⊆ V2, where g1 = f1|K .

Applying (1) to the function g − g1, we get a function f2 ∈ S(X,F)⊗Esuch that

f2(X) ⊆ (g − g1(K) and (g − g1 − g2)(K) ⊆ V3, where g2 = f2|K .

Continuing by induction, we get a sequence fn ∈ S(X,F) ⊗ E suchthat, for gk = fk|K , we have that

(g −n∑

k=1

gk)(K) ⊆ Vn+1 and fn(X) ⊆ (g −n−1∑

k=1

gk)(K) for all n.

Claim I The series∑∞

n=1 fn converges uniformly to a continuousfunction f on X . Indeed, let hn =

∑nk=1 fk and let V ∈ W closed

There exists an m such that Vm ⊆ V . For m < n < N , we have thathN − hn =

∑Nk=n+1 fk. But, for k > n, we have

fk(X) ⊆

(g −

k−1∑

i=1

gi

)(K) ⊆ Vk

and so

(hN − hn)(X) ⊆ Vn+1 + Vn+2 + . . .+ VN ⊆ Vn ⊆ V.

Thus the sequence hn is τu-Cauchy in Cb(X,E) and hence it convergesuniformly to a continuous function f ∈ Cb(X,E). It is easy to see thatf(X) is totally bounded in E and hence relatively compact.


Claim II f = g on K. Indeed, let y ∈ K and let V ∈ W be closedand balanced. There exists an m such that Vm ⊆ V . Let n ≥ m. Then

g(y)−n∑

k=1

fk(y) ∈ Vn+1 ⊆ V.

Since V is closed, we get that g(y)− f(y) ∈ V . This, being true for allclosed balanced neighborhoods V of 0 in E, implies that g(y) = f(y)since E is Hausdorff. Hence the result follows.

Corollary 12.1.10. If X is zero-dimensional, then S(X,F)⊗ E isτk-dense in C(X,E).

Theorem 12.1.11. Assume that X is zero-dimensional and let K bean upwards directed family of compact subsets of X covering X. Then(C(X,E), τK) is complete iff E is complete and every f ∈ F (X,E) suchthat f |A is continuous, for each A ∈ K, is continuous on X.

Proof. (⇒) Suppose that (C(X,E), τK) is complete. Since K coversX , it follows easily that E is complete. Let f ∈ F (X,E) be such thatf |A is continuous, for each A ∈ K. Consider the the set F of all pairs(A, V ), where A ∈ K and V ∈ W closed and balanced. We make F intoa directed set by defining

(A1, V1) ≥ (A2, V2) iff A2 ⊆ A1 and V1 ⊆ V2.

Given α = (A, V ) ∈ F , there exists (by Theorem 12.1.9) a function gα =gA,V in C(X,E) such that (f − gα)(A) ⊆ V . The net (gα)α∈F is Cauchyin (C(X,E), τK). Indeed, let V ∈ W be balanced and Ao ∈ K. Choose aclosed balanced Vo ∈ W such that Vo + Vo ⊆ V . Let αo = (Ao, Vo) andlet αi = (Ai, Vi), i = 1, 2, αi ≥ αo. Then (f − gαi

)(Ao) ⊆ Vi ⊆ Vo and so

(gα1− gα2

)(Ao) ⊆ Vo + Vo ⊆ V.

This proves that the net (gα)α∈F is Cauchy in (C(X,E), τK) and henceit converges to some g ∈ C(X,E). We will show that g = f . Indeed, letxo ∈ X and let V ∈ W. Choose a closed balanced Vo ∈ W such thatVo + Vo ⊆ V . Let Ao ∈ K containing xo. There exist A1 ∈ K containingAo and a closed balanced V1 ∈ W contained in Vo such that

(g − gα)(Ao) ⊆ Vo for all α ≥ α1 = (A1, V1).

Thus

g(xo)− gα1(xo) ⊆ (g − gα1

)(Ao) ⊆ Vo.

Also

f(xo)− gα1(xo) ⊆ (f − gα1

)(A1) ⊆ V1 ⊆ Vo


and hence g(xo)− f(xo) ∈ Vo − Vo ⊆ V . It follows that f = g and so fis continuous.

(⇐) Suppose that E is complete and every f ∈ F (X,E) is con-tinuous if f |A is continuous for all A ∈ K. Let (fα) be a Cauchy net in(C(X,E), τK). For each x ∈ X , the net fα(x) is Cauchy in E and henceconvergent. Define f(x) = lim fα(x). If V ∈ W is a closed balanced andAo ∈ K, there exists αo such that (fα1

−fα2)(Ao) ⊆ V for all α1, α2 ≥ αo.

Thus (fα − f)(Ao) ⊆ V for all α ≥ αo, which proves that fα → f uni-formly on Ao. This implies that f |A is continuous for each A ∈ K andhence f is continuous, Moreover, fα → f in (C(X,E), τK).

Corollary 12.1.12. Suppose that X is zero-dimensional. Then(C(X,E), τρ) is complete iff E is complete and X discrete.

Proof. (⇒) In view of the preceding Theorem, if E is complete andX discrete, then (C(X,E), τρ) is complete.

(⇐) Conversely, suppose that (C(X,E), τρ) is complete. Then E iscomplete. Let A ⊆ X and a a non-zero element of E. Let f = χAa. Forevery finite subset B of X , f |B is continuous and hence f is continuousby the Theorem 12.1.11. The set A is closed. Indeed let x ∈ A \ A andlet xα be a net in A converging to x. Then f(xα)→ f(x) = 0. Thereexists an open V ∈ W not containing a. Since f(x) = 0 ∈ V , there existsa α such that a = f(xα) ∈ V , a contradiction. Thus every subset of Xis closed and hence X is discrete.


2. Strict Topology on Cb(X,E)

In this section, we shall study the τP topology which is the non-Archimedian analogues of the usual strict topology studied in earlierchapters.

For ϕ ∈ F (X,F) and V ∈ W, we set

N(ϕ, V ) = f ∈ Cb(X,E) : (ϕf)(X) ⊆ V .

It is easy to see that:(1) If ϕ ∈ B(X,F), then N(ϕ, V ) is absorbing in Cb(X,E).(2) For λ a non-zero scalar, we have γN(ϕ, V ) = N(λ−1ϕ, V ) =

N(ϕ, λV ).(3) If V is balanced, then N(ϕ, V ) is balanced.(4) If V ⊆ V1 ∩ V2 and |ϕ| ≥ max|ϕ1|, |ϕ2| and if V is balanced,

then

N(ϕ, V ) ⊆ N(ϕ1, V1)⋂

N(ϕ2, V2).

(5) If V is convex, then N(ϕ, V ) is convex.Let now P be an upwards directed family of compact subsets of X

covering X . We denote by BoP(X,F) the collection of all bounded func-tions ϕ ∈ F (X,F) which P-vanish at infinity, i.e. for each ε > 0 thereexists an A ∈ P such that |ϕ(x)| < ε if x /∈ A. We denote by βP the lineartopology on Cb(X,E) for which the family of all N(ϕ, V ), ϕ ∈ BoP(X,F)and V ∈ W, is a base at 0. It follows easily that βP has as a base at0 the family of all N(ϕ, V ), with V ∈ W and ϕ ∈ BoP(X,F), ‖ϕ‖ ≤ 1.In case P is the family of all compact (resp. all finite) subsets of X , wewill denote the corresponding βP by βo (resp. βs). Let τu denote thetopology of uniform convergence on Cb(X,E).

Theorem 12.2.1. (1) τρ 6 τP 6 βP 6 τu on Cb(X,E).(2) τu, βP and βs have the same bounded sets.(3) τP and βP coincide on τu-bounded sets.(4) A sequence fn in Cb(X,E) is βP-convergent to some f iff it is

τu-bounded and fnτP→ f .

(5) If (Cb(X,E), βP) is bornological, then βP = τu.(6) If X is zero-dimensional, then(a) τu = βP iff X ∈ P .(b) τP = βP iff, for each sequence (An) in P , there exists A in P

which contains every An.(c) Coo(X,E) is βo-dense in Cb(X,E) iff X is locally compact.Proof. (1). Since P covers X and is upwards-directed, it follows

easily that τρ is coarser than τP . Also τP is coarser than βP since the

2. STRICT TOPOLOGY ON Cb(X,E) 215

F-characteristic function of any A ∈ P belongs to BoP(X,F). Finally, letV ∈ W be balanced and ϕ ∈ P, ‖ϕ‖ ≤ 1. Then

f ∈ Cb(X,E) : f(X) ⊆ V ⊆ N(ϕ, V ),

which proves that βP is coarser than τu.(2). Assume that there exists a subset G of Cb(X,E) which is βs-

bounded but not τu-bounded. Then there exists a balanced V ∈ W suchthat G is not absorbed by the set W = f ∈ Cb(X,E) : f(X) ⊆ V .Let λ ∈ F, |λ| > 1. Suppose that we have already chosen f1, f2, . . . , fnin G and distinct x1, x2, . . . , xn in X such that fk(xk) /∈ λ2kV , for k =1, . . . , n. The set D =

⋃nk=1 fk(X) is bounded in E. Hence there exists

|γ| > |λ|2(n+1) such that D ⊆ γV and G ⊆ γ · Nb(Z, V ) where Z =x1, x2, . . . , xn and

Nb(Z, V ) = f ∈ Cb(X,E) : f(Z) ⊆ V .

Since G is not absorbed by Nb(X, V ), there exists fn+1 in G which isnot in γNb(X, V ) and hence γ−1fn+1(xn+1) /∈ V , for some xn+1. Butthen xn+1 /∈ Z. Thus, we get by induction a sequence fn in G and asequence xn of distinct elements of X such that

fn(xn) /∈ λ2nV for all n.

Now define ϕ ∈ F (X,F) by ϕ(xn) = λ−n and ϕ(x) = 0 if x /∈ x1, x2, . . ..If S is the family of all finite subsets of X , then ϕ ∈ BoS(X,F) and soG ⊆ αN(ϕ, V ) for some non-zero scalar α. Let n be such that |λ|n > |α|.Now

α−1λ−nfn(xn) = α−1fn(xn)ϕ(xn) ∈ V,

and so fn(xn) ∈ αλnV ⊆ λ2nV , a contradiction. This clearly completesthe proof of (2).

(3). Let G be a τu-bounded set, fo ∈ G, ϕ ∈ BoP(X,F), with ‖ϕ‖ ≤1, and V ∈ W be balanced. There exists a balanced V1 ∈ W withV1 + V1 ⊆ V . Since G is τu-bounded, there exists a non-zero scalar λsuch that G(X) ⊆ λV1. Choose A ∈ P such that |ϕ| < |λ|−1 on X \ A.Now

[fo +Nb(A, V1)]⋂

G ⊆ fo +N(ϕ, V ).

Indeed, let f ∈ G be such that f − fo ∈ Nb(A, V1). If x ∈ A, thenf(x) − fo(x) ∈ V1 and so ϕ(x)[f(x) − fo(x)] ∈ V1 ⊆ V . If x /∈ A, then|ϕ(x)| < |λ|−1 and hence

ϕ(x)[f(x)− fo(x)] ∈ λ−1λV1 − λ

−1λV1 ⊆ V.

Thus f ∈ fo +N(ϕ, V ), which clearly proves that τP = βP on G.


(4) (⇒) If fn is βP-convergent, then it is βP-bounded and henceτu-bounded. Also fn is clearly τP-convergent.

(⇐)This follows from (3).(5) It follows from the fact that βP is coarser than τu and the topolo-

gies βP and τu have the same bounded sets.

(6). Suppose that X is zero-dimensional.(a). (⇒) If X ∈ P, then the constant function ϕ(x) = 1, for all

x ∈ X , is in BoP(X,F), and f ∈ Cb(X,E) : f(X) ⊆ V = N(ϕ, V ).Thus τu is coarser than βP and so τu = βP .

(⇐) Assume that τu = βP and let V ∈ W be a proper balancedset (such a neighborhood exists since E is Hausdorff and non-trivial).Choose a balanced V1 ∈ W and ϕ ∈ BoP(X,F), 0 < ‖ϕ‖ ≤ 1, such thatN(ϕ, V1) ⊆ Nb(X, V ). Let a ∈ E \V and choose a non-zero scalar λ suchthat a ∈ λV1. There exists A ∈ P such that |ϕ(x)| < |λ|−1 if x /∈ A.Now A = X . Indeed, suppose that there exists an x ∈ X \A. Since X iszero-dimensional, there exists a clopen neighborhood Z of x disjoint fromA. If f = χZa, then f ∈ N(ϕ, V1) but f /∈ Nb(X, V ), a contradiction.

(b) (⇒)Suppose that βP = τP and let 0 < |λ| < 1. Let (An) be asequence in P and ϕ =

∑∞n=1 λ

nχAn. Then ϕ ∈ BoP(X,F). Let V ∈ Wbe proper and balanced and a ∈ E \ V . There exists a balanced V1 ∈ Wand A ∈ P such that Nb(A, V1) ⊆ N(ϕ, V ). Now assume that there existsan x ∈ (∪An) \ A and let n be the smallest integer such that x ∈ An.Then |ϕ(x)| = |λ|n. Choose a clopen neighborhood Z of x disjoint fromA and let f = λ−nχZa. Clearly f ∈ Nb(A, V1) but f /∈ N(ϕ, V ) since|ϕ(x)λ−n| = 1 and a /∈ V . This proves that

⋃An ⊆ A.

(⇐) Assume that the condition is satisfied and let V ∈ W be balancedand ϕ ∈ BoP(X,F), with ‖ϕ‖ ≤ 1. Let 0 < |λ| < 1 and choose, for eachpositive integer n, an An ∈ P such that |ϕ(x)| < |λ|n when x /∈ An.By our hypothesis, there exists an A ∈ P containing each An. NowNb(A, V ) ⊆ N(ϕ, V ). In fact, let f ∈ Nb(A, V ) and x ∈ X . If ϕ(x) 6= 0,then |ϕ(x)| ≥ |λ|n, for some n, and so x ∈ An ⊆ A, which implies thatf(x) ∈ V and thus ϕ(x)f(x) ∈ V .

(c) (⇒) Assume that X is locally compact and let V ∈ W be bal-anced, ϕ ∈ BoP(X,F), with ‖ϕ‖ ≤ 1, and f ∈ Cb(X,E). There exists anon-zero scalar λ such that f(X) ⊆ λV and an A ∈ P such that |ϕ(x)| <|λ|−1 when x /∈ A. Every y ∈ A has a compact clopen neighborhood. Bythe compactness of A, there exists a compact clopen set Z containing A.If g = χZf , then g ∈ Coo(X,E). Moreover, if g(y) 6= f(y), then g(y) = 0and y /∈ A, which implies that −ϕ(y)f(y) ∈ −λϕ(y)V ⊆ V , since V is


balanced. It follows that g − f ∈ N(ϕ, V ), which proves that Coo(X,E)is βP-dense in Cb(X,E).

(⇐) Assume that some x ∈ X has no compact neighborhood. Iff ∈ Coo(X,E), then the set F = y : f(y) 6= 0 is open. Also F iscontained in a compact set B ∈ P. Thus x /∈ F and so f(x) = 0.Choose any non-zero element a of E and let V ∈ W be balanced notcontaining a. The set D = Nb(x, V ) is a τρ-neighborhood of 0 and hencea βP -neighborhood of zero. Let f ∈ Cb(X,E), f(y) = a for all y. Ifg ∈ Coo(X,E), then g(x)− f(x) = −f(x) /∈ V and so g − f /∈ D, whichimplies that f is not in the βP-closure of Coo(X,E). Hence the resultfollows.

Theorem 12.2.2. Assume that X is zero-dimensional and let G =(Cb(X,E), βP). Then G is complete iff E is complete and every boundedf ∈ F (X,E) such that f |A is continuous, for each A ∈ P, is continuouson X.

Proof. (⇒) Suppose that G is complete. It is easy to see that E iscomplete. Let f ∈ F (X,E) be bounded and such that f |A is continuous, for each A ∈ P. Consider the family ∆ of all pairs (A, V ), where A ∈ Pand V ∈ W is balanced. We make ∆ into a directed set by defining(A1, V1 ≥ (A2, V2) iff A2 ⊆ A1 and V1 ⊆ V2. For each α = (A, V ) in ∆,there exists a gα ∈ S(X,F)⊗ E such that

gα(X) ⊆ f(A) and (f − gα)(A) ⊆ V.

The net gα is τu-bounded. Also the net is τP-Cauchy. Indeed, letAo ∈ P and V ∈ W be balanced. Choose a closed balanced Vo ∈ W suchthat Vo + Vo ⊆ V . If αi = (Ai, Vi) ≥ αo = (Ao, Vo), i = 1, 2, then

(f − gαi)(Ao) ⊆ (f − gαi

)(Ai) ⊆ Vi ⊆ V

and so (gα1− gα2

)(Ao) ⊆ Vo + Vo ⊆ V which proves that gα is τP -Cauchy. Let D = g ∈ Cb(X,E) : g(X) ⊆ f(X). Then F = D − Dis uniformly bounded. Let now N be a βP -neighborhood of zero. Thereexists a τP-neighborhood N1 of 0 such that N ∩ F = N1 ∩ F since βPcoincides with τP on F . There exists a αo such that

gα − gα′ ∈ N1 if α, α′ ≥ αo.

For such α, α′ ≥ αo we have that gα − gα′ ∈ N . This proves that thenet gα is βP-Cauchy and it converges to some g ∈ Cb(X,E). Nowg(x) = lim gα(x) for each x ∈ X . But lim gα(x) = f(x). Indeed, LetV ∈ W be balanced and choose another one Vo ∈ W which is closed andsuch that Vo + Vo ⊆ V . Let xo ∈ X and choose Ao ∈ P containing xo.Since gα converges to g with respect to the topology τP , there exists


a α1 = (A1, V1), where Ao ⊆ A1 and V1 ⊆ Vo, such that, for α ≥ α1, wehave (g − gα)(Ao) ⊆ Vo. Thus,

g(xo)− gα1(xo) ∈ (g − gα)(Ao) ⊆ Vo.

Also f(xo) − gα1(xo) ∈ Vo and so g(xo) − f(xo) ∈ Vo + Vo ⊆ V . Thus

f(xo) = g(xo) since E is Hausdorff. Thus f = g is continuous.(⇐) Suppose that E is complete and the condition is satisfied. Let

(fγ) be a Cauchy net inG. For each x ∈ X , the net (fγ(x)) is Cauchy in Eand hence convergent. Define f(x) = lim fγ(x). For each ϕ ∈ BoP(X,F),the function ϕf is bounded. In fact, let V ∈ W and choose another oneV1 ∈ W, which is closed and balanced, such that V1 + V1 ⊆ V . Thereexists a γo such that

ϕ(fγ − fγ′)(X) ⊆ V1 if γ, γ′ ≥ γo.

Since V1 is closed, we get that ϕ(fγo − f)(X) ⊆ V1. Since ϕ and fγo arebounded, there exists |λ| ≥ 1 such that (ϕfγo)(X) ⊆ λV1. It follows that

(ϕf)(X) ⊆ V1 + λV1 ⊆ λV,

which proves that ϕf is bounded. Next we show that f is bounded.Assume the contrary. Then, there exists a balanced V ∈ W which doesnot absorb f(X). Hence, there are 0 < |λ1| < |λ2| < . . ., with lim |λn| →∞, and a sequence (xn) such that

f(xn) ∈ λ2n+1V \ λ

2nV for all n.

Define ϕ on X by ϕ(xn) = λ−1n and ϕ(x) = 0 if x /∈ x1, x2, . . .. Thenϕ ∈ BoP(X,F) but ϕf is not bounded since ϕ(xn)f(xn /∈ λnV . This con-tradiction proves that f is bounded. Next we prove that f is continuous.In fact, let V ∈ W be closed balanced and A ∈ P. We get easily thatthere exists γo such that

(fγ − f)(A) ⊆ V for all γ ≥ γo.

Thus fγ(x) → f(x) uniformly on A and so f |A is continuous. This, byour hypothesis, implies that f is continuous on X . Moreover fγ → f inG.

Theorem 12.2.3. If X is zero-dimensional, then (Cb(X,E), βP) ismetrizable iff E is metrizable and X ∈ P .

Proof. This follows from Theorem 12.2.1 and the fact that a HausdorffTVS over F is metrizable iff it has a countable base at 0 [BBNW75] (seealso Section A.2).


Theorem 12.2.4. βP has as a base at 0 the family of all setsN(λn, An, V ) of the form

N(λn, An, V ) =

∞⋂

n=1

f ∈ Cb(X,E) : f(An) ⊆ λnV ,

where 0 < |λn| → ∞, (An) a sequence in P and V ∈ W.Proof. (⇒) Let N = N(λn, An, V ) be as in the Theorem. We may

assume that V is balanced. Also we may assume that (An) is increasingsince P is upwards directed. As 0 < |λn| → ∞, for each n there existsa scalar γn such that |γn| = min|λk| : k ≥ n. Now |γn| → ∞. Letn1 < n2, . . . be such that

|γ1| = . . . = |γn1| < |γn1+1| = . . . = |γn2

| < |γn2+1| = . . .

and consider N1 = N(γnk, Ank

, V ). Then N1 ⊆ N .. [Indeed, let f ∈N(γnk

, Ank, V ) and set no = 0. For each positive integer k, there exists a

j such that nj−1 < k ≤ nj . If x ∈ Ak, then x ∈ Anjand so f(x) ∈ γkV ⊆

λkV since |γk| = |γnj|. Thus N1 ⊆ N .] Let αk = γnk

and Bk = Ank. The

function ϕ =∑∞

k=1 α−1k χBk

belongs to BoP(X,F). Moreover N(ϕ, V ) ⊆W . Indeed, let f ∈ N(ϕ, V ) and let x ∈ Anj = Bj. Let k be the smallestpositive integer with x ∈ Bk. Then |ϕ(x)| = |αk|−1. Now ϕ(x)f(x) ∈V . Since |ϕ(x)| ≥ |αj|

−1, we have that f(x) ∈ αjV . This proves thatN(ϕ, V ) ⊆ N(γnk

, Ank, V ) ⊆ N(λn, An, V ) and so N(λn, An, V ) is a βP-

neighborhood of zero.(⇐) Let No be any βP-neighborhood of zero. There exist a balanced

neighborhood V ∈ W and ϕ ∈ BoP(X,F), ‖ϕ‖ < 1, such that N(ϕ, V ) ⊆Wo. Let 0 < |λ| < 1. For each n, there exists An ∈ P such that|ϕ(x)| < |λ|n if x /∈ An. Set N2 = N(λ−n+1, An, V ). We will finish theproof by showing that N2 ⊆ N(ϕ, V ). So let f ∈ N2. If ϕ(x) 6= 0, thenthere exists an n such that |λ|n < |ϕ(x)| ≤ |λ|n−1. Now x ∈ An and sof(x) ∈ λ−n+1V , which proves that ϕ(x)f(x) ∈ V since |ϕ(x)λ−n+1| ≤ 1.

Theorem 12.2.5. If N is a βP-neighborhood of zero, then thereexists a balanced V ∈ W such that, for each non-zero scalar λ, thereexist A ∈ P and γ ∈ K, γ 6= 0, such that

f ∈ Cb(X,E) : f(X) ⊆ λV, f(A) ⊆ γV ⊆ N .

Proof. By Theorem 12.2.4, there exists a balanced V ∈ W, an increas-ing sequence (An) in P and a sequence (λn) in F, with 0 < |λn| → ∞,such that N1 = N(λn, An, V ) ⊆ N . Let now λ be a non-zero scalar


and choose n such that |λk| > |λ| when k > n. If γ ∈ F is such that|γ| = min|λ1|, |λ2|, . . . , |λn|, then

f ∈ Cb)(X,E) : f(X) ⊆ λV, f(An) ⊆ γV ⊆ N .

Theorem 12.2.6. Let X, Y be Hausdorff topological spaces and leth : X → Y be a continuous function. Then the induced linear map

T = Th : Cb(Y,E)→ Cb(X,E), f 7→ f h,

is βo − βo continuous.Proof. Let N be a βo-neighborhood of 0 in Cb(X,E). In view of

Theorem 12.2.4, there exist a sequence (λn) of scalars, where 0 < |λn| →∞, a sequence (An) of compact subsets subsets of X and V ∈ W suchthat N1 = N(λn, An, V ) ⊆ N . Each Bn = h(An) is compact in Y .Moreover,

N2 =∞⋂

n=1

f ∈ Cb(Y,E) : f(Bn) ⊆ λnV ⊆ T−1(N1)

and so T is continuous.

Theorem 12.2.7. If X is a zero-dimensional P -space and E se-quentially complete, then (Cb(X,E), βo) is sequentially complete.

Proof. Let fn be a βo-Cauchy sequence in Cb(X,E). Then fn isβo-bounded and hence τu-bounded. Also, fn is τk-Cauchy and hence

there exists f ∈ C(X,E) such that fnτk→ f . We show first that f is

bounded. Assume the contrary and let |λ| > 1. There exist a closedbalanced V ∈ W and a sequence xn of distinct elements of X suchthat f(xn) /∈ λnV for all n. Choose a closed balanced V1 ∈ W such thatV1 + V1 ⊆ V . The function ϕ(xn) = λ−n and ϕ(x) = 0 if x /∈ xn : n =1, 2, . . . belongs to Bo(X,F). Hence there exists no such that

[ϕ(fn − fm)](X) ⊆ V1 for all n,m ≥ no,

and so

[ϕ(fn − f)](X) ⊆ V1 for all n ≥ no.

Since fn is uniformly bounded, there exists k ≥ no such that fn(X) ⊆λkV1 for all n. Now

f(xk) = (f(xk)− fk(xk)) + fk(xk) ∈ λkV1 + λkV1 ⊆ λkV,

a contradiction. Thus f is bounded. Since τk coincides with βo on τu-

bounded sets and since fnτk→ f , we get that fn

βo→ f .


Theorem 12.2.8. Let X be zero-dimensional and let G be anS(X,F)-submodule of Cb(X,E). Then G is βP-dense in Cb(X,E) iff, for each x ∈ X, the set G(x) = f(x) : f ∈ G is dense in E.

Proof. (⇒) Assume that G is βP-dense and let x ∈ X . Consider thelinear map x : Cb(X,E)→ E given by

x(f) = f(x), f ∈ Cb(X,E),

which is clearly βP-continuous. Thus

E = x(Cb(X,E)) = x(G) ⊆ x(G) = G(x)

and so G(x) is dense in E.(⇐) Suppose that the condition is satisfied and let N = N(ϕ, V ),

where V ∈ W is open and balanced and ϕ ∈ BoP(X,F). Given f ∈Cb(X,E), there exists a non-zero scalar λ such that f(X) ⊆ λV and‖ϕ‖ ≤ |λ|. Let K ∈ P be such that |ϕ(x)| < |λ|−1 if x /∈ K. For eachx ∈ K, choose gx ∈ G such that gx(x) − f(x) ∈ λ−1V . Since X is zerodimensional, there exists a clopen neighborhood Zx of x such that

gx(y)− f(y) ∈ λ−1V for all y ∈ Zx.

By the compactness of K, there are x1, x2 . . . , xn ∈ K such that K ⊆⋃nk=1Zxk . Let

D1 = Zx1 and Dk = Zxk \k−1⋃

j=1

Dj for k = 2, 3, . . . , n.

Each hk = χDkgxk belongs to G and so g =

∑nk=1 hk ∈ G. Now g − f ∈

N(ϕ, V ). Indeed if x /∈⋃nk=1Dk, then g(x) = 0 and |ϕ(x)| < |λ|−1 which

implies thatϕ(x)[g)(x)− f(x)] = −ϕ(x)f(x) ∈ V

because V is balanced. For x ∈ Dk we have

ϕ(x)[g(x)− f(x)] = ϕ(x)[gxk(x)− f(x)] ∈ V .

Corollary 12.2.9. If X is zero-dimensional, then S(X,F)⊗E andCrc(X,E) are βP-dense in Cb(X,E).


3. Maximal Ideals in NA Function Algebras

In this section, we give a characterizations of maximal and β-closedideals in the algebra Cb(X,E) (k-closed ideals in the algebra C(X,E))when E is a topological algebra.

Recall that

the strict topology β on Cb(X,E) is the linear topology which has abase of neighborhoods of 0 in Cb(X,E) consisting of all sets of the form

N(ϕ,W ) = f ∈ Cb(X,E) : (ϕf)(X) ⊆W,

where ϕ ∈ Bo(X,F) and W ∈ W. It is shown in ([Kat84a], Proposition2.5) that, in case X is locally compact and 0-dimensional, the abovedefinition of strict topology is equivalent to the one given in ([Pro77], p.198-199), where ϕ is assumed to vary over Co(X,F) instead of Bo(X,F).

We first consider a version of the Stone-Werierstrass theorem forCb(X,F)-submodules in Cb(X,E).

Theorem 12.3.1. [KKK11] Assume that either X or E is 0-dimen-sional (in particular, E is a non-Archimedean TVS), and let A be avector subspace of Cb(X,E) which is a Cb(X,F)-module. If, for eachx ∈ X , A(x) is dense in E, then A is β-dense in Cb(X,E).

Proof. Let f ∈ Cb(X,E), and let ϕ ∈ Bo(X,F) and W an openbalanced neighborhood of 0 in E. Choose λ ∈ F with ||ϕ|| < |λ| andf(X) ⊆ λW . There exists a compact set K of X such that |ϕ(x)| < |λ|−1

for x /∈ K. For each x ∈ K, there is a gx ∈ A such that

f(x)− gx(x) ∈ λ−1W.

Let

G(x) = y ∈ X : f(y)− gx(y) ∈ λ−1W = (f − gx)

−1(λ−1W ).

If E is 0-dimensional, we may assume that W is clopen, and thereforeG(x) is also clopen in X . If X is 0-dimensional, there exists clopenneighborhood N(x) of x such that N(X) ⊆ G(x). We see that in anycase, there exists a clopen neighborhood Vx of x such that

Vx ⊆ G(x) = y ∈ X : f(y)− gx(y) ∈ λ−1W.

By the compactness of K, there are x1, x2, ..., xn in Y such that K ⊆∪nk=1Vxk . Let

D1 = Vx1 and Dxk+1= Vxk+1

\ ∪ni=1 Vxi .

The sets D1, ..., Dn are pairwise disjoint and clopen. Let ϕk be the F-characteristic function of Dk. The function g =

∑nk=1 ϕkgxk belongs to

3. MAXIMAL IDEALS IN NA FUNCTION ALGEBRAS 223

A. Moreover, [ϕ(f − g)](X) ⊆ W . [Indeed, let x ∈ X. If x ∈ Dk, theng(x) = gxk(x) and so

ϕ(x)[f(x)− g(x)] ∈ ϕ(x)λ−1W ⊆W.

If x /∈ ∪nk=1Dk, then g(x) = 0 and x /∈ Y . Thus

ϕ(x)[f(x)− g(x)] = ϕ(x)f(x) ∈ ϕ(x)λW ⊆W.]

Thus f ∈ Aβ.

A useful equivalent form of the above is the following theorem whichcharacterizes β-closed submodules in Cb(X,E).

Theorem 12.3.2. [KKK11] Assume that either X or E is 0-dimensional (in particular, E is a non-Archimedean TVS), and let Abe a vector subspace of Cb(X,E) which is a Cb(X,F)-module. Then, for

any f ∈ Cb(X,E), f ∈ Aβiff f(x) ∈ A(x) for all x ∈ X . In particular,

A is β-closed iff, for each x ∈ X , f(x) ∈ A(x) for all f ∈ A.

Proof. Assume that f ∈ Aβ, and let x ∈ X . If ϕ is the characteristic

function of x, then ϕ ∈ Bo(X,F) and so there exists a g ∈ A such that:

f(x)− g(x) ∈ [ϕ(f − g)](x) ⊆W ,

which implies that f(x) ∈ A(x). The converse part follows from Theorem12.3.1.

Recall that if E is an associative algebra over a valued field F and τa topology on E, the pair (E, τ) is called a topological algebra if (E, τ)is a TVS and the multiplication (x, y) → xy from E × E into E isjointly continuous (i.e. for any τneighborhood W of 0 in E, there existsa τneighborhood H of 0 in E such that H2 ⊆ W, where H2 = xy : x, y ∈H). (E, τ) is called topological simple if E has no two-sided closed idealsother than 0 and E. It is easy to verify that if (E, τ) is a topologicalalgebra, then (Cb(X,E), β) is also a topological algebra.

Theorem 12.3.3. [KKK11] Let E be a topological algebra with iden-tity over F. Assume that either X or E is 0-dimensional (in particular,E is non-Archimedean) Let Ω be the set of all pairs α = (N,W ), whereN is a non-empty closed subset of X and W a proper closed two-sidedideal in E. For αi = (Ni,Wi), i = 1, 2, define α1 6 α2 iff N2 ⊆ N1 andW1 ⊆ W2. Also let Jα = S(N,W ) = f ∈ Cb(X,E) : f(N) ⊆ W forα = (N,W ). Then:

(a) Each Jα is a proper two-sided ideal in Cb(X,E) which is ρ-closedand hence β-closed.


(b) Jα1⊆ Jα2

iff α1 6 α2. Thus the map α→ Jα is one-to-one andorder preserving.

(c) A β-closed two-sided ideal J in Cb(X,E) is maximal iff thereexist a unique xo ∈ X and a unique maximal closed two-sided ideal Wo

in E such that J = S(xo,Wo).(d) If E is topologically simple and J is a β-closed two-sided ideal

in Cb(X,E), then:(i) There exists a closed subset N of X such that

J = I(N) = f ∈ Cb(X,E) : f(N) = 0.

(ii) J is maximal iff there exists a (unique) x ∈ X such that J =I(x).

Proof. (a) It is easy to see that Jα is a two-sided ideal in Cb(X,E)which is a-closed since W is closed. Moreover Jα is a proper subset ofCb(X,E). Indeed let a ∈ E\W . Then the constant function f(x) = a isnot in Jα.

(b) It is clear that Jα1⊆ Jα2

when α1 6 α2. Conversely, supposethat Jα1

⊆ Jα2. If x ∈ N2\N1, then there exists a clopen neighborhood

V of x which is disjoint from N1. Let a ∈ E\W2 and consider thefunction g = χA ⊗ a, where χA is the F-characteristic function of V .Then g ∈ Jα1

\Jα2, a contradiction. Thus N2 ⊆ N1. Also if b ∈ W1\W2,

then the constant function g(x) = b is in Jα1\Jα2

, a contradiction. ThusW1⊆W2 and (b) follows.

(c) We first note that J is a Cb(X,F)-module. Indeed, let f ∈ J andh ∈ Cb(X,F). If u is the identity of E, then the function g = h⊗u belongsto Cb(X,E) and so gf ∈ J . Since gf = hf , our claim follows. Now

suppose that J is maximal and let f ∈ Cb(X,E)\J . Since f /∈ J = Jβ,

there exists, by Theorem

12.3.2, xo ∈ X such that f(xo) /∈ J (xo) = Wo. Since Wo is a properideal and J ⊆ S(xo,Wo), we have that J = S(xo,Wo). The ideal Wo

is maximal. [Indeed, let W1 be a proper closed two-sided ideal whichcontains Wo. Then J ⊆ S(xo,W1) 6= Cb(X,E), and so S(xo,Wo) =S(xo,W1), which implies that W1 = Wo.] The uniqueness of xo and Wo

follows from (b).Conversely, suppose that J = J(xo,Wo), where Wo is a maximal closed

ideal in E and let J1 be a proper closed two-sided ideal in Cb(X,E)

which contains J . Let f ∈ Cb(X,E)\J1. Since f /∈ J1 = J1β, there

exists, by Theorem 12.3.2, y ∈ X such that f(y) /∈ J1(y) = W . NowJ(xo,Wo) ⊆ J1 ⊆ J(y,W ). But then Wo⊆W and y = xo (by (b)). Since Wo

3. MAXIMAL IDEALS IN NA FUNCTION ALGEBRAS 225

is maximal, we have that Wo = W and so J =J1 = J(y,W ), which provesthat J is maximal.

(d) Suppose that E is topological simple. Then:(i) Let J be a β-closed two-sided ideal in Cb(X,E). Then J is a

Cb(X,F)-module. Define

N = x ∈ X : f(x) = 0 for all f ∈ J .

Then N is closed in X. Further, J ⊆ I(N). We now show that I(N) ⊆J . [Let g ∈ I(N), but g /∈ J . Since J is β-closed, by Theorem 12.3.2,

J = Jβ= f ∈ Cb(X,E) : f(x) ∈ J (x) for all x ∈ X;

so there exists an xo ∈ X such that g(xo) /∈ J (xo). Therefore J (xo) 6= E.

Since E is simple and J (xo) is a two-sided ideal in E, we have J (xo) =0. In particular, f(xo) = 0 for all f ∈ J and so xo ∈ N . But

g(xo) /∈ J (xo) and so g(xo) 6= 0 (as 0 ∈ J (xo)). This is a contradiction,since g ∈ I(N) and xo ∈ N imply that g(xo) = 0. Therefore I(N) ⊆ J .]Thus J = I(N).

(ii) It follows from (c) since the only maximal closed ideal in E is0.

Remark. Some results related to the above ones for the ”compact-open topology” on C(X,E), due to Prolla ([Pro82], p. 218-222), may bestated without proof as follows:

Theorem 12.3.4. [Pro82]Assume that either X or E is 0-dimensional(in particular, E is a non-Archimedean TVS), and let A be a vectorsubspace of C(X,E) which is a C(X,F)-module. Then, for any f ∈

C(X,E), f ∈ Akiff f(x) ∈ A(x), for all x ∈ X .

Theorem 12.3.5. [Pro82] Let E be a topological algebra with identityover F. Assume that either X or E is 0-dimensional (in particular, Eis non-Archimedean). Then:

(a) For any non-empty closed subset N of X and a proper closedtwo-sided ideal W in E, S(N,W ) is a proper two-sided ideal in C(X,E)which is k-closed.

(b) A k-closed two-sided ideal J in C(X,E) is maximal iff there exista unique xo ∈ X and a unique maximal closed two-sided ideal Wo in Esuch that

J = S(xo,Wo) = f ∈ C(X,E) : f(xo) ∈ Wo.

(c) If E is simple and J is a k-closed two-sided ideal in C(X,E),then:


(i) There exists a closed subset N of X such that

J = I(N) = f ∈ C(X,E) : f(N) = 0.

(ii) J is maximal iff there exists a (unique) x ∈ X such that J =I(x).



Section 12.1. In this section, a generalized form of compact-opentopology τK is considered on the non-Archimedean function space C(X,E).Earlier authors assumed X a zero-dimensional topological space and Ea non-Archimedean valued field F or a non-Archimedean LCS over F.

In fact, the first result on the Weierstrass type approximation theoremin the non-Archimedean area was proved by Dieudonne [Dieu44] in 1944for C(X,F), where F is a p-adic field. This result was extended in 1950by I. Kaplansky [Kap50] to the case of F a non-Archimedean valued field(i.e. a rank one valued field). These results were completed by Chernoff,Rasala and Waterhouse [CRW68] in 1968 who extended them to any fieldF with a krull valuation (or Archimedean valuations other than the usualabsolute value of C). When E is a non-Archimedean normed space, Prolla[Pro77, Pro78] obtained a Stone-Weierstrass theorem which was laterextended by Carneiro [Car79] to weighted function spaces CV (X,E).In his work, Carneiro considered locally F-convex spaces E over a localfield (F, | · |), i.e. over a locally compact valued field (F, | · |). Later,Soares [So80 ], Prolla [Pro82] and Prolla and Verdoodt [PV97] obtainedfurther generalizations where the field (F, |·|) is not assumed to be locallycompact.

Here we have considered the case where the functions take their val-ues in a TVS over a non-Archimedian field F. The main results dealwith completeness, completion, extension of function and denseness ofsubspaces, as given in [Kat11].

Section 12.2. This section makes a parallel study of the strict topol-ogy βP on Cb(X,E) assuming X a zero-dimensional topological space andE a non-Archimedean TVS, similar to that in Section 12.1, as given in[Kat11].

As mentioned earlier, the strict topology was introduced for the firsttime by Buck in 1958 on the space of all bounded continuous functions ona locally compact space X . Several other authors have extended Buck’sresults by taking as X an arbitrary completely regular space and consid-ering spaces of continuous functions on X which are either real-valuedor have values in a classical locally convex space or even in a classicaltopological vector space. In the non-Archimedean case some authorsstudied strict topologies on spaces of continuous functions, on a zero-dimensional topological space, with values either in a non-Archimedeanvalued field F or in a non-Archimedean locally convex space over F. In


this paper we will consider the case where the functions take their valuesin a non-Archimedean linear topological space.

Section 12.3. In this section, we first consider a Stone-Weierstrasstype theorem in the non-Archimedean setting for the spaces (Cb(X,E), β)and (C(X,E), k), as given in [KKK11, Pro82].]. As applications, we givea characterizations of maximal and closed ideals in the algebra when Eis a topological algebra.

APPENDIX A

Topology and Functional Analysis

229

230 A. TOPOLOGY AND FUNCTIONAL ANALYSIS

1. Topological Spaces

In this section we present some basic definitions and results on topo-logical spaces.

Definition. A topology on a non-empty set X is a collection τ ofsubsets of X having the following properties:

(i) X and the empty set ∅ belong to τ .(ii) The union of an arbitrary number of sets in τ belongs to τ .(iii) The intersection of a finite number of sets in τ belongs to τ .In this case (X, τ) is called a topological space and each set U in τ

is called an open set. A subset A of X is said to be closed set if itscomplement X\A is an open set. The closure of a set A in X , denotedby A (or cl(A)), is defined as the intersection of all closed set containingA (i.e., A is the smallest closed set containing A). The interior of a setA in X , denoted by A0 (or int(A)), is defined as the union of all open setcontained in A (i.e., A0 is the largest open set contained in A). Further,A is open iff A = A0; A is closed iff A = A. A is said to be dense in Xif A = X .

Definition. Let (X, τ) be a topological space.(1) For any x ∈ X, a subset V of X is called a neighborhood of

x if there exists an open set W such that x ∈ W ⊆ V. In particular,every open set U containing x is called an open neighborhood of x. Thecollection Ux of all neighborhoods of x is called the neighborhood systemat x.

(2) A base of neighborhoods of x (or a local base) at x is a subcollectionBx of Ux such that, for each U ∈ Ux, there exists some V ∈ Bx such thatV ⊆ U ; in this case, the elements of Bx are called basic neighborhoods ofx.

Theorem A.1.1. Let (X, τ) be a topological space. For any x ∈ X,let Bx be a base of neighborhoods of x. Then:

(NB1) x ∈ U for all U ∈ Bx.(NB2) If U, V ∈ Ux, there is some W ∈ Bx such that W ⊆ U ∩ V .(NB3) If V ∈ Bx, there is some Vo ∈ Bx such that if y ∈ Vo, there is

some W ∈ By with W ⊆ V (i.e. V contains a basic neighborhood of eachy ∈ Vo),

Furthermore,(NB4) G ⊆ X is open iff G contains a basic neighborhood of each of

its points.

1. TOPOLOGICAL SPACES 231

Conversely, given a set X and that for each x ∈ X is given a non-empty collection Bx of subsets of X satisfying (NB1)−(NB3) and if wedefine ”open” using (NB4), there exists a unique topology µ on X suchthat Bx is a base of µneighborhoods of x for each x ∈ X.

Definition. Let (X, τ) be a topological space.(1) A subcollection B ⊆ τ is called a base ( or basis) for τ if, for

any U ∈ τ and x ∈ U , there exists V ∈ B such that x ∈ V ⊆ U (orequivalently each U ∈ τ is a union of sets in B).

(2) A subcollection S ⊆ τ is called a subbase (or subbasis) for τ if thefinite intersection of sets in S form a base for τ .

Remark. Any collection B of subsets of a set X is a base for sometopology on X iff (i) X = ∪B : B ∈ B, and (ii) for any B1, B2 ∈ Band x ∈ B1 ∩ B2, there is some B3 ∈ B with x ∈ B3 ⊆ B1 ∩ B2. On theother hand, any collection of subsets of a set X is a subbase for sometopology on X . In fact, this is the smallest topology containing the givencollection of sets.

Definition. (1) If X is a non empty set, then a function d : X×X →R is called a pseudometric on X if, for any x, y, z ∈ X,

(M1) d(x, y) ≥ 0;(M2) if x = y, then d(x, y) = 0;(M3) d(x, y) = d(y, x) (symmetry);(M4) d(x, y) ≤ d(x, z) + d(z, y) (triangle inequality).If, in addition, d satisfies: d(x, y) = 0 ⇒ x = y, then d is called a

metric on X . If d is a pseudometric (resp. metric) on a set X , then thepair (X, d) is called a pseudometric (resp. metric) space. For any x ∈ Xand r > 0, the sets

B(x, r) = y ∈ X : d(y, x) < r and B[x, r] = y ∈ X : d(y, x) ≤ r

are called open ball and closed ball, respectively, with center x and radiusr.

(2) A subset U of a pseudometic space (X, d) is called a d-open set if,for each x ∈ U , there exists r > 0 such that B(x, r) ⊆ U. The collectiontd of all d-open sets is a topology on X , called the pseudometric topologyinduced by d. Clearly, the collection B = B(x, r) : x ∈ X, r > 0 is abase for the topology td.

(3) A topological space (X, τ) is said to be pseudometrizable (resp.metrizable) if there exists a pseudometric (resp. metric) d on X suchthat τ = td.


Definition. Let (X, d) be a metric space.(1) A sequence xn : n ∈ N is said to be convergent to x ∈ X if,

given any ε > 0, there exists an integer no such that

d(xn, x) < ε for all n ≥ no;

in this case, we write xn → x.(2) xn is said to be a Cauchy sequence if, given any ε > 0, there

exists an integer no such that

d(xn, xm) < ε for all n,m ≥ no.

Clearly, every convergent sequence is a Cauchy sequence.

Definition. Let (X, d) be a metric space, and let A ⊆ X . Then A iscalled complete if every Cauchy sequence in A converges to a point in A.

Definition. Let (X, τ) be a topological space and A ⊆ X .(1) A point x in A is called an interior point of A if there exists an

open set U in X such that x ∈ U ⊆ A.(2) A point x ∈ A is called an isolated point of A if it has a neigh-

borhood which does not contain any other point of A.(3) A point x ∈ X is called a limit point of A if each neighborhood

of x contains a point of A other than x. The set of all limit points of Ais denoted by Ad.

Note that, for any A ⊆ X , its closure A = A ∪ Ad; so A is closed iffevery limit point of A is in A. Further, every x ∈ A is either an isolatedpoint of A or a limit point of A.

Definition. (1) A relation ≥ on a set D is called a direction (or adirected relation) if:

(i) α ≥ α,for all α ∈ D (reflexive),(ii) if α ≥ β, β ≥ γ, then α ≥ γ (transitive);(iii) if α, β ∈ D, there exists a γ ∈ D such that γ ≥ α, γ ≥ β

(directive).In this case, (D,≥) is called a directed set.(2) A net in a set X is a mapping f of a directed set D into X . We

denote this net by f(α) : α ∈ D or xα : α ∈ D, where xα = f(α) ∈X . If B is a subset of a directed set (D,≥), with the induced orderingis directed, then (B,≥) is called a directed subset of (D,≥). A subnetxβ : β ∈ B in X is a directed subset of a net xα : α ∈ D.

(3) A net xα : α ∈ D in a topological space (X, τ) is said to beconvergent to x ∈ X if, given any neighborhood U of x, there exists


αo ∈ D such that

xα ∈ U for all α ≥ αo,

(i.e., xα ∈ U eventually). In this case, x is called the limit of the net andwe write xα −→ x.

Theorem A.1.2. Let X be a topological space and A ⊆ X. Then:(a) For any x ∈ X, x ∈ A iff there exists a net xα ⊆ A such that

xα −→ x.(b) A is closed iff, for any net xα ⊆ A with xα → x ∈ X, we have

x ∈ A.

Definition. Let X and Y be topological spaces.(1) A function f : X → Y is said to be continuous at xo ∈ X if, for

each neighborhood Vo of f(xo) in Y , there exists a neighborhood Uo ofxo in X such that f(Uo) ⊆ Vo. f is said to be continuous on X if it iscontinuous at each x ∈ X .

(2) A function f : X → Y is called a homeomorphism if f is contin-uous, one-one, onto and f−1 : Y → X is continuous.

It is often convenient to use the following equivalent conditions forcontinuity.

Theorem A.1.3. Let X and Y be topological spaces and f : X → Ya function.

(a) The following conditions are equivalent:(i) f : X → Y is continuous on X.(ii) For each open set V ⊆ Y, f−1(V ) is open in X.(iii) For each closed set W ⊆ Y, f−1(W ) is closed in X.(iv) For any net xα ⊆ X with xα → x in X, f(xα)→ f(x) in Y .(b) If X and Y are metric spaces, then f : X → Y is continuous on

X iff, for any sequence xn ⊆ X with xn → x in X, f(xn) → f(x) inY .

(c) If τ1 and τ2 are two topologies on a set X, then the identity mapi : (X, τ1)→ (X, τ2) is continuous iff τ2 ⊆ τ1.

Remark. Clearly, if X and Y are topological spaces, then any func-tion f : X → Y is continuous on X in each of the following cases: (1) Xhas the discrete topology τ = P (X), the power set of X ; (2) Y has theindiscrete topology τ = X,∅.

Definition. A function f : X −→ Y is called an open function (resp.a closed function) if, for any open set (resp. closed set) U in X, f(U) isopen (resp. closed) in Y .


In general, f is continuous < f is open < f is closed. However, iff : X −→ Y is one-one and onto, then f is open iff f is closed iff f−1

is continuous; hence f is a homeomorphism iff f is continuous and openiff f is continuous and closed.

Theorem A.1.4. (Map Gluing theorem) Let X and Y be topologicalspaces, and let X = A∩B, where A and B are closed subsets of X. Letf : A → Y and g : B → Y be continuous functions such that f = g onA ∩B. Then the function h : X → Y defined by

h(x) =

f(x) if x ∈ A,g(x) if x ∈ B,

is continuous on X.Proof. Let W be a closed set in Y . Then f−1(W ) is closed in A and

hence also in X , since A is closed in X . Similarly, f−1(W ) is closed bothin B and X . Hence h−1(W ) = f−1(W ) ∩ g−1(W ), and thus h−1(W ) isclosed in X . This implies that, h is continuous on X .

Definition. A mapping ϕ : (X, d) −→ (Y, ρ) is called an isometryof (X, d) into (Y, ρ) if

ρ(ϕ(x1), ϕ(x2)) = d(x1, x2) for all x1, x2 ∈ X.

( i.e. ϕ preserves distances). In this case, X and ϕ(X) are said tobe isometric.Clearly, every isometry is one-one, continuous and an openmapping. Hence an isometry ϕ : (X, d) → (Y, ρ) is a homeomorphism⇐⇒ ϕ is onto. However, a homeomorphism ϕ : (X, d)→ (Y, ρ) need notbe an isometry.

Definition. If X, Y, Z are topological spaces, then a function f :X × Y → Z is called

(i) separately continuous on X × Y if, given any (x, y) ∈ X × Y andany neighborhood G of f(x, y) in Z, there exist neighborhoods H of x inX and J of y in Y such that f(H × y) ⊆ G and f(x × J) ⊆ G.

(ii) jointly continuous on X×Y if, given any (x, y) ∈ X×Y and anyneighborhood G of f(x, y) in Z, there exist neighborhoods H of x in Xand J of y in Y such that f(H × J) ⊆ G.

Clearly, jointly continuity⇒ separately continuity; the converse neednot hold.

Definition. A topological space (X, τ) is called:(1) a T1-space if, for any two points x 6= y in X , there exist two open

sets U, V ⊆ X such that x ∈ U, y ∈ V and x /∈ V, y /∈ U ;


(2) Hausdorff if, for any two points x 6= y in X, there exist two opensets U, V ⊆ X such that x ∈ U, y ∈ V and U ∩ V = ∅;

(3) regular if, for any closed set A in X and x ∈ X with x /∈ A, thereexist two open sets U, V ⊆ X such that x ∈ U, A ⊆ V and U ∩ V = ∅;

(4) normal if for any two closed sets A,B ⊆ X with A∩B = ∅, thereexist two open sets U, V ⊆ X such that A ⊆ U,B ⊆ V and U ∩ V = ∅;

(5) completely regular if, for any closed set A ⊆ X and x ∈ X withx /∈ A, there exists a continuous function f : X → [0, 1] such thatf(x) = 0 and f(y) = 1 for all y ∈ A.

Definition. Let X be a topological space, and let A ⊆ X .(1) A collection U = Gα : α ∈ I of open subsets of X is said to

be an open cover of A if A ⊆ ∪α∈IGα. If U and V are two covers of X ,then V is said to be a refinement of U if each V ∈ V is contained in someU ∈ U .

(2) A is called compact if every open cover of A has a finite subcover.(3) A is called countably compact if every countable open cover of A

has a finite subcover.(4) A is called sequentially compact if every sequence in A has a

convergent subsequence with limit in A.(5) A is called relatively compact if its closure A is compact.(6) If A is a subset of a pseudometric space (X, d), then A is called

precompact (or totally bounded) if, for each ε > 0, there exists a finitesubset D = x1, ..., xn of X such that

A ⊆n⋃

i=1

B(xi, ε).

Definition. A topological space (X, τ) is called:(1) locally compact if each x ∈ X has a relative compact neighbor-

hood;(2) pseudocompact if every continuous real-valued function on X is

bounded;(3) σ-compact if it can be written as a countable union of compact

sets;(4) Lindelof if every open cover of A has a countable subcover;(5) hemicompact if X can be expressed as a countable union of com-

pact sets Kn such that each compact subset of X is contained in someKn;


(6) paracompact if every open cover U of X has refinement V which islocally finite (i.e. each x ∈ X has a neighborhood which intersects onlya finite number of members of V);

(7) a k-space if, for any U ⊆ X , U is closed in X whenever U ∩K isclosed in K for each compact K ⊆ X ;

(8) a kR-space if a function f : X → R is continuous on X wheneverf | K is continuous for every compact subset K of X ;

(9) separable if it has a countable dense subset;(10) zero-dimensional (or 0-dimensional) if, for each x ∈ X and each

neighborhood U of x, there exists a clopen (i.e., both closed and open)set V such that x ∈ V and V ⊆ U ;

(11) totally disconnected if, for any x ∈ X, Cx = x , where Cx isthe component of x (i.e., the maximal connected subset of X containingx).

(12) X is called extremely disconnected (or Stonian) if the closure ofevery open set is open.

We now summarize some properties of the above spaces in the fol-lowing theorem.

Theorem A.1.5. (a) A topological space X is Hausdorff iff everyconvergent net in X has a unique limit.

(b) A closed subset A of a compact space X is compact.(c) A compact subset A of a Hausdorff space X is closed.(d) Every compact Hausdorff space is normal.(e) A subset A of a topological space X is compact iff every net in A

has a convergent subnet with limit in A.(f) Every compact space and every sequentially compact space is count-

ably compact and hence pseudocompact.(g) A pseudometric space is compact iff it is sequentially compact iff

it is countably compact iff it is precompact and complete.(h) Every locally compact space and every first countable (in particu-

lar, metric) space is a k-space.(i) The continuous image of a compact set is compact.(j) If X is a k-space and Y any topological space, then a function

f : X → Y is continuous on X iff f | K is continuous for every compactsubset K of X; hence every k-space is a kR-space.

(k) Every locally compact σ-compact Hausdorff space is hemicompact.(l) Every compact metric space is separable.(m) Every open subspace of a separable space is separable; an arbitrary

subspace of a separable metric space is also separable.(n) The continuous image of a separable space is separable.


(o) Every zero-dimensional T1-space is totally disconnected and com-pletely regular.

(p) A locally compact Hausdorff is 0-dimensional iff it is totally dis-connected.

(q) Continuous image of a totally disconnected space need not be to-tally disconnected.

Examples. (1) Q, P and discrete spaces are zero-dimensional andtotally disconnected.

(2) Q is not extremely disconnected.

Note. Clearly, a finite subset A of a topological space X is alwayscompact; but need not be closed. So it need not be relatively compact(e.g. if X is neither T1 nor regular).

Example. Let X = N with the topology

τ = N ∪ An : n ∈ N,

where An = 1, 2, ...n− 1 for every n ∈ N. This is T0 but not T1 (hencealso not Hausdorff). In this case, the only closed compact subsets of Xand ∅, so the only relatively compact set is ∅.

Theorem A.1.6. Let X be any topological space and Y a Hausdorfftopological space, and let f, g : X → Y be continuous. Then:

(a) The set A = x ∈ X : f(x) = g(x) is closed in X.(b) If D ⊆ X and f = g on D, then f = g on D. In particular, if

D is dense in X and f = g on D, then f = g on X.

Notation. For any function ϕ : X → R, A ⊆ X, and t ∈ R, we shalloften write ϕ(A) = t to mean that ϕ(A) = t, that is, ϕ(x) = t for allx ∈ A.

Theorem A.1.7. (Urysohn lemma

) Let X be a normal space, and let A and B be closed subsets of Xwith A ∩ B = ∅. Then there exists a continuous function ϕ : X → [0, 1]such that ϕ(A) = 0 and ϕ(B) = 1.

Theorem A.1.8. (a) Let X be a completely regular space, K acompact subset of X, and B a closed of X with K ∩ B = ∅. Thenthere exists a continuous function ϕ : X → R such that ϕ(K) = 0 andϕ(B) = 1.

(b) Let X be a locally compact Hausdorff space, K a compact subsetof X, and U a neighborhood of K. Then there exists a continuousfunction ϕ : X → [0, 1] such that ϕ(K) = 1 and the support of ϕ is


compact and contained in U . (The support of ϕ is the closure of the setx ∈ X : ϕ(x) 6= 0).

In particular, every normal T1-space and every locally compact Haus-dorff space is completely regular. Further, every 0-dimensional T1-spaceis completely regular.

Definition. A topological space X is called a Baire space if wheneverX = ∪∞n=1An with each An closed, then, for at least one m, int(Am) 6= ∅.

Theorem A.1.9. (Baire) (a) A topological space X is a Baire spaceiff, for any countable family Un of open dense subsets in X, ∩∞n=1Un 6=∅.

(b) Every locally compact Hausdorff space is a Baire space.(c) Every complete metric space is a Baire space.

Theorem A.1.10. (Tietze extension theorem) Let X be a normalspace, and let A be a closed subset of X. Then, for any continuousfunction f : A → [0, 1] (or R), there exists a continuous function g :X → [0, 1] (or R) such that g = f on A. (Here g is called a continuousextension of f from A to X).

Definition. Let (Xα, τα) : α ∈ I be an indexed family of topo-logical spaces. The product topology on the Cartesian product Π

α∈IXα is

defined as the topology τΠ obtained by taking as its base the collection ofall sets of the form Π

α∈IUα, where each Uα is τα-open in Xα and Uα = Xα

except for a finite number of indices α’s.

Remark. If Xα : α ∈ I is a family of compact spaces, then ΠαεIXα

with the product topology τΠ is a compact space (Tychonoff theorem).

Definition. If (X, τ) is a completely regular Hausdorff space, letC = Cb(X,R), the set of all continuous and bounded functions f : X →R. For each f ∈ C, define a closed bounded (hence compact) interval inR by

If = [ infx∈X

f(x), supx∈X

f(x)].

Define a map e : X → Πf∈C

If (with the product topology) by

e(x) = (f(x))f∈C , x ∈ X.

Then βX = e(X) is a compact space, called the Stone-Cech compactifi-cation of X.


Theorem A.1.11. (Stone-C ech) Let X be a completely regularHausdorff space and βX its Stone-Cech compactification. Let Y be acompact Hausdorff space or Y = R. Then every continuous function

f : X → Y has a unique continuous extension f : βX → Y .

Definition. (1) A subset A of a topological space X is called a zeroset [GJ60] if there exists a continuous function ϕ : X → R and that

A = ϕ−1(0) = x ∈ X : ϕ(x) = 0.

In this case, we write A = Z(ϕ). The complement of a zero set is calleda cozero set.

(2) A subset D of X is called a Gδ-set if it is a countable intersectionof open sets; D is called an Fσ-set if it is a countable union of closed sets.

(3) X is called a P -space ([GJ60], p. 62-63) if every zero set (orequivalently, every Gδ-set) in X is open.

Clearly, the complement of a Gδ-set is an Fδ-set, and vice versa. Forany function f : X → R, the set D of all points of discontinuity of f is anFσ-set and the set of all points of continuity of f is a Gδ-set. It followsfrom ([GJ60], 4K(3)) that every compact subset of a P -space is finite.

Definition. [GJ60, Nag65]) A subset A of a topological space Xis said to have finite covering dimension if there exists a non-negativeinteger n such that for each finite open cover U of A there exists an openrefinement V of U which is a cover of A and such that any point of Abelongs to a most n + 1 member of V. The least such n is called thecovering dimension of A. If no such finite n exists, then we say that Ahas an infinite covering dimension.

Theorem A.1.12. (a) For each n ≥ 1, the covering dimensions ofRn is n.

(b) Every compact subset of a P -space has finite covering dimension.(c) If a normal space X has finite covering dimensions, then so has

its Stone-Cech compactification βX.Recall that, if X is a topological space, then a function ϕ : X → R is

continuous at xo ∈ X if, for any ε > 0, there exists an open neighborhoodN(xo) of xo such that

|ϕ(y)− ϕ(xo)| < ε for all y ∈ N(xo),

or equivalently,

ϕ(xo)− ε < ϕ(y) < ϕ(xo) + ε for all y ∈ N(xo). (*)


Definition. If X is a topological space, then a function ϕ : X → Ris said to be

(a) upper semicontinuous at xo ∈ X if, for any r ∈ R with ϕ(xo) < r,there exists an open neighborhood N(xo) of xo such that

ϕ(y) < r for all y ∈ N(xo);

(b) lower semicontinuous at xo ∈ X if, for any r ∈ R with ϕ(xo) > r,there exists an open neighborhood N(xo) of xo such that

ϕ(y) > r for all y ∈ N(xo);

(c) upper (resp. lower) semicontinuous on X if it is upper (resp.lower) semicontinuous at each point of X , or equivalently, if, for eachr ∈ R, the set x ∈ X : ϕ(x) < r (resp. x ∈ X : ϕ(x) > r) is open([Will70], p. 49).

Clearly, by (∗), a function ϕ : X → R is continuous at xo ∈ X iff itis both upper and lower semicontinuous at xo ∈ X.

Theorem A.1.13. (a) A function ϕ : X → R is continuous iff itis both upper and lower semicontinuous.

(b) The characteristic function χA of a subset A of X is upper (resp.lower) semicontinuous iff A is closed (resp. open).

(c) Every upper (resp. lower) semicontinuous function assumes itssupremum (resp. infimum) on a compact set. In particular, every non-negative upper semicontinuous function on a compact set is bounded.

Definition: Let X and Y be topological spaces and T : X → Y amapping. Then:

(a) The set G(T ) = (x, Tx) : x ∈ X is called the graph of T inX × Y.

(b) T is said to have a closed graph if G(T ) is a closed subset of X×Y.(Here X × Y has the product topology.)

The following theorem gives equivalent condition for a mapping T :X → Y to have a closed graph.

Theorem A.1.14. Let X and Y be topological spaces and T : X×Ya mapping. Then the following are equivalent

(i) T has a closed graph.(ii) If xα is a net in X with xα → x ∈ X and Txα → y ∈ Y , then

y = T (x)(iii) For each x ∈ X and y 6= T (x) in Y , there exist neighborhoods

U of x in X and V of y in Y such that

T (U) ∩ V = ∅.


Proof. (i) ⇒ (ii). Let xα = α ∈ I ⊆ X with xα → x ∈ X andTxα → y ∈ Y . Then clearly (xα, Txα)→ (x, y) in X×Y (in the producttopology). Since (xα, Txα) = α ∈ I ⊆ G(T ) and, by (i), G(T ) isclosed, we have (x, y) ∈ G(T ). Hence y = Tx.

(ii) ⇒ (i). Suppose (ii) holds. To show that G(T ) ⊆ G(T ), let

(x, y) ∈ X × Y with (x, y) ∈ G(T ). Then there exists a net (xα, Txα) :α ∈ I ⊆ G(T ) such that (xα, Txα) → (x, y). This implies that xα → xand Txα → y. Hence, by hypothesis (ii), y = Tx. So (x, y) = (x, Tx) ∈

G(T ). Thus G(T ) = G(T ) and so T has a closed graph.

The next result is concerned with the questions: Under what con-ditions continuity of T implies that T has a closed graph ? Conversely,under what conditions, T has a closed graph implies that T is continuous?

Theorem A.1.15. Let X and Y be topological spaces with Y Haus-dorff, and let T : X → Y be a mapping. If T is continuous, then T hasa closed graph.

Proof. Suppose T is continuous. To show that T has a closed graph,let (x, y) ∈ X×Y with (x, y) ∈ G(T ). Then there exists a net (xα, Txα) :α ∈ I ⊆ G(T ) such that (xα, Txα) → (x, y). This implies that xα → xand Txα → y. Since T is continuous and xα → x, we have Txα → Txin Y . Since Y is Hausdorff, Txα has a unique limit and so y = Tx.

Hence (x, y) = (x, Tx) ∈ G(T ). Thus G(T ) = G(T ); i.e. T has a closedgraph.

The converse of the above theorem is known as the ”closed graphtheorem” and will be considered in Section A.3.

Definition: Let E be a vector space over the field K (= R or C).Then a function ‖.‖ : E → R is called a norm on E if

(N1) ‖x‖ ≥ 0 for all x ∈ E;(N2) ‖x‖ = 0 iff x = 0;(N3) ‖λx‖ = |λ| ‖x‖ for all x ∈ E and λ ∈ K;(N4) ‖x+ y‖ ≤ ‖x‖+ ‖y‖ for all x, y ∈ E.

Definition. (1) A vector space E with a norm ‖.‖ is called a normedvector space or, simply, a normed space. Clearly, every normed space(E, ‖.‖) is a metric space with metric given by:

d(x, y) = ‖x− y‖, x, y ∈ E.

(2) A normed space E is called a Banach space if it is complete withrespect to the metric induced by the norm.


Definition. Let A be an algebra over K (= R or C) which is also anormed space with norm || · ||. Then (A, || · ||) is called

(a) a normed algebra if the norm || · || is submultiplicative, i.e.

||xy|| ≤ ||x||.||y|| for all x, y ∈ A;

(b) a Banach algebra if A is a complete normed algebra.

2. TOPOLOGICAL VECTOR SPACES 243

2. Topological Vector Spaces

In the sequel, we consider all vector spaces over the field K (= R orC) with the usual absolute value.

Notations. Let E be a vector space and A,B ⊆ E , x ∈ E, α, β ∈ K.We denote

x±A = x± a : a ∈ A; A± B = a± b : a ∈ A, b ∈ B;

αA = αa : a ∈ A; αA+ βB = αa+ βb : a ∈ A, b ∈ B.

Clearly (α + β)A ⊆ αA + βA, but αA + βA ⊆ (α + β)A need nothold, in general.

Definition. If E is a vector space and τ is a topology on E, thenthe pair (E, τ) is called a topological vector space (TVS, in short) if thefollowing conditions hold:

(TVS1) The operation of addition (x, y) → x + y of E × E → E isjointly continuous; i.e., given any x, y ∈ E and any τneighborhood U ofx+y in E, there exist τneighborhoods V of x and W of y in E such thatV +W ⊆ U.

(TVS2) The operation of scalar multiplication (λ, x) → λx of K ×E → E is jointly continuous; i.e., given any x ∈ E and λ ∈ K and anyτneighborhood U of λx in E, there exist a τneighborhood V of x in Eand a neighborhood D of λ in K such that DV ⊆ U.

Examples: (1)

Any normed space (E, ‖.‖) is a TVS since the conditions (TVS1)and (TVS2) follow, respectively, from

(i) If x, y ∈ E and xn → x, yn → y, then xn + yn → x+ y in E.(ii) If x ∈ E, λ ∈ K and λn → λ, xn → x, then λnxn → λx in E.(2) Any vector space E with the indiscrete topology τI = E,∅ is

a TVS.(3) A vector space E with the discrete topology τD = P (E) is not a

TVS unless E = 0 (since the scalar multiplication need not be contin-uous).

Definition: Let E be a vector space over the field K (= R or C). Asubset A of a vector space E is called:

(i) absorbing if, for each x ∈ E, there exists a number r = r(x) > 0such that

x ∈ λA for all λ ∈ K with |λ| ≥ r;

(or equivalently, there exists a number t = t(x) > 0 such that λx ∈ A forall λ ∈ K with |λ| ≤ t);


(ii) balanced if λx ∈ A for all x ∈ A and λ ∈ K with |λ| ≤ 1;(iii) convex if tx+ (1− t)y ∈ A for all x, y ∈ A and 0 ≤ t ≤ 1.

Examples. (1) Any vector subspace A of E is balanced and convex,but it need not be absorbing.

(2) If (E, ||.||) is a normed vector space, then, for each r > 0, boththe open and closed balls

B(0, r) = x ∈ E : ||x|| < r, B[0, r] = x ∈ E : ||x|| ≤ r

are absorbing, balanced and convex subset of E.

Lemma.A.2.1. Let (E, τ) be a TVS.(a) For any fixed a ∈ E, the translation map fa : E → E given by

fa(x) = x+ a = a+ x, x ∈ E,

is a homeomorphism of E into itself.(b) For any fixed λ ( 6= 0) ∈ K, the scalar multiplication map gλ :

E → E given bygλ(x) = λx, x ∈ E,

is also a homeomorphism of E into itself.Proof. It is easy to see that f−1a = f−a and g−1λ = g1/λ.(a) Clearly fa

is one-one and onto. Since f−1a = f−a, by (TVS1), both fa and f−1a

are continuous. Hence fa is a homeomorphism of E into itself.(b) Since λ 6= 0, it easily follows that gλ is one-one and onto. Since

g−1λ = g1/λ, by (TVS2), both gλ and g−1λ are continuous. Hence gλ is alsoa homeomorphism of E into itself.

Lemma A.2.2. Let (E, τ) be a TVS, and let a ∈ E and λ( 6= 0) ∈ K.(a) If U is an open (respectively, a closed) set in E, then a+ U and

λU are also open (respectively, closed) in E.(b) U is a neighborhood of 0 iff a + U is a neighborhood of a.(c) U is a neighborhood of 0 iff λU is a neighborhood of 0.(d) A collection W of subsets of E is a base of neighborhoods of 0

in E iff the collection

a+W = a+ U : U ∈ W

is a base of neighborhoods of a in E.

Proof. (a) By above Lemma, the translation map fa : E → E isa homeomorphism; in particular, fa is both an open and closed map.Hence, if U is an open (resp. a closed) set in E, then fa(U) = a+U is alsoopen (resp. closed) in E. Next, the map gλ : E → E is a homeomorphism;


hence, if U is an open (resp. a closed) set in E, then gλ(U) = λU is alsoopen (resp. closed) in E.

(b) & (c). These follow immediately from (a).(c) By (b), U is a neighborhood of 0 iff a+ U is a neighborhood of a

in E. Hence W is a base of neighborhoods of 0 in E iff a +W is a baseof neighborhoods of a in E.

The following theorem shows that every TVS has a base W of neigh-borhoods of 0 consisting of closed, balanced and absorbing sets.

Theorem A.2.3. Let (E, τ) be a TVS, and let W be a base ofneighborhoods of 0 in a TVS E. Then:

(A1) For every U, V ∈ W, there exists W ∈ W such that W ⊆ U∩V .(A2) For each U ∈ W, there exists V ∈ W such that V + V ⊆ U.(A3) For each U ∈ W and λ( 6= 0) ∈ K, λU ∈ W.(A4) Each U ∈ W is absorbing.(A5) For each U ∈ W, there exists a balanced W ∈ W such that

W ⊆ U.(A6) For each U ∈ W, there exists a closed neighborhood W ∈ W

such that W ⊆ U.Conversely, if W is a non-empty collection of subsets of E satisfy

(A1)−(A5) then there exist a topology τ on E making (E, τ) a TVS witha base W of τ -neighborhoods of 0.

Proof. (A1) This follows from the property (NB2) of a local base ofneighborhoods of a topological space.

(A2) Let U ∈ W. By (TVS1), the mapping f : (x, y) → x + y iscontinuous at (x, y) = (0, 0) ∈ E ×E. Since f(0, 0) = 0+ 0 = 0 and U isa neighborhood of 0, there exist neighborhoods V1 and V2 of 0 such thatV1 + V2 ⊆ U. Put V = V1 ∩ V2. Then V ∈ W with V + V ⊆ V1 + V2 ⊆ U.

(A3) Let U ∈ W and λ( 6= 0) ∈ K. Choose an open neighborhood V of0 such that V ⊆ U. By (TVS2), the function gλ : E −→ E defined bygλ(x) = λx (x ∈ E) is a homeomorphism and hence an open map. Hencegλ(V ) = λV is an open set. Now λV ⊆ λU , and so λU is a neighborhoodof 0. Thus λU ∈ W.

(A4) Let U ∈ W, and let xo ∈ E. By (TVS2), the mapping hxo : K→E, defined by hxo(λ) = λxo (λ ∈ K) is continuous at λ = 0. So thereexist t > 0 such that hxo(B[0, t]) ⊆ U, where B[0, t] = µ ∈ K : |µ| ≤ t.Hence µxo ∈ U for all |µ| ≤ t, and so U is absorbing.

(A5) Let U ∈ W. The function g : (λ, x) → λx is continuous at(λ, x) = (0, 0) by (TVS2). So there exist a neighborhoods V of 0 in E andB[0, r] of 0 in K such that W = B[0, r]V ⊆ U . Clearly, W is a balanced.Further,W is a neighborhood of 0 (since rV ⊆ B[0, r]V = W and rV is


a neighborhood of 0 by (A3)). Thus, we have a balanced W ∈ W suchthat W ⊆ U.

(A6) Let U ∈ W. By (A2) and (A5), we can choose a balanced V ∈ Wsuch that V +V ⊆ U. We now show that V ⊆ U. [Let x ∈ V .Then, sincex+ V is a neighborhood of x, (x+ V )∩ V 6= ∅. Choose y ∈ (x+ V )∩ V.Then y = x+ v1 = v2,where v1,v2 ∈ V. So

x = v2 − v1 ∈ V − V = V + V ⊆ U.

This shows that V ⊆ U.] Thus W = V ∈ W is closed with W ⊆ U .Conversely, let V = V ⊆ E : V contains some U ∈ W. Clearly,

W ⊆ V. We first show that, for each x ∈ E, Bx = x + V satisfies(NB1)− (NB3).

(NB1) By (A4), each U ∈ W is absorbing and so 0 ∈ U ; hence 0 ∈ Vfor all V ∈ V. Therefore x = x+ 0 ∈ x+ V for all V ∈ V.

(NB2) Let x + V1, x + V2 ∈ Bx, where V1, V2 ∈ V. We show that(x+ V1) ∩ (x+ V2) ∈ x+ V. Now

(x+ V1) ∩ (x+ V2) = x+ V1 ∩ V2.

Choose U1, U2 ∈ W such that U1 ⊆ V1, U2 ⊆ V2. By (A1), there exists aU3 ∈ W such that U3 ⊆ U1 ∩ U2. Hence x+ U3 ⊆ x+ V1 ∩ V2.

(NB3) Let x + V ∈ Bx, where V ∈ V. Choose U ∈ W with U ⊆ V .By (A2), there exists aW ∈ W such thatW+W ⊆ U . Now, if y ∈ x+W,then y +W ∈ By = y + V and

y +W ⊆ x+W +W ⊆ x+ V.

Now (NB1)-(NB3) are satisfied and so, by a general result on baseof neighborhoods in Topology (Theorem A.1.1), there exists a uniquetopology τ on E such that Bx = x + V forms a base of τneighborhoodsof x for each x ∈ E. Clearly, for any x ∈ E and U ∈ W, x + U isa τneighborhood of x (since U ∈ W ⊆ V). Hence x + W is a baseof τneighborhoods of x for each x ∈ E. In particular, W is a base ofτneighborhoods of 0. It remains to verify (TVS1) and (TVS2).

(TVS1) By (A2), it follows that the map (x, y)→ x+ y is continuousat (x, y) = (0, 0). Let xo, yo ∈ E, and let U ∈ W. Then xo + yo + Uis a τneighborhood of xo + yo. By (A2), there exists a V ∈ W suchthat V + V ⊆ U . Then xo + V is a τneighborhood of xo, yo + V is aτneighborhood of yo, and

(xo + V ) + (xo + V ) = xo + yo + V + V ⊆ xo + yo + U.

Hence the map (x, y)→ x+ y is continuous at (x, y) = (xo, yo) ∈ E ×E.


(TVS2). Let λo ∈ K and xo ∈ E. We show that the map g : (λ, x)→λx is continuous at (λ, x) = (λo, xo). Let W ∈ W, so that λoxo +W is aτneighborhood of λoxo. Choose a balanced V ∈ W such that V +V ⊆W(by (A2) and (A5)). Since V is absorbing (by A4), there exists a r > 0such that xo ∈ rV . By (A3), U = r

r|λo|+1V ∈ W. We now show that

B(λo, 1/r)(xo + U) ⊆ λoxo +W . Let λ ∈ B(λo, 1/r) and x ∈ xo + U.Then, since V is balanced and

|λ| ≤ |λ− λo|+ |λo| <1

r+ |λo| =

1 + r|λo|

r,

so λx− λoxo = λ(x− xo) + (λ− λo)xo

∈ λ ·r

r|λo|+ 1V + (λ− λo)rV

⊆ V + V ⊆W.

Consequently, the map g : (λ, x)→ λx is continuous at (λ, x) = (λo, xo).Thus (E, τ) is a TVS.

We now summarize topological properties of a TVS.

Theorem A.2.4. Let (E, τ) be a TVS. Then:(a) (E, τ) is regular.(b) (E, τ) is Hausdorff iff

⋂U∈W

U = 0, where W is a base of τ -

neighborhoods of 0.(c) (E, τ) is a T1-space iff (E, τ) is Hausdorff iff (E, τ) is completely

regular.(d) (E, τ) is locally compact iff (E, τ) has a compact τ -neighborhood

of 0 iff (E, τ) is homeomorphic to Kn for some n ≥ 1 (i.e. (E, τ) isfinite dimensional).

(e) (E, τ) is compact iff E = 0.(f) (E, τ) is both connected and locally connected.If a vector space E is non-trivial (i.e. E 6= 0) with the indiscrete

topology τI = E,∅, then (E, τI) is a TVS which is not a T1-space,hence also not a Hausdorff space. But it is regular and normal.

Definition. A subset A of a TVS E is called:(a) bounded if, for any neighborhood W of 0 in E, there exists r > 0

such that

A ⊆ λW for all λ ∈ K with |λ| ≥ r,

or, equivalently, A ⊆ rW if W is balanced.


(b) precompact (or totally bounded) if, for each neighborhood W of 0,there exists a finite subset D = x1, ..., xn of E such that

A ⊆ D +W = ∪ni=1(xi +W ).

Clearly, compactness ⇒ precompactness ⇒ boundedness; the reverseimplications need not be true.

Definition. A net xα : α ∈ I in a TVS E is said to be convergentto x ∈ E if, given any neighborhood U of 0 in E, there exists an αo ∈ Isuch that

xα − x ∈ U for all α ≥ αo;

in this case, we write xα → x. xα is said to be a Cauchy net if, givenany neighborhood U of 0 in E, there exists an αo ∈ I such that

xα − xβ ∈ U for all α, β ≥ αo.

Definition. Let E be a TVS, and let A ⊆ E. Then A is called:(i) complete if every Cauchy net in A converges to a point in A;(ii) sequentially complete if every Cauchy sequence in A converges to

a point in A;(iii) quasi-complete (or boundedly complete) if every bounded Cauchy

net in A converges to a point of E (i.e. if every bounded closed subset ofE is complete).

Clearly, completeness ⇒ quasi-completeness ⇒ sequential complete-ness; the reverse implications need not hold.

Theorem A.2.5. A subset A of a TVS E is compact iff it is completeand precompact.

Definition. Let E be a TVS. Then a complete TVS E is called a

completion of E if there exists a topological isomorphism ϕ : E → E such

that ϕ(E) is dense inE. Every TVS has a unique Hausdorff completion([KN63], p. 63-64).

Definition. A TVS (E, τ) is called:(i) a locally convex space (or a locally convex TVS ) if it has a base of

convex neighborhoods of 0;(ii) metrizable if τ is induced by a metric d on E;(iii) an F -space if it is a complete metrizable TVS.In general, E is locally convex < E is metrizable.


Examples. (1) (i) ([Rud91], p. 86) For 0 < p < 1, the sequencespace

ℓp(N) = x = xn ⊆ K :

∞∑

n=1

|xn|p <∞,

the set of all p-summable sequences in K, is a metrizable TVS (in fact,an F -space) with metric given by

dp(x, y) =∞∑

n=1

|xn − yn|p , x = xn, y = yn ∈ ℓp(N).

but it is not locally convex.(ii) ([Rud91], p. 36-37) For 0 < p < 1,the space

Lp[a, b] = f : [a, b] −→ K:∫ b

a

|f(t)|p dt <∞,

the set of all Lebesgue p-integrable functions, is also an F -space but notlocally convex.

(2) For X a topological space, let E = C(X) with the compact-opentopology k, i.e. the topology which has a base W of kneighborhoods of0 consisting of all sets of the form

U(K, ε) = f ∈ C(X) : supx∈K|f(x)| ≤ ε,

where K ⊆ X is compact and ε > 0. Then each U(K, ε) is absorb-ing, balanced and convex, and so (C(X), k) is locally convex. However,(C(X), k) is not metrizable. In fact, if X is a locally compact Hausdorffspace, then (C(X), k) is metrizable iff X is hemicompact (i.e. X canbe expressed as a countable union of compact sets Kn such that eachcompact subset of X is contained in some Kn).

Definition. Let E be a vector space over K.(1) A function q : E → R is called a seminorm on E if it satisfies(S1) q(x) ≥ 0 for all x ∈ E;(S2) q(x) = 0 if x = 0;(S3) q(λx) = |λ|q(x) for all x ∈ E and λ ∈ K (absolutely homoge-

neous);(S4) q(x+ y) ≤ q(x) + q(y) for all x, y ∈ E (subadditive).

(2) A function q : E → R is called a k-norm on E, where 0 < k ≤ 1,if it satisfies

(K1) q(x) ≥ 0 for all x ∈ E;(K2) q(x) = 0 if and only if x = 0;

(K3) q(λx) = |λ|k q(x) for all x ∈ E and λ ∈ K;


(K4) q(x+ y) ≤ q(x) + q(y) for all x, y ∈ E.(3) A function q : E → R is called an F -seminorm on E if it satisfies(F1) q(x) ≥ 0 for all x ∈ E;(F2) q(x) = 0 if x = 0;(F3) q(λx) ≤ q(x) for all x ∈ E and λ ∈ K with |λ| ≤ 1;(F4) q(x+ y) ≤ q(x) + q(y) for all x, y ∈ E (subadditive);(F5) if λn → 0, then q(λnx)→ 0 for all x ∈ E;(F6) if q(n)→ 0, then q(λxn)→ 0 for all λ ∈ K.

(4) If q : E → R satisfies q(λx) = λq(x) for all λ ≥ 0, then q is calledpositively homogeneous.

(5) If a seminorm (resp. F -seminorm) q satisfies q(x) = 0 impliesx = 0, then q is called a norm (resp. F-norm) on E.

Remarks. (1) If q is a seminorm or F -seminorm on a vector space E,it may happen that q(x) = 0 for some x 6= 0. Hence, if q is a seminormor F -seminorm on a vector space E, then it defines a pseudometric dq onE given by

dq(x, y) = p(x− y), x, y ∈ E.

(2) It is easy to verify that if q is a seminorm on a vector space E,then, for each r > 0, the q-balls

Bq(0, r) = x ∈ E : q(x) < r, Bq[0, r] = x ∈ E : q(x) ≤ r

are absorbing, balanced and convex subset of E.

Definition: Let A be an absorbing subset of a vector space E. Thenthe function ρA : E → R defined by

ρA(x) = infr > 0, x ∈ rA, x ∈ E,

is called the Minkowski functional of A.

Remark. It is easy to see that if, in addition, A is either balancedor convex, then

(i) for any x ∈ E and ε > 0, x ∈ (ρA(x) + ε)A.(ii)

x ∈ E : ρA(x) < 1 ⊆ A ⊆ x ∈ E : ρA(x) ≤ 1.

Lemma A.2.6. Let A be an absorbing subset of a vector space E.Then:

(a) 0 ≤ ρA(x) <∞ with ρA(0) = 0.(b) (i) ρA(tx) = tρA(x) for all x ∈ E and t ≥ 0 (positively

homogeneous);(ii) A is balanced iff

ρA(λx) = |λ|ρA(x) for all x ∈ E and λ ∈ K (absolutely homogeneous).


(c) A is convex iff

ρA(x+ y) ≤ ρA(x) + ρA(y) for all x, y ∈ E (subadditive).

Hence, if A is an absorbing, balanced and convex subset of a vectorspace E, then ρA is a seminorm on E.

Note. If A is not convex, then ρA need not be subadditive. However,we have:

Lemma A.2.7. Let A be an absorbing subset of a vector space E.If H is a balanced subset of E with H +H ⊆ A, then

ρA(x+ y) ≤ ρH(x) + ρH(y) for all x, y ∈ E.

Proof. Let x, y ∈ E and let ε > 0. Then there exist r, s > 0 such that

ρH(x) < r < ρH(x) +ε

2and x ∈ rH,

ρH(y) < s < ρH(y) +ε

2and y ∈ sH.

so

x+ y ∈ rH + sH = (r + s)[r

r + sH +

s

r + sH ]

⊆ (r + s)[H +H ] (since H is balanced)

⊆ (r + s)A.

Hence

ρA(x+ y) ≤ r + s ≤ ρH(x) +ε

2+ ρH(y) +

ε

2= ρH(x) + ρH(y) + ε.

Since ε is arbitrary,

ρA(x+ y) ≤ ρH(x) + ρH(y).

Lemma A.2.8. Let E be a vector space and p, q : E → R+ bepositively homogeneous (e.g. p, q be seminorms on E or p = ρG, q = ρHbe Minkowski functionals of neighborhoods G and H of 0 in a TVS E).Then

p(x) ≤ q(x) ∀ x ∈ E ⇔ q(x) ≤ 1 implies p(x) ≤ 1 ∀ x ∈ E.

Proof. (⇒) Suppose p(x) ≤ q(x) ∀ x ∈ E. If q(x) ≤ 1, thenp(x) ≤ q(x) ≤ 1.

(⇐) Suppose q(x) ≤ 1 implies p(x) ≤ 1 for all x ∈ E. Fix x ∈ E, weconsider two cases:


Case I. Suppose q(x) 6= 0. Then q( xq(x)

) = q(x)q(x)

= 1; and so, by

hypothesis,

p(x

q(x)) ≤ 1, or

p(x)

q(x)≤ 1, or p(x) ≤ q(x).

Case II. Suppose q(x) = 0. Then q(x) < ε for every ε > 0, or

q(x

ε) ≤ 1 for every ε > 0.

Hence, by hypothesis, for any ε > 0,

p(x

ε) < 1, or p(x) < ε

Therefore, p(x) = 0. So, in this case p(x) = 0 = q(x).Thus, in each case, p(x) ≤ q(x) for all x ∈ E.

Lemma A.2.9. Let A be an absorbing and balanced (or convex)subset of a TVS (E, τ). Then:

(a) A0 ⊆ x ∈ E : ρA(x) < 1 ⊆ A ⊆ x ∈ E : ρA(x) ≤ 1 ⊆ A.(b) ρA is τ -continuous iff A is a τneighborhood of 0.(c) If A is a τneighborhood of 0, then:

(i) A0 = x ∈ E : ρA(x) < 1.(ii) A = x ∈ E : ρA(x) ≤ 1.

We shall later see that if A is a “shrinkable” (not necessarily convex)τneighborhood of 0, then ρA is τ -continuous on E. From the abovelemmas, we obtain:

Theorem A.2.10. If (E, τ) is a locally convex space, then its topol-ogy τ can be defined by a family of seminorms, namely, the Minkowskifunctionals of balanced convex τneighborhoods of 0. Conversely, if Pis a family of seminorms on a vector space E over K, then P defines alocally convex topology τ (say) on E and a base of τneighborhoods of 0consists of all sets of the form

x ∈ E : max1≤j≤n

pj(x) ≤ ε, , where ε > 0 and pj ∈ P, n ∈ N.

For general TVSs, we have:

Theorem A.2.11. If (E, τ) is TVS, then its topology τ can be de-fined by a family of F -seminorms. If τ is generated by a countable familyof F -seminorms, then τ can be generated by a single F -norm and so itis metrizable.

Proof. ([Wae71], p. 2-3) Let V ∈ W be any balanced τneighborhoodof 0 in E. Define V1, ...., Vk, ... inductively in such a way that each Vk isbalanced, V ⊇ V1 + V1, and Vk ⊇ Vk+1 + Vk+1. Let D = p/2n : p is


rational, n ∈ N, the set of dyadic rationals. Fix any t = q2N∈ D, 0 ≤

t < 1. Then t =∑N

k=1εk2N

with εk = 0 or 1 for all k. Put

Wt =

N∑

k=1

εkVk ; that is, Wt =∑

εk=1

Vk.

Then Wt is balanced, and clearly Wt1+Wt2 ⊆ Wt1+t2 if t1 + t2 < 1. Weput Wt = E for t ≥ 1, and define

qV (x) = inft : x ∈ Wt, x ∈ E.

Then qV is an F -seminorm on E. Thus the family qV : V ∈ W generatesthe topology τ. If τ is generated by a countable family qn : n ∈ N ofF -seminorms, then clearly τ can be generated by a single F -norm q givenby

q(x) =

∞∑

n=1

2−nqn(x)

1 + qn(x), x ∈ E.

Thus (E, τ) is metrizable.

The general criterion for the metrizability of a TVS may be statedas:

Theorem A.2.12. (Metrization Theorem

) ([KN63], p. 48-49) (a) A TVS (E, τ) is pseudo-metrizable iff it hasa countable base of τneighborhoods of 0. In this case, τ may be definedby a pseudo-metric d which is translation invariant and each ball about0 is balanced; i.e.,

(i) d(x+ z, y + z) = d(x, y) = d(x− y, 0) for all x, y, z ∈ E,(ii) d(λx, 0) ≤ d(x, 0) for all x ∈ E and λ ∈ K with |λ| ≤ 1;

(b) A TVS (E, τ) is metrizable iff it is Hausdorff and has a countablebase of τneighborhoods of 0. In this case, τ may be given by an F -normq, where q(x) = d(x, 0) for all x ∈ E.

Definition. A TVS E is said called locally bounded if E has abounded neighborhood of 0. In this case, if V is a bounded neighborhoodof 0 in E, then clearly, 1

nV : n ≥ 1 is a countable base of neighborhoods

of 0 in E. Then, by above theorem, every Hausdorff locally bounded TVSis metrizable.

Theorem A.2.13. (Kolmogoroff) (i) The topology of a Hausdorfflocally bounded TVS E can always be generated by a k-norm for somek, 0 < k ≤ 1.


(ii) A Hausdorff TVS (E, τ) is normable iff it has a bounded convexneighborhood of 0. (Hence every Hausdorff locally bounded locally convexspace is normable.)

Theorem A.2.14. Let E and F beTVSs. Then:(a) A linear mapping T : E → F is continuous on E iff T is contin-

uous at 0 ∈ E iff T is uniformly continuous on E.(b) If f : E → K a non-zero linear functional, then f is continuous

on E iff f−1(0) is closed in E iff f is bounded on some neighborhoodU of 0 in E.

Definition: The set of all linear functionals on a vector space (or aTVS) E is a vector space, called the algebraic dual of E and is denotedby E ′. If E is a TVS, the set of all continuous linear functionals on E isalso a vector space, called the topological dual of E and is denoted by E∗.Clearly, E ′ is a vector space over K. Further, E∗ is a vector subspace ofE ′.

Definition. Let (E, τ) be a TVS and M a vector subspace of E.(1) The quotient space E/M is defined by

E/M = x+M : x ∈ E,

which is a vector space under the operations

(x+M) + (y +M) = (x+ y) +M, λ(x+M) = λx+M,

where x, y ∈ E and λ ∈ K.(2) The canonical map (or quotient map) π : E → E/M is defined by

π(x) = x+M, x ∈ E.

(3) The quotient topology τM on E/M is defined by

τM = A ⊆ E/M : π−1(A) ∈ τ.

Theorem A.2.15. ([Scha71], p. 20, 31) Let (E, τ) be a TVS andM a vector subspace of E. Then:

(a) (E/M, τM) is a TVS.(b) π : (E, τ)→ (E/M, τM) is a linear, continuous and open map.(c) τM is Hausdorff iff M is closed.(d) If (E, τ) is locally convex whose topology is given by a family sE of

seminorms, then (E/M, τM) is also a locally convex space and its topologyis given by the family p : p ∈ sE of seminorms, where p : E/M → Ris given by

p(x+M) = infp(x+m) : m ∈ M, x+M ∈ E/M.


(e) If (E, τ) is metrizable (resp. normable) and M is closed, then(E/M, τM) is also metrizable (resp. normable).

(f) If (E, τ) is complete and metrizable (resp. a Banach space) andM is closed, then (E/M, τM) is complete and metrizable (resp. a Banachspace).

Remarks. (1) If M is a closed vector subspace of a TVS E, thenthe seminorm p : E/M −→ R defined above in (d) becomes a norm onE/M , since

p(x+M) = 0⇒ x ∈M =M ⇒ x+M =M, the zero of E/M.

(2) ([Rud91], p. 33) Let p be a seminorm on a vector space E andNp = ker (p) = p−1(0). Then

(i) inf p(x+m) : m ∈ Np = p(x).(ii) If p : E/Np −→ R is defined by

p(x+Np) = p(x), x+Np ∈ E/Np,

then p is a norm on E/Np.

Definition. (1) IfM be a proper maximal vector subspace of a vectorspace E and xo ∈ E, then the set H = xo +M is called a hyperplanethrough xo; in particular, M is a hyperplane through 0.

(2) Two vector subspacesM and N of E are said to be complementaryif E =M ⊕N (i.e. if E =M +N with M ∩N = 0).

(3) The co-dimension of vector subspace M of E is defined as the di-mension of its complementary subspace N (or equivalently, the dimensionof the quotient space E/M).

Theorem A.2.16. Let E be a vector space and H ⊆ E. Then:(a) H is a hyperplane iff H = ϕ−1(α) = x ∈ E : ϕ(x) = α for

some non-zero linear functional ϕ on E and α ∈ K.(b) H is a hyperplane through 0 iff M is of co-dimension one iff

M = ϕ−1(0) for some non-zero linear functional ϕ on E.(c) If E is a TVS, then a hyperplane H = ϕ−1(α) in E is either

closed or dense in E. Further, H = ϕ−1(α) is closed iff ϕ is continuouson E.

Theorem A.2.17. (Hahn-Banach) Let (E, τ) be a TVS and M avector subspace of E.

(a) If f is a continuous linear functional on M and p a continuousseminorm on E with

|f(x)| ≤ p(x) for all x ∈M ,


then there exists a g ∈ E∗ such that

g = f on M and |g(x)| ≤ p(x) for all x ∈ E.

(b) If E is locally convex and f a continuous linear functional on M ,then there exist a g ∈ E∗ and a continuous seminorm p on E such that

g = f on M and |g(x)| ≤ p(x) for all x ∈ E.

Corollary A.2.18. Let E be a Hausdorff locally convex space (inparticular, a normed space) over K. Then E∗ separates points of E (i.e.for any xo 6= 0 in E, there exist a g ∈ E∗ such that g(xo) 6= 0).

Note that if E is not Hausdorff or not locally convex, then E∗ neednot separate the points of E.

Examples. For 0 < p < 1, the spaces ℓp(N) and Lp[a, b] are F -spaces but not locally convex spaces (as seen before). However, (ℓp(N))∗

separates points of ℓp(N) and (Lp[a, b])∗does not separate points of Lp[a, b]

(as (Lp[a, b])∗ = 0).

Remark. In fact, if E is a TVS, then E∗ 6= 0 iff E containsa non-empty open convex subset U 6= E ([Scha71], p. 47). For morematerial on non-locally convex spaces, see the monographs by Wael-broeck ([Wae71, Wae73]), Adasch, Ernst and Keim [AEK78], Khaleelulla[Khal81], Kalton, Peck and Roberts [KPR84], Rolewicz [Rol85]; see alsothe papers by Klee ([Kle60a, Kle60b]) and Kakol ([Kako85, Kako87,Kako90, Kako92]).

Definition. Let (E, τ) be a TVS. For any finite (or singleton) setA ⊆ E∗, the mapping pA : E → R defined by

pA(x) = sup|f(x)| : f ∈ A, x ∈ E,

is a seminorm pA on E. If A is the collection of all finite subsets of E∗,then the collection pA : A ∈ A of seminorms defines a locally convextopology on E, called the weak topology and denoted by w = w(E,E∗) orσ(E,E∗). Similarly, for any finite (or singleton) set B ⊆ E, the mappingqB : E∗ → R defined by

qB(f) = sup|f(x)| : x ∈ B, f ∈ E∗.

is a seminorm qB on E∗. If B is the collection of all finite subsets ofE, then the collection qA : B ∈ B of seminorms defines a locallyconvex topology on E∗, called the weak ∗ topology on E and denoted byw∗ = w(E∗, E) or σ(E∗, E).

Theorem A.2.19. ([Hor66], p. 185-209) (a) w(E,E∗) ≤ τ.


(b) (E,w(E,E∗)) is Hausdorff iff E∗ separates points of E; in par-ticular, if (E, τ) is a Hausdorff locally convex space, then (E,w(E,E∗))is Hausdorff.

(c) (E,w(E,E∗))∗ = (E, τ)∗.(d) If (E, τ) a Hausdorff locally convex space, then:

(i) (Mazur) For any convex subset M of E, M is τ -closed iff Mis w(E,E∗)-closed; in particular, M

w=M

τ.

(ii) (Mackey) For any subset M of E, M is τ -bounded iff M isw(E,E∗)-bounded.

(e) If (E, τ) is finite-dimensional, then τ = w(E,E∗).We next consider strong and strong∗ topologies on E and E∗, respec-

tively. We first note that, for any set B ⊆ E, the mapping qB : E∗ → Rdefined by

qB(f) = sup|f(x)| : x ∈ B, f ∈ E∗,

is a seminorm on E∗ iff B is w(E∗, E)-bounded in E∗ (see [RR64], p.45;[Hor66], p. 195).

Definition. Let (E, τ) be a TVS. For any w(E∗, E)-bounded setA ⊆ E∗, the mapping pA : E → R defined by

pA(x) = sup|f(x)| : f ∈ A, x ∈ E,

is a seminorm pA on E. If A is the collection of all w(E∗, E)-boundedsubsets of E∗, then the collection pA : A ∈ A of seminorms definesa locally convex topology on E, called the strong topology and denotedby β = β(E,E∗). Clearly, w(E,E∗) ≤ β(E,E∗). Similarly, for any τ -bounded (or equivalently, w(E∗, E)-bounded) set B ⊆ E, the mappingqB : E∗ → R defined by

qB(f) = sup|f(x)| : x ∈ B, f ∈ E∗.

is a seminorm qB on E∗. If B is the collection of all τ -bounded subsetsof E, then the collection qB : B ∈ B of seminorms defines a locallyconvex topology on E∗, called the strong∗ topology and denoted by β∗ =β(E∗, E). Clearly, w(E∗, E) ≤ β(E∗, E).

Remarks. (1) ([Wila78], p. 119) If (E, || · ||) is a normed space, thenthe strong∗ topology β(E∗, E) on E∗ is the norm topology given by

||f || = sup|f(x)| : ||x|| ≤ 1, f ∈ E∗.

However, in this case, the strong topology β(E,E∗) on E need not thenorm topology for E; in general, || · || ≤ β(E,E∗). If E is a Banach space,then β(E,E∗) = || · ||.


(2) The weak and strong topologies are often useful in giving counter-examples. For instance, the Banach space ℓ1 = ℓ1(N) with the weaktopologyw = w(ℓ1, ℓ∞) is sequentially complete but not complete; ℓ1 withthe weak∗ topology w∗ = w(ℓ1, co) is quasi-complete but not complete.(Recall that c∗o

∼= ℓ1 and ℓ∗1 = ℓ∞ as normed spaces.) Hilbert space isweakly quasi-complete but not weakly complete.

Definition: Let (E, τ) be a TVS. A subset H of E∗ is said to beequicontinuous if, given any ε > 0, there exists a neighborhood U of 0 inE such that

|f(x)| < ε for all x ∈ U and f ∈ H.

Definition. Let (E, τ) is a TVS such that E∗ separates points of E.For each set U ⊆ E, the set

Up = f ∈ E∗ : qU(f) = sup|f(x)| : x ∈ U ≤ 1

is called the polar of U in E∗.

Theorem A.2.20. ([Hor66], p. 190, 195] Let (E, τ) be a TVS andU ⊆ E. Then:

(i) Up is convex, balanced and w(E∗, E)-closed in E∗

.(ii) Up is absorbing iff U is w(E,E∗)-bounded iff qU is a seminorm

on E∗.

Theorem A.2.21. ([Hor66], p. 200]) Let (E, τ) be a TVS. Then(i) [Hor66, p. 200] A subset H of E∗ is equicontinuous iff there exists

a τ -neighborhood U of 0 in E such that H ⊆ Up. In particular, for anyτneighborhood U of 0 in E, Up is an equicontinuous subset of E∗.

(ii) Every equicontinuous subset H of E∗ is w(E∗, E)-bounded.

Theorem 1.2.22. (Bourbaki-Alaoglu) ([Hor66], p. 201) Let (E, τ)be a TVS. Then any τ -equicontinuous subset H of E∗ is relatively w(E∗, E)-compact. In particular, for any τneighborhood U of 0 in E, Up isw(E∗, E)-compact.

Corollary A.2.23. (Banach-Alaoglu) Let E be a normed space.Then the norm closed unit ball

B∗[0, 1] = f ∈ E∗ : ‖f‖ ≤ 1

of E∗ is w(E∗, E)-compact.

Note. ([Ko69], p. 282) A norm closed unit ball of a normed space Eneed not be w(E,E

∗

)-compact in E; take, for example, E = ℓ1.


Theorem A.2.24. ([Sim63], p. 234]) Let (E, ||·||) is a Banach space,and let S∗ = (B∗1 , w

∗), a compact Hausdorff space. For any x ∈ X, definea continuous map x : S∗ → K by

x(f) = f(x), f ∈ S∗.

Then the map π : x −→ x is an isometric isomorphism of E onto aclosed vector subspace of the Banach space C(S∗).

Remarks. (1) By above theorem, any general Banach space (E, || ·||) can be considered as a closed vector subspace of the function space(C(X), || · ||∞), where X is a compact Hausdorff space.

(2) [Jam74, p. 273] It is fair to ask whether C[0, 1] is typical of manyother spaces C(X). In this regard, we mention a famous result of A.A.Milyutin (1966) which states that: if X is a uncountable compact metricspace, then C(X) is isomorphic to C(I).

Definition. A sequence xn in a TVS E is called a topological basisof E if, for each x ∈ E, there is a unique sequence λn of scalars suchthat

x = limk→∞

k∑

n=1

λnxn (in the topology of E);

that is, for any neighborhood U of 0 in E, there exists an integer N ≥ 1such that

k∑

i=1

λixi − x ∈ U for all k ≥ N.

In this case we write x =∑∞

i=1 λixi. For each m = 1, 2, ..., the linearfunctionals fm : E → K defined by

fm (x) = λm for x =∞∑

i=1

λixi ∈ E,

are called the associated coefficient functionals

of the basisxn . Note that fm(xn) = δmn. If all fm are continuous,the basis xn is called a Schauder basis.

Definition. (1) A TVS E is said to have the approximation propertyif the identity map on E can be approximated uniformly on precom-pact sets by continuous and linear maps of finite rank (i.e. with rangecontained in finite dimensional subspaces of E).


(2) A TVS E is said to be admissible if the identity map on E canbe approximated uniformly on compact sets by continuous maps of finiterank.

Every locally convex space and every F -space with a basis (e.g. ℓp, 0 <p < 1) is admissible ([Kl60a, Kl60b, Shu72]).

3. SPACES OF CONTINUOUS LINEAR MAPPINGS 261

3. Spaces of Continuous Linear Mappings

Definition. Let E and F be TVSs over the same field K. Then alinear mapping T : E → F is called bounded if it maps bounded sets ofE into bounded sets of F.

Lemma A.3.A. Every continuous linear mapping T : E → F isbounded.

Proof. Let S be a bounded set in E, and let W be a neighborhood of0 in F . By continuity of T, T−1(W ) is a neighborhood of 0 in E. SinceS is a bounded set in E, there exists r > 0 such that S ⊆ λT−1(W ) forall |λ| ≥ r. Then T (S) ⊆ λW for all |λ| ≥ r. Hence T (S) is a boundedset in F .

In general, a bounded linear map T : E → F need not be continu-ous. We shall see below that, under some additional hypothesis, everybounded linear mapping T : E → F is continuous.

Theorem A.3.2. Let E be a metrizable TVS and F any TVS. Thenany bounded linear map T : E → F is continuous (i.e. T is continuousiff T is bounded).

Proof. First suppose that E is a metrizable TVS and F any TVS,and let T : E → F be a bounded linear map. In this case, there exists acountable base U1, U2, ... of neighborhood of 0 in E. Let

V1 = U1, V2 = U1 ∩ U2, ...., Vn = U1 ∩ U2 ∩ .... ∩ Un, .....

Then V1, V2, .... is also a base of neighborhood of 0 with V1 ⊇ V2 ⊇ ....If T is not continuous, there is some neighborhood W of 0 in F such thatT−1(W ) is not a neighborhood of 0 in E. Hence, for each n ≥ 1, Vn "nT−1(W ) and so Vn\nT−1(W ) 6= ∅. For each n ≥ 1, choose xn ∈ Vn\ nT−1(W ). Then xn is a bounded set in E. [Indeed, let V be anyneighborhood of 0 in E. There exists a Vm with Vm ⊆ V . Now, for anyn ≥ m, xn ∈ Vn ⊆ Vm. Choose r > 1 such that x1, ...., xm−1 ⊆ rVm.Hence xn ∈ rVm ⊆ rV for all n ≥ 1.] However, Txn is not boundedin F since, for all n ≥ 1, xn /∈ nT−1(W ) and hence T (xn) /∈ nW . Thiscontradicts the fact that T is bounded.

Notations. The set of all bounded (resp. continuous) linear map-pings T : E → F is denoted by BL(E, F ) (resp. CL(E, F )). Clearly,BL(E, F ) and CL(E, F ) are vector spaces with the usual pointwiseoperations and CL(E, F ) ⊆ BL(E, F ). Note that E ′ = L(E,K) andE∗ = CL(E,K). Further, if F = E, BL(E) = BL(E,E) and CL(E) =


CL(E,E) are algebras under composition (i.e. (ST )(x) = S(T (x))) andhave the identity I : E → E given by I(x) = x (x ∈ E).

Definition. (1) A TVS E is called ultrabornological [Iya68] if everybounded linear map from E into any TVS F is continuous. In thiscase, CL(E, F ) = BL(E, F ). By above theorem, every metrizable TVSis ultrabornological.

(2) A TVS (E, τ) is called ultrabarrelled ([RobW58, Edw65, Hus65])if, any linear topology τ ′ on E, having a base of neighborhoods of 0 formedof τ -closed sets, is weaker than τ. Equivalently, a TVS E is ultrabarrelledif every closed linear map from E into any complete metrizable TVS iscontinuous [Iya68].

(3) E is called quasi-ultrabarrelled if every bounded linear map fromE into any complete metrizable TVS is continuous.

Every Baire TVS (in particular, F -space) is ultrabarrelled ([Edw65],p. 428). Further, every ultrabornological and every ultrabarrelled TVSis quasi-ultrabarrelled.

We next consider some linear topologies on CL(E, F ), known as theG-topologies.

Definition. Let E and F be TVSs, and let G be a collection ofsubsets of E and WF a base of closed balanced neighborhoods of 0 in F .We topologize CL(E, F ), as follows. For any D ∈ G and W ∈ WF , let

U(D,W ) = T ∈ CL(E, F ) : T (D) ⊆W.

Then the family U(D,W ) : D ∈ G,W ∈ WF forms a subbase fora topology on CL(E, F ), called the topology of uniform convergence onmembers of G, or briefly, the G-topology and is denoted by tG .

Note that, for any D ∈ G and W ∈ WF ,(i) U(D, λW ) = λU(D,W ) (λ ∈ K);(ii) U(D,W ) is balanced (resp. convex) if W is so;(ii) U(D,W ) is absorbing iff T (D) is absorbed by W for each T ∈

CL(E, F ).

Theorem A.3.4. ([Scha71], p. 79-80) (a) (CL(E, F ), tG) is a TVSiff T (D) is bounded in F for each D ∈ G and T ∈ CL(E, F ). In particu-lar, if G is the collection of all bounded subsets of E, then (CL(E, F ), tG)is a TVS.

(b) In addition, if E = ∪D : D ∈ G, then (CL(E, F ), tG) is Haus-dorff; if F is locally convex, so is (CL(E, F ), tG).

The most important cases of the G-topology are as follows:


Definition. (1) If G is the collection of all bounded subsets of E, thenthe G-topology on CL(E, F ) is called the topology of uniform convergenceon bounded sets ( or the uniform topology) and is denoted by tu.

(2) If G is the collection of all precompact subsets of E, then theG-topology on CL(E, F ) is called the topology of precompact convergenceand is denoted by tpc.

(3) If G is the collection of all compact subsets of E, then the G-topology on CL(E, F ) is called the topology of compact convergence andis denoted by tc.

(4) If G is the collection of all finite subsets of E, then the G-topologyon CL(E, F ) is called the topology of pointwise convergence and is de-noted by tp.

Clearly, tp ≤ tc ≤ tpc ≤ tu. Since each T ∈ CL(E, F ) maps boundedsets of E into bounded sets in F , it follows from the above theorem thatCL(E, F ) with each of these topologies is a TVS. We also mention that,if F = K, we have CL(E, F ) = CL(E,K) = E∗ and so the weak topologyw(E∗, E) and the strong topology β(E∗, E) on E∗are particular cases ofthe G-topologies.

Theorem A.3.5. ([Edw65], p. 87; [Sch71], p. 85) Let E and FTVSs.

(a) If E is ultrabornological (in particular, metrizable) and F iscomplete (resp. quasi-complete), then (CL(E, F ), tu) is complete (resp.quasi-complete).

(b) If E is a Baire space (in particular, an F-space) and F is quasi-complete, then (CL(E, F ), tu) is quasi-complete.

Proof. We only prove the first part of (a). Let Tα : α ∈ I be aCauchy net in (CL(E, F ), tu). Then, for each x ∈ E, Tα(x) : α ∈ I isa Cauchy net in F , as follows. [Let x ∈ E, and consider any W ∈ WF .Then U(x,W ) is a tu-neighborhood of 0 in CL(E, F ). Since Tα is aCauchy net in (CL(E, F ), tu), there exists αo ∈ I such that

Tα(x)− Tβ(x) ∈ W for all α, β ≥ αo.

Hence Tα(x) is a Cauchy net in F .] Since F is complete, the mappingT : E → F given by

T (x) = limαTα(x), x ∈ E,

is well-defined. Clearly T is linear. To show that Tαtu−→ T , let D be a

bounded set in E and W ∈ WF . Since Tα is tu-Cauchy, there existsαo ∈ I such that

Tα(x)− Tβ(x) ∈ W for all x ∈ D and α, β ≥ αo.


Since W is closed, fixing α ≥ αo and taking limβ, we have

Tα(x)− T (x) ∈ W for all x ∈ D and α ≥ αo.

HenceTα − T ∈ U(D,W ) for all α ≥ αo.

Therefore Tαtu−→ T .

We next show that T is continuous. Since E is ultrabornological, bydefinition, CL(E, F ) = BL(E, F ) and so it suffices to show that T isbounded (i.e. T maps bounded sets into bounded sets). Let D be anybounded set in E. To show that T (D) is bounded in F , let W ∈ WF .Choose a closed and balanced V ∈ WF such that V + V ⊆ W . Since

Tαtu−→ T , there exists αo ∈ I such that

Tα(x)− T (x) ∈ V for all x ∈ D and α ≥ αo.

Since Tαo , being continuous, is bounded, there exists r ≥ 1 such thatTαo(D) ⊆ rV. Now, for any x ∈ D,

T (x) = (T (x)− Tαo(x)) + Tαo(x) ∈ V + rV ⊆ r(V + V ) ⊆ rW ;

that is,T (D) ⊆ rW . Hence T is bounded or, equivalently, T is continu-ous. Thus T ∈ CL(E, F ), and so (CL(E, F ), tu) is complete.

Remark. In general, (CL(E, F ), tp) need not be complete. In fact,completeness results in the other G-topologies are mostly of negativecharacter. If (E, τ) is a locally convex space, then (E∗, w(E∗, E)) iscomplete iff E is finite dimensional; hence, if E is an infinite dimensionalmetrizable locally convex space, then (E∗, w(E∗, E)) is never complete.However, if E is complete, then (E∗, w(E∗, E)) is quasi-complete.

Definition: Let E and F be TVSs. Then a subset F of CL(E, F )is said to be

(i) equicontinuous if for each neighborhood V of 0 in F , there existsa neighborhood U of 0 in E such that T (U) ⊆ V for all T ∈ F ;

(ii) uniformly bounded if, each bounded set B in E,

∪T (B) : T ∈ F is a bounded set in F ;

(iii) pointwise bounded if, each x ∈ E,

T (x) : T ∈ F is a bounded set in F.

Clearly, if F ⊆ CL(E, F ) is uniformly bounded, then it is pointwisebounded. Further, we obtain:

Lemma A.3.6. ([Scha71], p. 83, 85) Let E and F be TV Ss, andlet F ⊆ CL(E, F ) be equicontinuous. Then:


(a) F is uniformly bounded (i.e. tu-bounded).(b) The topologies tpc and tp coincide on F .

Proof. (a) Let D be a bounded set in E andW a neighborhood of 0in F . Since F is equicontinuous, there exists a neighborhood G of 0 inE such that

T (G) ⊆W for all T ∈ F .

Since D is a bounded set in E, there exists r > 0 such that D ⊆ rG.Then

T (D) ⊆ rT (G) ⊆ rW for all T ∈ F .

Thus ∪T (D) : T ∈ F is a bounded set in F .(b) Since tp ⊆ tpc, we need to prove that tpc ⊆ tp on F . Let To ∈ F ,

and let K be a precompact subset of E and W ∈ WF . We show thatthere exists a finite set D ⊆ K and V ∈ WF such that

[To + U(D, V )] ∩ F ⊆ To + U(K,W ). (1)

Choose a balanced V ∈ W with V +V +V ⊆W. Since F is equicontinuousat 0, there exists a balanced neighborhood G of 0 in E such that

T (G) ⊆ V for all T ∈ F . (2)

Now K is precompact, there exists a finite set D = yi : 1 ≤ i ≤ m ⊆ Ksuch that

K ⊆ D +G.

Now, let T ∈ [To+U(D, V )]∩F . Then T = To+S, where S ∈ U(D, V )]∩F . By (2) and (3),

S(K) ⊆ S(D + G) ⊆ S(D) + S(G)

⊆ V + (T − To)(G) ⊆ V + V + V ⊆W.

Thus T = To + S ∈ To + U(K,W ). This proves (1).

Theorem A.3.7. (Principle of uniform boundedness) [RobW58] Let(E, τ) be an ultrabarrelled TVS and F any TVS. Let F ⊆ CL(E, F ) bea collection such that, for each x ∈ E, the set F(x) = T (x) : T ∈ Fis bounded in F . Then F is equicontinuous; hence, for any bounded setD in E, ∪T (D) : T ∈ F is a bounded set in F .

Proof. ([Edw65], p. 465) Let WE and WF denote a base of closedneighborhoods of 0 in E and F , respectively. (Let V be a closed neigh-borhood of 0 in F . We show that there exists a neighborhood V ∗ of 0 inE such that T (V ∗) ⊆ V for all T ∈ F .)

For each V ∈ WF , let

V ∗ = ∩T−1(V ) : T ∈ F.


Then V ∗ is τ -closed and absorbing in E, as follows. [Since the inverseimage of a closed set under a continuous mapping is a closed set and alsothe intersection of any number of closed sets is closed, it follows that V ∗

is τ -closed in E. Since F is pointwise bounded, for each x ∈ E, thereexists rx ≥ 1 such that

T (x) ∈ rxV for all T ∈ F . (2)

i.e. x ∈ rx(∩T−1(V ) : T ∈ F) = rxV∗. This implies that V ∗ is

absorbing.] Then the collection

U = V ∗ : V ∈ WF

may be taken as a base of neighborhoods of 0 for a vector topology τ ∗onE. Since (E, τ) is ultrabarrelled, τ ∗ ≤ τ and so each V ∗ is a neighborhoodof 0 in E. Thus T (V ∗) ⊆ V for all T ∈ F , and so F is equicontinuous.

Theorem A.3.8. (Banach-Steinhaus) Let E be an ultrabarrelledTVS and F is a quasi-complete TVS. Let Tα : α ∈ I be a net inCL(E, F ) such that, for each x ∈ E,

T (x) = limα

Tα(x)

exists. Then T is continuous on E and Tα → T uniformly on precompactsubsets of E. Further, if the net Tα : α ∈ I is replaced by a sequenceTn, it is enough that F be sequentially complete.

Proof. (see [Edw65], p. 465) It is easy to verify that T is linear.Since Tα : α ∈ J is pointwise bounded, by the principle of uniformboundedness, Tα : α ∈ J is equicontinuous. We now show that T iscontinuous. Let W be a closed neighborhood of 0 in F . Since Tα :α ∈ J is equicontinuous, there exists a neighborhood U of 0 in E suchthat Tα(U) ⊆ W for all α ∈ J . Since W is closed, for each x ∈ U ,limα Tαx ∈ W ; that is, for each x ∈ U , Tx ∈ W . So T (U) ⊆ W andhence T is continuous. Next, by Lemma A.3.6(b), the topologies tpc andtp coincide on A; hence Tα → T uniformly on precompact subsets ofE.

We next state the open mapping and the closed graph theorems.

Definition. Let E and F be TVSs. A mapping T : E → F iscalled open if T (U) is open in F for every open set U in E. T is calledopen at x ∈ E if T (U) contains a neighborhood of T (x) whenever U is aneighborhood of x in E. It is clear that T is open iff T is open at everypoint of E. Because of the invariance of vector topologies, it follows thata linear mapping T : E → F is open iff it is open at 0 ∈ E.


Theorem A.3.9. (Open mapping theorem) [RobW58] Let E be anF-space and F an ultrabarrelled TVS, and let T : E → F be a continuousand onto linear mapping. Then T is an open mapping. If, in addition Tis one-one, then T−1 : F → E is also continuous.

Proof. See ([Edw65], p. 437; [Hus65, p. 53]).As before, a mapping T : E → F is said to have a closed graph if

its graph G(T ) = (x, Tx) : x ∈ E is closed in E × F or, equivalently,for any net xα in E with xα → x ∈ E and T (xα) → y ∈ F, we havey = T (x). Every continuous map T : E → F has a closed graph. Theconverse holds under some additional hypotheses in the following form.

Theorem A.3.10. (Closed graph theorem) [RobW58] Let E be anultrabarrelled TVS and F an F-space. If T : E → F is linear and hasclosed graph, then T is continuous.

Proof. See ([Edw65], p. 437; [Hus65], p. 53).


4. Shrinkable Neighborhoods in a TVS

In this section, we study the notion of a shrinkable neighborhood of0 in a TVS E, as given by Klee [Kl60a, Kl60b].

Definition. A neighborhoodW of 0 in a TVS E is said to be shrink-able if

tW ⊆W 0 for all 0 ≤ t < 1.

Lemma A.4.1. Any convex neighborhood V of 0 in a TVS E isshrinkable.

Proof. Let ρV be the Minkowski functional of V . Then, since ρV iscontinuous, we have

V 0 = x ∈ E : ρV (x) < 1, V = x ∈ E : ρV (x) ≤ 1.

Let y ∈ V and 0 ≤ t < 1. Now

1− ρV (ty) ≥ ρ(y)− tρV (y) = (1− t)ρV (y) > 0.

Hence ρV (ty) < 1 and so ty ∈ V 0. Thus tV ⊆ V 0.

Important facts about shrinkable neighborhoods are:(A) Every Hausdorff TVS E has a base of shrinkable neighborhoods

of 0.(B) If W is a shrinkable neighborhood of 0 in a TVS E, then its

Minkowski functional ρW is continuous and positively homogeneous.The proofs of these important results, due to Klee [Kl60a], may not

be found in the text books and therefore we include them here. We firstneed to prove the following three results.

Theorem A.4.2. Every Hausdorff TVS is linearly homeomorphicwith a linear subspace of a product of metric linear spaces.

Proof. Let E be a Hausdorff TVS andW the class of all open balancedneighborhoods W of 0 in E. For each W ∈ W, let Wo,W1,W2, .... be asequence of members of W such that

Wo = W, and Wn +Wn ⊆Wn−1 for n = 1, 2, ....

Let MW =∞∩n=o

Wn. Then clearly

MW = [−1, 1]MW and MW +MW ⊆MW ,

and so MW is a linear subspace of E. Let πW be the quotient map of Eonto E/MW . Let E/MW be topologized by taking πW (Wn) : n ≥ 1 asa base of neighborhoods of 0. It is easily verified that E/MW is a TVSand πW is continuous. Further,

π−1W (πW (Wn)) =Wn +MW ⊆Wn +Wn ⊆Wn−1,

4. SHRINKABLE NEIGHBORHOODS IN A TVS 269

and so it follows that∞∩n=o

πW (Wn) = MW. Thus the topology of E/MW

is Hausdorff; further it is metrizable since it has a countable base ofneighborhoods of 0. Let P = Π

W∈WE/MW (the product space), let π :

E → P be defined by

(π(x))W = πW (x), x ∈ E, W ∈ W.

Then π(E) is a linear subspace of P and π is a linear homeomorphism.

This next two lemmas are easy to verify.

Lemma A.4.3. Suppose E is a Hausdorff TVS,M a vector subspaceof E, and W a shrinkable neighborhood of 0 in E. Then W ∩M is ashrinkable neighborhood of 0 in M .

Lemma A.4.4. Suppose Eα : α ∈ I is a family of HausdorffTVSs, Wα is a shrinkable neighborhood of 0 in Eα for each α ∈ I, andWα = Eα for all but finitely many α ∈ I. Then the set Π

a∈IWα is a

shrinkable neighborhood of 0 in the product space Πa∈I

Eα.

Theorem A.4.5. In a Hausdorff TVS E, every neighborhood of 0contains a shrinkable neighborhood of 0.

Proof. In view of the above results, we may assume that E is a metriz-able TVS. According to a classical result of Edelheit and Mazur (see[DS58], p.91), the topology of E can be generated by an invariant metricd which is strictly monotonic in the sense that

d(tx, 0) < d(sx, 0) for 0 ≤ t < s and x 6= 0.

We need to show that, for any ε > 0, B(0, ε) = x ∈ E : d(x, 0) < ε isshrinkable. [Let y ∈ B(0, ε) and 0 ≤ t < 1. To show that ty ∈ [B(0, ε)]0,we show that there exists a δ > 0 such that B(ty, δ) ⊆ B(0, ε). Sincet < 1, d(ty, 0) < d(y, 0) and so δ = d(y, 0)−d(ty, 0) > 0. Let z ∈ B(ty, δ).Then

d(z, 0) ≤ d(z, ty) + d(ty, 0) < δ + d(ty, 0) = d(y, 0) < ε.

Hence z ∈ B(ty, δ) ⊆ B(0, ε). Thus ty ∈ [B(0, ε)]0 and so B(0, ε) isshrinkable.]

Theorem A.4.6. Let W be a balanced neighborhood of 0 in a Haus-dorff TVS E and let ρ = ρW be the Minkowski functional of W . Then:

(a) ρ is upper semicontinuous iff W 0 = x ∈ E : ρ(x) < 1;(b) ρ is lower semicontinuous iff W = x ∈ E : ρ(x) ≤ 1.(c) ρ is continuous iff W is shrinkable.


Proof. We first note that

W 0 ⊆ x ∈ E : ρ(x) < 1 ⊆W ⊆ x ∈ E : ρ(x) ≤ 1 ⊆W. (1)

(a) From (1) it follows that W 0 = x ∈ E : ρ(x) < 1 iff the latter setis open. Since ρ is positively homogeneous, openness of the set x ∈ E :ρ(x) < 1 is equivalent to that of the set x ∈ E : ρ(x) < t for all t > 0,and that in turn is equivalent to the upper semicontinuity of ρ.

(b) Again by (1) we note that W = x ∈ E : ρ(x) ≤ 1 iff the latterset is closed. Since ρ is positively homogeneous, closedness of the setx ∈ E : ρ(x) ≤ 1 is equivalent to that of the set x ∈ E : ρ(x) ≤ rfor all r > 0, and that is equivalent to the lower semicontinuity of ρ.

(c) Suppose ρ is continuous. To show that W is shrinkable, let 0 ≤t < 1. Then

y ∈ t.W = tx : ρ(x) ≤ 1

⇒ y = tx, ρ(x) ≤ 1

⇒ ρ(y) = tρ(x) < 1

⇒ y ∈ z ∈ E : ρ(x) < 1 = W 0,

as required.Conversely, suppose that W is shrinkable. We first show that ρ is

lower semicontinuous or that A = x ∈ E : ρ(x) < 1 is open. Lety ∈ A. Let t = max1

2, ρ(y) < 1. Then

ρ(1

ty) =

1

tρ(y) ≤ 1, and so y ∈ tW ⊆W 0 ⊆ A.

Hence A is open. To establish upper semicontinuity of ρ, we need toshow that x ∈ E : ρ(x) > 1 is open or that B = x ∈ E : ρ(x) ≤ 1 isclosed. We claim that B = W . Clearly, B ⊆ W . Suppose W B, andlet z ∈ W\B. Then r = ρ(z) > 1, and so ρ( 1

r1/2z) = (r)1/2 > 1. Now,

since 1r1/2

< 1 and W is shrinkable,

1

r1/2z ∈

1

r1/2W ⊆ W 0 ⊆ x ∈ E : ρ(x) < 1,

and so ρ( 1r1/2

z) < 1, a contradiction. Hence B = W , and so B is closed.Thus ρ is both lower and upper semicontinuous, and hence it is continu-ous.

Theorem A.4.7. Let W be a closed shrinkable neighborhood of 0in a TVS E, ρ = ρW the Minkowski functional of W , and h : E → E isgiven by

h(x) =

x if x ∈ W,

xρ(x)

if x ∈ E\W.

4. SHRINKABLE NEIGHBORHOODS IN A TVS 271

Then h is continuous.Proof. Let A = W = x ∈ E : ρ(x) ≤ 1 and B = x ∈ E : ρ(x) ≥

1. Then A and B are closed with E = A ∪ B. Define f : A → R andg : B → R by

f(x) = x, x ∈ A,

g(x) =x

ρ(x), x ∈ B.

Hence, by the Map Gluing Theorem, h is continuous on E.


5. Non-Archimedean Functional Analysis

In this section, we consider the notions of non-Archimedean TVSs,locally F-convex spaces and almost non-Archimedean TVSs.

Definition. Let F be any field with unit element e. Then a function| · |: F→ R is called a valuation on F if, for any a, b ∈ F,

(i) |a| ≥ 0, with |a| = 0 iff a = 0;(ii) |ab| = |a|.|b|;(iii) |a+ b| ≤ |a|+ |b|.In this case, the pair (F, | · |) is called a valued field. Clearly, |e| = 1.

Definition. A valued field (F, | · |) is called a non-Archimedean field(in short, an NA field) if

|a+ b| ≤ max|a|, |b| for all a, b ∈ F (Strong Triangular inequality).(ST)

Remark. Recall that an important property of the field R is that ithas the Archimedean Property in the sense that, given any x > 0 in R,there exists an n ∈ N such that x ≤ n; equivalently, N is not boundedin R. The proof of this fact is based on the supremum (or completeness)axiom of R. However, if (F, | · |) is an NA valued field, then, using (ST)inequality, for any integers n ∈ N,

| ne |=| e + e+ ....... + e |≤ max| e |, | e |, ..., | e | =| e |= 1;

hence the set Ne is bounded in (F, | · |). Thus (F, | · |) does nothave the Archimedean property. This justifies the use of the term non-Archimedean for such fields. The formula (ST) is essential to the entirenon-Archimedean theory.

Examples (1) Clearly, R and C with the usual absolute value arevalued fields but not non-Archimedean.

(2) Any field F with the trivial valuation

| a |=

1 if a 6= 00 if a = 0

is an NA field. All other valuations on F are called non-trivial valuations.(3) Let F = Q, the field of rationals, and let p be any fixed prime

number. Every x ∈ Q, x 6= 0,can be written uniquely as x = pk · ab,

where k ∈ Z = 0,±1,±2, ........ and a, b cannot be divided by p. Definex→| x |pby

| x |p=

p−k if x 6= 0,0 if x = 0.

5. NON-ARCHIMEDEAN FUNCTIONAL ANALYSIS 273

Then | . |p is an NA valuation on Q, called the p-adic valuation on Q.Note that (Q, | x |p), being a metric space, has a completion, denoted byQp. Then Qp is called the field of p-adic numbers.

Remark. Let E be a vector space over a valued field (F, | · |). Thenthe definition of a TVS E over F is identical with that of a TVS E over K.However, the result on the characterization of a base of neighborhoods of0 with the desired properties needs some modification. In fact, in mostcases, we require that the valued field (F, | . |) is non-trivial. The reasonis that, for any real r > 0, we can choose a, b ∈ F such that 0 < |a| ≤ rand |b| ≥ r. [Choose c ∈ F such that |c| 6= 0 and |c| 6= 1. Clearly, if0 < |c| ≤ r, then |c−1| ≥ r; if |c| ≥ r, then 0 < |c−1| ≤ r.]

Definition. A subset S of vector space E over F is called a non-Archimedean set (in short, an NA set) if S + S ⊆ S. By induction, forany k ∈ N, the k-fold sum

S + S + ......... + S ⊆ S.

A TVS (E, τ) over a valued field (F, | · |) is called a non-ArchimedeanTVS (in short, an NA TVS) if it has a base of τneighborhoods of 0consisting of NA sets.

Examples. (1). Every NA valued field (F, | · |) is an NA TVS overitself.

(2). Any indiscrete TVS is an NA TVS.(3). The trivial TVS E = 0 is an NA TVS.

Theorem

A.5.3. Let (E, τ) be a TVS over a valued field (F, | · |) . If V is anyNA neighborhood of 0 in E, then V is clopen (i.e., both closed and open).

Proof. V is open, as follows. Let y ∈ V . Then y + V ⊆ V + V ⊆V (since V is NA), so y has a neighborhood y+V with y+V ⊆ V. HenceV is open.

V is closed, as follows. We need to show that E \ V is open. Let x ∈E\V. Then V ∩ (x− V ) = ∅. [Suppose V ∩ (x− V ) 6= ∅, and let z ∈V ∩(x− V ) . Then z = y1, z = x−y2 with y1, y2 ∈ V, and so y1 = x−y2.Hence

x = y1 + y2 ∈ V + V ⊆ V,

a contradiction since x ∈ E \ V .] Therefore, x− V ⊆ E\V. Since x− Vis a neighborhood of x, E \ V is open. Thus V is closed.

Recall that a topological space X ( 6= ∅) is said to have dimension 0at a point x ∈ X if x has a local base Bx of neighborhood consisting of


sets which are both open and closed; X is said to be 0-dimensional if Xhas a dimension 0 at each point of X .

Theorem A.5.4. Every NA TVS (E, τ) is a 0-dimensional topolog-ical space.

Proof. Since (E, τ) is an NA TVS, it has a baseW of τneighborhoodsof 0 consisting of NA sets. Then, by above theorem, each V ∈ W is bothopen and closed. Hence, for each x ∈ E, Bx = x+ V : V ∈ W consistsof both open and closed neighborhoods of x.

Theorem

A.5.5. Let (E, τ) be an NA TVS over a non-trivially valued field(F, | · |). Then there exist a base of neighborhoods of 0 consisting of NA(hence clopen) ’balanced’ sets.

6. TOPOLOGICAL ALGEBRAS AND MODULES 275

6. Topological Algebras and Modules

Definition. Let A be an algebra over K (=R or C) and τ a topologyon A. Then the (algebra) multiplication (x, y)→ xy from A×A→ A iscalled:

(i) separately continuous if, for any τneighborhood W of 0 in A andany x ∈ A, there exists a τneighborhood V of 0 in A such that xV ⊆Wand V x ⊆W .

(ii) hypocontinuous if, for any τneighborhood W of 0 in A and anyτ -bounded set D in A, there exists a τneighborhood V of 0 in A suchthat DV ⊆W and V D ⊆W .

(iii) jointly continuous if, for any τneighborhood W of 0 in A, thereexists a τneighborhood U of 0 in A such that U2 ⊆ W, where U2 =U.U = xy : x, y ∈ U.

Clearly, joint continuity ⇒ hypocontinuity ⇒ separate continuity.

Definition: Let A be an algebra over K (=R or C) and τ a topologyon A such that (A, τ) is a TVS. Then the pair (A, τ) is called a topolog-ical algebra if it has a separately continuous multiplication. A completemetrizable topological algebra is called an F -algebra.

Recall that a topological space X is called a Baire space if wheneverX = ∪∞n=1An with each An closed, then, for at least one m, int(Am) 6= ∅.

Theorem A.6.1. (Arens [Are47])

Every Baire metrizable topological algebra (A, τ) has jointly con-tinuous multiplication. In particular, every F -algebra (A, τ) has jointlycontinuous multiplication.

Proof. ([Hus83], p. 5-6) Since (A, τ) is metrizable, there exists ametric d on A such that τ = τd. To show that the map (x, y) −→ xy isjointly continuous at (0, 0) ∈ A×A, let W be any τd-neighborhood of 0in A. We shall find a τd-neighborhood U of 0 in A such that U.U ⊆W .

For each n ≥ 1, let B(0, 1n) = x ∈ A : d(x, 0) < 1

n. Then B(0, 1

n)

is a countable base of τd-neighborhoods of 0. Choose a closed balancedτneighborhood V of 0 in A such that V + V ⊆ W . Let An = x ∈ A :xB(0, 1

n) ⊆ V . Clearly AnB(0, 1

n) ⊆ V . We now show that (i) each An

is closed; (ii) A = ∪∞n=1An.(i) An is closed. [Let x ∈ An. Then there existsxk ⊆ An such that

xkτd−→ x. Then xkB(0, 1

n) ⊆ V for all k ≥ 1. By separate continuity of

multiplication, for any y ∈ B(0, 1n), xky −→ xy ∈ V . SinceV is closed,

V = V , and so xy ∈ V . Therefore xB(0, 1n) ⊆ V , and so x ∈ An. Hence

An is closed.]


(ii) A = ∪∞n=1An. [Clearly ∪∞n=1An ⊆ A. To show A ⊆ ∪∞n=1An, letx ∈ A. Since B(0, 1

n) is a base of τd-neighborhoods of 0, by sep-

arate continuity of (x, y) −→ xy at (x, 0), there exists a B(0, 1N) ∈

B(0, 1n)such that xB(0, 1

N) ⊆ V . Hence x ∈ AN ⊆ ∪∞n=1An. This

proves that A ⊆ ∪∞n=1An.]Since A is a Baire space and A = ∪∞n=1An with each An closed, by

the Baire Category theorem, int(Am) 6= ∅ for some m ≥ 1. Let xo ∈int(Am). Then there exists ε > 0 such that

B(xo, ε) ⊆ Am. (1)

Now U = B(0, ε)∩B(0, 1m) is a τd-neighborhood of 0 in A. We now show

that U.U ⊆ W . Clearly, U.U ⊆ B(0, ε)B(0, 1m). We need to show that

B(0, ε)B(0, 1m) ⊆ W. [For any x ∈ B(0, ε), d(x + xo, xo) = d(x, 0) < ε

and so, by (1),

x+ xo ∈ B(xo, ε) ⊆ Am;

hence, by definition of Am,

xB(0,1

m) = [(x+ xo)− xo]B(0,

1

m) ⊆ (x+ xo)B(0,

1

m)− xoB(0,

1

m)

⊆ V − V ⊆W.

So B(0, ε)B(0, 1m) ⊆W.] Thus U.U ⊆W.

Note: (1) In general, separate continuity of multiplication need notimply joint continuity even in algebras which are normed spaces but notcomplete (see [Rud91], p. 247, 272).

(2) Every ultrabarrelled topological algebra has hypocontinuous mul-tiplication (see Theorem A.6.28 below).

(3) [Fra05, p. 8] If (A, τ) is a topological algebra, then its completion

A (i.e. the completion of the underlying TVS (A, τ)) need not be topo-logical algebra unless the multiplication of (A, τ) is jointly continuous.

(4) In some books/papers, a topological algebra is assumed to havejointly continuous multiplication; but we shall not use this notion here.

Definition. A subset U of an algebra A is called idempotent (ormultiplicative or an m-set) if U2 ⊆ U .

Definition. A topological algebra (A, τ) is called:(i) a locally convex algebra if it has a base W=WA of neighborhoods

of 0 consisting of convex sets; or equivalently, its topology τ is generated


by a family sA of seminorms such that, for each p ∈ sA, there exists ap1 ∈ sA such that

p(xy) ≤ p1(x)p1(y) for all x, y ∈ A;

(ii) a locally m-convex algebra if it has a base W of neighborhoodsof 0 consisting of convex and idempotent sets (i.e. the sets U such thatU2 ⊆ U); or equivalently, if its topology τ is generated by a family sA ofseminorms which are submultiplicative, i.e. for each p ∈ sA,

p(xy) ≤ p(x)p(y) for all x, y ∈ A;

(iii) a locally idempotent algebra if it has a base W of neighborhoodsof 0 consisting of idempotent sets;

(iv) a locally bounded algebra if it has a bounded neighborhood of 0.

Examples. (1) Every normed algebra and, more generally, everylocally m-convex algebra is locally convex.

(2) Every locally m-convex algebra is locally idempotent and everylocally idempotent algebra is a topological algebra with jointly continuousmultiplication.

(3) The notion of locally idempotent algebras is a strict generalizationof the notion of locally m-convex algebras. This is shown by the algebraℓq+ (0 ≤ q < 1) consisting of all two-sided complex sequences a = ansuch that ‖a‖p =

∑∞n=1 |an|

p is convergent for each p with q < p ≤ 1 (see

[Zel60], p.348).

Theorem A.6.2. ( [Zel73], p. 32) Let (A, d) be an F -algebra withidentity e. Then the following conditions are equivalent

(a) There exists an equivalent metric ρ on A such that

ρ(xy, 0) ≤ ρ(x, 0).ρ(y, 0) for all x, y ∈ A.

(b) A is a locally bounded algebra.(c) The topology of A can be generated by a submultiplicative k-norm

‖.‖k, 0 < k ≤ 1, with ‖e‖k = 1.

Note. (1) If A is a locally m-convex algebra with identity e, thenits topology can be given by a family pα : α ∈ I of submultiplicativeseminorms such that pα(e) = 1 for all α ∈ I ([Fra05]. p. 16).

(2) If A is a locally convex algebra with identity e, we do not knowwhether its topology can be given by a family pα : α ∈ I of seminormswhich satisfy pα(e) = 1 for all α ∈ I.

Definition: Let A be an algebra. A map ∗ : x→ x∗ from A into Ais called an involution on A if it satisfies:


(i) (x+ y)∗ = x∗ + y∗,(ii) x∗

∗

= x,(iii) (λx)∗ = λx∗, λ is the complex conjugate of λ ∈ K,(iv) (xy)∗ = y∗x∗,(v) e∗ = e if A has identity e.

An algebra A equipped with an involution ∗ is called an involutivealgebra or a ∗-algebra.

Definition. A normed algebra (A, ||.||) is called:(1) a normed ∗-algebra if A has an involution such that ||x∗|| =

||x|| for all x ∈ A; in this case, clearly the involution map x → x∗ iscontinuous.

(2) a B∗-algebra (or equivalently, a C∗-algebra) if A is a Banach ∗-algebra whose norm ||.|| satisfies

||xx∗|| = ||x∗x|| = ||x||2 for all x ∈ A;

In this case, the norm ||.|| is called a C∗-norm.

Definition. A complete locally m-convex ∗-algebra (A, τ) is called alocally C∗-algebra if its topology τ is determined by a family pα : α ∈ Iof submultiplicative seminorms satisfying

pα(x∗x) = [pα(x)]

2 for all x ∈ A.

The seminorms pα are called C∗-seminorms.

Example. (Arens Algebra)

([Are46]; [Mal86], p. 12)

Let

Lω[0, 1] =∞∩p=1Lp[0, 1],

where each Lp[0, 1], p ≥ 1, is a normed space with respect to the norm:

||f ||p = (o |f(t)|p dt)

1/p, f ∈ Lp[0, 1].

Then:(a) Lω[0, 1] is an algebra and ||f ||1 ≤ ||f ||2 ≤ ...∀f ∈ Lω[0, 1]. Note

that: ||fg||p ≤ ||f ||q||g||r for 1/p = 1/q + 1/r; hence Lω[0, 1] is non-normed algebra.

(b) For any p ≥ 1 and ε > 0, let Bp(0, ε) = f ∈ Lω[0, 1] : ‖f‖p < ε.Then the collection Bp(0, ε) : p ∈ N, ε > 0 forms a base of neighbor-hoods of 0 for a Hausdorff locally convex topology τω on Lω[0, 1].


(c) (Lω[0, 1], τω) is a complete metric space w.r.t. the metric:

dw(f, g) =∞∑

p=1

2−p||f − g||p

1 + ||f − g||p, f, g ∈ Lω[0, 1].

(d) (Lω[0, 1], τω) is not locally m-convex.

Definition: Let A be an algebra with identity e and let x ∈ A. Thenx is said to be invertible (or regular) if there exists some y ∈ A, calledthe inverse of x, such that xy = yx = e; in this case, we write y = x−1.

Definition: Let A be an algebra. For any x, y ∈ A, the q-productx y of x, y is defined by

x y = x+ y − xy.

An element x ∈ A is said to be q-invertible if there exists some y ∈ A,called the q-inverse of x such that xy = y x = 0; in this case, we writey = x−1.

Definition: Let A be an algebra, and let x ∈ A.(1) If A has an identity e, then the spectrum of x is defined by

σA(x) = λ ∈ C : λe− x is not invertible in A.

(2) If A is without identity, then the spectrum of x is defined as

σA(x) = λ ∈ C : λ 6= 0,x

λis not q-invertible in A ∪ 0.

(3) The spectral radius of x is defined as

rA(x) = sup|λ| : λ ∈ σA(x).

Clearly, 0 ≤ rA(x) ≤ ∞, rA(0) = 0, rA(αx) = |α|rA(x) (α ∈ C), andrA(e) = 1 if A has an identity e.

Example. Let X be a locally compact Hausdorff space.(1) If X is compact, for any f ∈ Co(X) = C(X),

σC(X)(f) = f(x) : x ∈ X = f(X).

(2) If X is non-compact, then

σCo(X)(f) = f(X) ∪ 0 for all f ∈ Co(X);

σC(X)(f) = f(X) ⊆ f(βX) = f(X) = σCb(X)(f) for all f ∈ Cb(X),

where βX is the Stone-Cech compactification of X .


Definition. A topological algebra A with identity (resp. withoutidentity) is said to be a Q-algebra if the set G(A) of all invertible elementsof A (resp. Gq(A) of all quasi-invertible elements of A) is open in A.

Examples. (1) Every complete locally bounded Hausdorff topologi-cal algebra (hence every Banach algebra) is a Q-algebra ([Mal86], p.42).

(2). An incomplete normed algebra need not be a Q-algebra, sinceneither G(A) nor Gq(A) need be an open set.

(3). A complete metrizable locally m-convex algebra need not be a Q-algebra (e.g.(C(R), k), with k the compact-open topology, is a completemetrizable locally m-convex algebra but not a Q-algebra ([Hus83], p. 8).

Theorem A.6.3. ([Mal86], p. 43-44, 59) Let A be a topologicalalgebra. Then the following are equivalent:

(a) A is a Q-algebra, i.e. Gq(A) is open in A.(b) [Gq(A)]0 6= ∅.(c) Gq(A) is a neighborhood of 0 in A.(d) S(A) = x ∈ A : rA(x) ≤ 1 is a neighborhood of 0 in A.While considering the spectrum and related notions in the sequel, we

shall consider all algebras over the complex field C.

Definition: A topological algebra A with identity (resp. withoutidentity) is said to have a continuous inversion (resp. continuous quasi-inversion) if the inversion map x → x−1 from G(A) → G(A) (resp. thequasi-inversion map x→ x−1 from Gq(A)→ Gq(A)) is continuous.

In the sequel, for convenience, we shall consider mostly the results forthe more general notion of ”quasi-inversion”.

Theorem A.6.4. (a) ([Mal86], p. 52) Every locally m-convex algebra(in particular, every normed algebra) A has a continuous quasi-inversion.Conversely:

(b) ([Pa78], p. 277) Let A be a commutative locally convex F -algebrawith a continuous quasi-inversion. Then A is a locally m-convex algebra.

Example. [Are46] The Arens algebra Lω[0, 1] = ∩p≥1Lp[0, 1] is a lo-

cally convex F -algebra, but not a locally m-convex algebra and it doesnot have a continuous inversion.

Definition: ([Mal86], p. 85; [Hus83], p.10) Let (A, τ) be a locallym-convex algebra whose topology τ is generated by a family sA = p :p ∈ sA of submultiplicative seminorms. For each seminorm p ∈ sA,Np = p−1(0) is a two-sided ideal of A and so A/Np is a normed algebrawith the norm: ||πp(x)||p = p(x), where πp : A→ A/Np is the canonical


map given by

πp(x) = xp = x+Np, x ∈ A.

Let Ap = A/Np denote the completion of the normed algebra A/Np.Then Ap is a Banach algebra and each πp can be regarded as the mapfrom A to Ap. Each πp is a continuous homomorphism.

We define the partial ordering ≤ on sA by

p, q ∈ sA, p ≤ q ⇔ pp(x) ≤ pq(x) for all x ∈ A.

Then, for p ≤ q, we have Nq ⊆ Np and so there exists an algebra ho-momorphism ϕpq : A/Nq → A/Np which has the unique extension toϕpq : Aq → Ap for p ≤ q, where ϕpq(πq(x)) = πp(x) for all x ∈ A andp ≤ q. Clearly, each ϕpq is a continuous homomorphism. Let

lim←−

A/Np = (πp(x)) ∈ Πp∈sAA/Np : ϕpq(πq(x)) = πp(x) for p ≤ q, x ∈ A,

lim←−

Ap = (πp(x)) ∈ Πp∈sAAp : ϕpq(πq(x)) = πp(x) for p ≤ q, x ∈ A.

Clearly, lim←−

A/Np (resp. lim←−

Ap) is a subalgebra of the product Πp∈sA

A/Np (resp. Πp∈sAAp) of normed (resp. Banach) algebras. lim←−

A/Np

(resp. lim←−

Ap) is called the projective limit of A/Np’s (resp. Ap’s).

Theorem A.6.5. ([Mal86], p. 88; [Mi52], p. 13; [Zel73], p.33) Let(A, τ) be a locally m-convex algebra whose topology τ is generated by afamily sA = p : p ∈ sA of submultiplicative seminorms. Then

A ⊆ lim←−

A/Np ⊆ lim←−

Ap ⊆ A,

within topological algebraic isomorphism. If, in addition, A is complete,then

A = lim←−

A/Np = lim←−

Ap,

within topological algebraic isomorphism.Proof. With the notations in the above definition, we can define

ϕ : A → lim←−

A/Np by ϕ(x) = (πp(x))p, i.e. the pth projection of ϕ(x)

equals πp(x), x ∈ A. It is clear that ϕ(x) = 0 iff πp(x) = 0 for all p ∈ sAiff p(x) = 0 for all p ∈ sA iff x = 0, because A is Hausdorff. Hence ϕ isa one-one map. Further, since the topology τ of A is identical with theCartesian product topology of its image ϕ(A) in

∏p∈sA

A/Np, it follows that

ϕ is a homeomorphism of (A, τ) onto the subalgebra ϕ(A) of∏p∈sA

A/Np

and so A is isomorphic and homeomorphic to a subalgebra of a Cartesian


product of a family of normed algebras. Hence A can be identified witha subalgebra of Πp∈sA A/Np, the product of normed algebras.

If A is metrizable, then the indexing set sA is countable and so eachmetrizable locally m-convex algebra is identifiable with a subalgebra ofa countable product of normed algebras.

Theorem A.6.6. ([Hus83], p. 11; [Zel73], p. 33) Let (A, τ) be alocally m-convex F -algebra. Then A is isomorphic and homeomorphicto a closed subalgebra of Cartesian product of Banach algebras.

Theorem

A.6.7. ([Mal86], p. 85; [Hus83], p.10) Let (A, τ) be a complete lo-cally m-convex algebra whose topology τ is generated by a family sA ofsubmultiplicative seminorms. Then:

(a) σA(x) = ∪p∈sAσAp(xp), x ∈ A.(b) rA(x) = supp∈sA rAp(xp) = supp∈sA limn→∞[p(x

n)]1/n, x ∈ A.In the sequel, most of the results stated for locally m-convex algebras

actually hold for the more general class of locally convex spaces havingcontinuous inversion/ quasi-inversion (see [Mal86]).

Theorem A.6.8. (1) (

[Mal86], p. 58; [Fra05], p. 46) If A is a locally m-convex algebra,then, for each x ∈ A, σA(x) is a non-empty (possibly non- compact)subset of C; hence the spectral radius rA(x) is finite if exists.

(2) (

[Mal86], p. 60; [Fr05], p. 74) If A is a Q-algebra, then, for eachx ∈ A, σA(x) is a (possibly empty) compact subset of C; hence the spectralradius rA(x) is finite if exists.

(3) If A is a locally m-convex Q-algebra, then, for each x ∈ A, σA(x)is a non-empty compact subset of C ; hence rA(x) is finite and exists.

Definition A.6.9. Let A be an algebra. A subalgebra I of A iscalled a left ideal (resp. right ideal) if AI ⊆ I (resp. IA ⊆ I; it iscalled a two-sided ideal (or simply, an ideal) if it is both a left and rightideal. A left (resp. right, two-sided) I in A is called a proper ideal ifI 6= 0 and I 6= A. A proper left, right or two-sided ideal M in A iscalled a maximal left, right or two-sided ideal if, for any ideal (left, rightor two-sided, respectively) L, M ⊆ L ⊆ A implies that L =M or L = A.

Definition. ([Zel73], p. 27-28; [BD73], p.45) Let A be an algebra.A left (resp. right) ideal I in A is called a modular left (resp. modular


right) ideal if there exists an element u = urI (resp. u = ulI) in A, calleda modular right (resp. modular left) identity for A, such that

xu− x ∈ I (resp. xu − x ∈ I) for all x ∈ A.

A two-sided ideal I in A is called amodular ideal if there exists an elementu = uI ∈ A, called a modular identity for A, such that

ux− x ∈ I, xu− x ∈ I for all x ∈ A.

Remarks. (1) ([Zel73], p. 27-28) A two-sided ideal I in A is modulariff A/I has an identity (the identity of A/I is the coset containing uI).By Zorn’s lemma, each proper modular ideal is contained in a maximalmodular ideal. Clearly, if an algebra A has already an identity e, thenevery two-sided ideal in A is a modular ideal with e a modular identityfor A; hence in this case, every ideal in A is modular.

(2) If A be a topological algebra, then the closure of every modularleft (resp. right, two-sided) modular ideal in A is a modular left (resp.right, two-sided) ideal.

Theorem

A.6.10. ([Mal86], p. 67; [Fra05], p. 74-75) Every maximal modularleft, right or two-sided ideal in a Q-algebra A is closed. In particular,every maximal ideal in a Q-algebra A with identity in A is closed.

Proof. (see [Mal86], p. 67) We only prove for M a maximal modulartwo-sided ideal in A with modular identity u for M . By definition M isproper and u /∈ M . Suppose M is not closed then M 6= M , and so, bymaximality of M , M = A. Since A is a Q-algebra, Gq(A) is open and sou+ Gq(A) is an open neighborhood of u. Since u ∈ A =M,

(u+Gq(A)) ∩M 6= ∅.

Let z ∈ (u + Gq(A)) ∩M . Then there exist some w ∈ Gq(A), m ∈ Msuch that

z = (u+ w) = m, or u−m = −w ∈ Gq(A).

Hence u−m has a q-inverse y (say) in A. Then

(u−m)y = 0, or u−m+y−(u−m)y = 0, or u−m+y−uy+my = 0,

or u = m+ (uy − y)−my ∈M +M +M =M.

This is a contradiction since u /∈M . Hence M =M, and so M is closed.If A has an identity e, then each maximal ideal M in A is modular

with modular identity e. Hence, by above argument, M is closed.


Note. ([Mal86], p. 67) In general, a maximal non-modular ideal in alocally m-convex algebra A may be dense in A and so need not be closed.

Definition: (a) If A and B are two algebras, then a linear mappingϕ : A→ B is called an algebra homomorphism if

ϕ(xy) = ϕ(x)ϕ(y) for all x, y ∈ A.

In particular, a non-zero algebra homomorphism ϕ : A → K is called amultiplicative linear functional or a character on A.

Note. (1) If ϕ is a multiplicative linear functional on an algebra Awith identity e, then ϕ is non-zero iff ϕ(e) = 1. [Suppose ϕ is non-zero.Clearly, e2 = e implies [ϕ(e)]2 = ϕ(e) and so either ϕ(e) = 0 or ϕ(e) = 1.If ϕ(e) = 0, then, for any x ∈ A, ϕ(x) = ϕ(xe) = ϕ(x)ϕ(e) = 0; hence ϕis identically zero, a contradiction.]

(2) ([Mal86], p. 68) Any multiplicative linear functional ϕ on analgebra is onto. [Let λ ∈ C. Choose x ∈ A with ϕ(x) = α 6= 0. Theny = λ

αx ∈ A with ϕ(y) = λ.]

Recall that the co-dimension of a vector subspaceM of a vector spaceE is defined as the dimension of its complementary subspace (or equiv-alently, the dimension of the quotient space E/M). Further, a vectorsubspace M of a vector space E is maximal iff M is of co-dimension oneiff M = ϕ−1(0) for some non-zero linear functional ϕ on E.

Theorem A.6.11. Let A be an algebra. Then:(a) If ϕ is a multiplicative linear functional on A, then M = ϕ−1(0)

is a maximal modular ideal of co-dimension one in A.(b) If M ⊆ A is a maximal modular ideal of co-dimension one, then

there exists a unique multiplicative linear functional ϕ on A such thatM = ϕ−1(0). (This need not hold if M is not of co-dimension one.)

(c) There is a one-one correspondence between the set ∆(A) of multi-plicative linear functional on A and the setM(A) of all maximal modularideals of co-dimension one in A determined by parts (a) and (b).

Proof. We give the proof only for the case of A an algebra withidentity e.

(a) If ϕ is a nonzero multiplicative linear functional, then it is easy tosee thatM = ϕ−1(0) is a maximal ideal. To see thatM is of co-dimension(i.e. A/ϕ−1(0) is one-dimensional), let x ∈ A. Then

ϕ(x−ϕ(x)

ϕ(e)e) = ϕ(x)− ϕ(x) = 0, or that x−

ϕ(x)

ϕ(e)e ∈ ϕ−1(0) =M ;


hence

x ∈M +ϕ(x)

ϕ(e)e ⊆M ⊕ Ce.

Therefore, A = M ⊕ Ce (direct sum). Hence M = ϕ−1(0) is a maximalideal of codimension one.

(b) Let M be a maximal ideal of A of codimension one. Then e /∈Mbecause otherwise M = A, which contradicts the fact thatM is a propersubset of A by definition. Hence A = M ⊕ Ce. Thus, for each x ∈ A,there is a unique λ ∈ C and m ∈ M with x = m + λe. Now defineϕ : A→ C by

ϕ(x) = λ for x = m+ λe ∈M ⊕ Ce = A.

It is easy to verify that ϕ is a multiplicative linear functional with M =ϕ−1(0). Further, ϕ(e) = 1, and so ϕ is nonzero.

(c). This follows from (a) and (b).

Note. (1) If M is a maximal ideal of an algebra A, then M neednot be of co-dimension one; in fact, M may be of infinite co-dimension([Zel73], p. 37, Exercise 8.4.a.).

(2) However, if A is a complex Banach algebra, every maximal modu-lar idealM of A is automatically closed and of co-dimension one ([BD73],p. 79); also then the corresponding multiplicative linear functional ϕ isautomatically continuous. Consequently, in this case, an analogue of theabove theorem holds for commutative Banach algebras without mention-ing the condition of co-dimension one.

We now state two results on the automatic continuity of multiplicativelinear functionals.

Theorem A.6.12. ([BD73], p. 77; [Zel73], p. 37)

Let ϕ be a multiplicative linear functional on a Banach algebra A.Then:

(i) ϕ is continuous on A and ||ϕ|| ≤ 1.(ii) If A has an identity e with ||e|| = 1, then ||ϕ|| = 1.

Theorem A.6.13. ([Mal86], p. 72-73)

If A is a topological algebra with the property that every maximalmodular two-sided ideal in A is closed, then every multiplicative linearfunctional ϕ on A is continuous. In particular, each multiplicative linearfunctional ϕ on a Q-algebra A is continuous.

Remark.


([Zel73], p. 37) A multiplicative linear functional ϕ need not becontinuous on a non-complete complex normed algebra A.

Open Problem. As noted above, every multiplicative linear func-tional ϕ on a Banach algebra (or, more generally, on a Q-algebra) iscontinuous. In general, a multiplicative linear functional on a locallym-convex algebra A need not be continuous. It is still an open problem(known as the Michael-Mazur problem): whether every multiplicativelinear functional on a commutative locally m-convex F -algebra A is con-tinuous (see ([Mic52, Hus83, Dal00, Fra05]).

Notation. The set of all multiplicative linear functionals on an al-gebra A is denoted by ∆(A). Clearly, ∆(A) ⊆ A′, the algebraic dual ofA, but ∆(A) need not be a vector subspace of A′. If A is a topologicalalgebra, let ∆c(A) denote the set of all (non-zero) continuous multiplica-tive linear functionals on A. In general, ∆(A) and ∆c(A) may be empty.For example, if A is the Arens algebra

Lω[0, 1], then ∆(A) = ∅ (see [Zel73], p. 65). However, we have:

Theorem A.6.14. ([Zel73], p. 38, 65; [Fra05], p. 55) If A is a com-mutative complete locally m-convex algebra (in particular, a commutativeBanach algebra) with identity, then ∆c(A) 6= ∅.

Theorem A.6.15. Let A be a commutative locally m-convex algebra.Then there is a one-one correspondence between the set ∆c(A) and theset Mc(A) of all closed maximal modular ideals of co-dimension one inA given by ϕ↔M , where M = ϕ−1(0).

Definition. Let A be an algebra. The intersection of all maximalmodular left ideals of A is called the radical (or Jacobson radical) of Aand is denoted by rad(A).

Theorem A.6.16. ([Fra05], p. 60-61) Let A be an algebra. Then:(a) rad(A) is an ideal.(b) rad(A) ⊆ x ∈ A : rA(x) = 0.(c) If A is a commutative complete locally m-convex algebra, then

rad(A) = x ∈ A : rA(x) = 0.

Also, in this case, if the topology of A is generated by a family sA ofcontinuous seminorms,

rad(A) = x ∈ A : limn→∞

[p(xn)]1/n = 0 for all p ∈ sA.

(d) If A is a topological algebra with ∆c(A) 6= ∅, then

rad(A) ⊆ ∩ker(ϕ) : ϕ ∈ ∆c(A) = x ∈ A : ϕ(x) = 0 for all ϕ ∈ ∆c(A);


if, in addition, A is a commutative complete locally m-convex Q-algebra,

rad(A) = ∩ker(ϕ) : ϕ ∈ ∆c(A);

hence, in this case, rad(A) is closed.Note. For a non-locally m-convex F -algebra A, rad(A) need not be

closed.

Definition. An algebra A is called semisimple if rad(A) = 0;A is called simple if it has no two-sided ideal other than 0 and A.The quotient algebra A/rad(A) is semisimple. Every locally C∗-algebraand every topological algebra with an orthogonal basis is semisimple.(Recall that a topological basis xn for a topological algebra A is calledorthogonal if xnxm = 0 if n 6= m and x2n = xn [Hus83, p. 62]. If A isa commutative complete locally m-convex Q-algebra with ∆c(A) 6= ∅,then, by above theorem, A is semisimple iff

x ∈ A : ϕ(x) = 0 for all ϕ ∈ ∆c(A) = 0.

Definition: Let A be a topological algebra with ∆c(A) 6= ∅, andlet w(A∗, A) be the weak∗-topology of A∗. Then the relative weak∗-topologies s = w(A∗, A)|∆c(A) of A

∗ on ∆c(A) is called the Gelfand topol-ogy. ∆c(A) with this topology is called the maximal ideal spaces of A (orthe topological spectrum of A) and is denoted by (∆c(A), s). In general,(∆c(A), s) need not be compact or even locally compact.

Recall that, if A is a TVS, the every equicontinuous subset of A∗ isrelatively w(A∗, A)-compact (Alaoglu-Bourbaki theorem; see [Hor66], p.201).

Theorem A.6.17. Let A be a topological algebra. Then:(a) ( [Mal86], p. 140) (∆c(A), s) is a completely regular Hausdorff

space.(b) ([Mal86], p. 75) If A is a Q-algebra, then (∆c(A), s) is equicon-

tinuous and hence locally compact.(c) ([Mal86], p. 143) If A is an F -algebra, then (∆c(A), s) is compact

iff A is a Q-algebra.(d) ([Hus83], p. 12) If A is a complete locally m-convex algebra, then

∆c(A) ∪ 0 is a w(A∗, A)-closed subset of A∗.

(e) ([Hus83], p. 12) If A is a complete locally m-convex algebra withidentity, then 0 is an isolated point of ∆c(A) ∪ 0 and ∆c(A) is aw(A∗, A)-closed subset of A∗.

Definition A.6.18. ([Mal86], p. 142)


Let A be a topological algebra. Then ∆c(A) is called locally equicon-tinuous, as a subset of A∗ if, for each f ∈ ∆c(A), there exists a neigh-borhood U of f in ∆c(A) such that U is an equicontinuous subset of(∆c(A), s) ⊆ (A∗, w(A∗, A)).

Definition A.6.19. ([Mal86], p.73) Let A be a topological algebra.For each x ∈ A, define x : ∆c(A)→ C by

x(ϕ) = ϕ(x), ϕ ∈ ∆c(A).

Then x is continuous on ∆c(A); hence x ∈ C(∆c(A)) = C(∆c(A), s)).

The function x is called the Gelfand transform of x on A. Setting A =x : x ∈ A, we have A ⊆ C(∆c(A)). The map G : x → x is called theGelfand representation of A into C(∆c(A)).

Theorem A.6.20. ([Mal86], p. 143) Let A be a topological algebra.Then:

(a) (∆c(A), s) is locally equicontinuous iff ∆c(A) is locally compact

and G : A→ A ⊆ C(∆c(A)) is continuous.(b) If A is a locally m-convex Q-algebra and (∆c(A), s) is locally

compact, then ∆c(A) is locally equicontinuous.

Theorem A.6.21. ([Mal86], p. 140, 266) Let A be a topologicalalgebra. Then:

(a) The map G : x → x is a continuous algebraic homomorphism of

A onto the subalgebra A of C(∆c(A)).(b) G is one-one iff A is semisimple.The commutative version of the Gelfand-Naimark theorem for topo-

logical algebras is as follows.

Theorem

A.6.22. (Gelfand-Naimark) ([Mic52], p. 34-35; [Mal86], p. 488)(a) If A is a commutative complete semisimple locally m-convex algebra,then A is both algebraically and topologically isomorphic to a separatingsubalgebra A of (C(c(A), k), k being the compact-open topology; if A

has an identity e, then A contains the identity function.(b) If A is a commutative locally C∗-algebra and that the Gelfand

map G : x → x is continuous, then A is algebraically ∗-isomorphic to(Co(+

c (A), k), where +c (A) = c(A) ∪ 0.

Definition. Let A be a topological algebra.


(1) A net eλ : λ ∈ I in A is called a left ( resp. right) approximateidentity of A if

limλeλx = x (resp. lim

λxeλ = x) for all x ∈ A.

(2) eλ : λ ∈ I is called a two-sided approximate identity (or anapproximate identity) of A if it is both a left and right approximateidentity.

Every locally C∗-algebra A has a uniformly bounded approximateidentity eλ : λ ∈ I.

Definition. An algebra A is called(a) left faithful if, for any a ∈ A, aA = 0 ⇒ a = 0.(b) right faithful if, for any a ∈ A, Aa = 0 ⇒ a = 0.(c) faithful if it is both left and right faithful.

Other equivalent terms for “faithful ” are “proper”, “without order”,“without annihilators”.

Theorem A.6.23. An algebra A is faithful in each of the followingcases:

(a) A has an identity e.(b) A is a topological algebra having an approximate identity eλ :

λ ∈ I (e.g. A is a C∗-algebra or locally C∗-algebra).(c) A is a topological algebra with an orthogonal basis.(d) A is a commutative semisimple Banach algebra.

Proof. (a) This is clear.(b) Suppose A is a topological algebra with an approximate identity

eλ : λ ∈ I. Let a ∈ A with ax = 0 for all x ∈ A. Then aeλ = 0 for allλ ∈ I; hence a = lim

λaeλ = 0. Similarly, Aa = 0 ⇒ a = 0.

(c) Suppose A is a topological algebra with an orthogonal basis xn.Let a ∈ A with ax = 0 for all x ∈ A.We can write a =

∑n≥1 tnxn. Then,

since each xm ∈ A,

axm = 0, or∑

m≥1

tn(xnxm) = 0, or tmx2m = 0,

and so tm = 0 for all m ≥ 1. Hence a =∑

m≥1 tmem = 0. Similarly,Aa = 0 ⇒ a = 0.

(d) Suppose A is a semisimple. Let a ∈ A with ax = 0 for allx ∈ A. Then, a.a = 0 and so,by induction, an = 0 for all n ≥ 1. Hence


limn→∞

‖an‖1

n = 0,showing that

a ∈ x ∈ A : limn→∞

‖xn‖1

n = 0 = rad(A).

Since A is semisimple, rad(A) = 0 and so a = 0.

Example. Let A be any vector space (resp. TVS) over C withA 6= 0. Then A becomes a commutative algebra (resp. topologicalalgebra) without an identity under the trivial multiplication: x.y = 0 forall x, y ∈ A. Then, for any non-zero x ∈ A, xA = 0. However x 6= 0,and so A is not faithful.

Recall that if E is a TVS, then CL(E) = CL(E,E) is an algebraunder composition and has identity I : E → E given by I(x) = x(x ∈ E).

Theorem A.6.24. Let E be a TVS (in particular, a topologicalalgebra). Then (CL(E), tu) and (CL(E), tp) are topological algebras.

Proof. We need to show that the composition map (S, T ) → ST isseparately continuous. Fix So ∈ CL(E), and let Tα : α ∈ I ⊆ CL(E)

with Tαtu−→ T ∈ CL(E). To show that SoTα

tu−→ SoT , let U(D,W ) beany tu-neighborhood of 0 in CL(E). By continuity of So : E → E, there

exists V ∈ WE such that So(V ) ⊆W. Since Tαtu−→ T , there exists αo ∈ I

such that(Tα − T )(D) ⊆ V for all α ≥ αo.

Then, for any α ≥ αo,

So(Tα − T )(D) ⊆ So(V ) ⊆W.

Hence SoTαtu−→ SoT.

Next, fix To ∈ CL(E), and let Sα : α ∈ I ⊆ CL(E) with Sαtu−→

S ∈ CL(E). To show SαTotu−→ STo, let U(D,W ) be any uneighborhood

of 0 in CL(E). Since To : E → E is continuous, To is bounded and soTo(D) is a bounded set in E. Then U(To(D),W ) is a tu-neighborhood of

0 in CL(E). Since Sαtu−→ S, there exists αo ∈ I such that

Sα − S ∈ U(To(D),W ) or (Sα − S)To(D) ⊆W for all α ≥ αo.

Hence SαTotu−→ STo. Thus (CL(E), tu) is a topological algebra. By the

same argument, (CL(E), tp) is also a topological algebra.

Recall that a subcollection B of the collection b(E) of all boundedsubsets of a TVS E is called fundamental system of bounded sets if, foreach bounded set D ⊆ E, there exists a Di ∈ B such that D ⊆ D1

([Hus65], p. 20).


Theorem A.6.25. Let E be an F-space (in particular, F-algebra). IfE has a countable fundamental system of bounded sets, then (CL(E), tu)is a complete metrizable topological algebra with jointly continuous mul-tiplication.

Proof. Let D = D1, D2, .... be a countable fundamental systemof bounded sets in E. Since E is metrizable, it has a countable baseWE = W1,W2, ... (say) of neighborhoods of 0. Then clearly the col-lection U(Di,Wj) : i, j = 1, 2, 3, .... is a countable base of neighbor-hoods of 0 in (CL(E), tu). Also (CL(E), tu) is Hausdorff. Hence, bythe metrization theorem of TVSs, (CL(E), tu) is metrizable. Therefore(CL(E), tu) is a complete metrizable topological algebra. Thus, by theAren’s theorem, (CL(E), tu) is a topological algebra with jointly contin-uous multiplication.

Definition: Let (A, q) be an F -normed algebra. For any T ∈ CL(A),let

‖T‖q = supx∈A,q(x)≤1

q(T (x)). (∗)

In general, ‖T‖q need not exist since the set x ∈ A : q(x) ≤ 1may not be bounded in the F -algebra (A, q). However, for q a k-norm(0 < k ≤ 1), the existence and other useful properties of ||.||q are sum-marized in the following theorem (see [Rol85], p. 101-102; [Bay03], p.3-5; [ARK11]).

Theorem A.6.26. Let (A, q) be a k-normed algebra, where 0 < k ≤1. Then:

(a) A linear mapping T : A→ A is continuous ⇔ ||T ||q <∞.(b) ‖.‖q is a k-norm on CL(A).

(c) For any T ∈ CL(A),

‖T‖q= supx∈A,x 6=0

q(Tx))

q(x).

(d) For any T ∈ CL(A), q(Tx) ≤ ‖T‖q .q(x) for all x ∈ A.(e) For any S, T ∈ CL(A), ‖ST‖q ≤ ‖S‖q ‖T‖q ; hence (CL(A), ‖.‖q)

is an k-normed algebra.(f) If A is complete, then (CL(A), ‖.‖q) is also complete.

(g) If A has a minimal approximate identity eλ : λ ∈ I, then, forany a ∈ A,

‖La‖q = ‖Ra‖q = q(a),

where La, Ra : A→ A are the maps given by La(x) = ax and Ra(x) =xa, x ∈ A.


Proof. (a) This is clear.(b) Clearly ‖T‖q ≥ 0 and ‖T‖q = 0 iff T = 0. For any T ∈ CL(A)

and λ ∈ K,

‖λT‖q = supq(x)≤1

q(λT (x))q(λT (x))

q(x)= |λ|k sup

q(x)≤1

q(T (x)) = |λ|k ‖T‖q .

It is easy to verify that, for any S, T ∈ CL(A), ‖S + T‖q ≤ ‖S‖q+ ‖T‖q.(c) For any x ( 6= 0) ∈ A, if y = x

[q(x)]1k, then

q(y) = q[x

q(x)1

k

] =1

[q(x)1

k ]kq(x) = 1;

hence

supx∈A,x 6=0

q(Tx))

q(x)= sup

x∈A,x 6=0

q[T ( x

q(x)1k.q(x)

1

k )]

q(x)

= supx∈A,x 6=0

[q(x)1

k ]k.q[T ( x

q(x)1k)]

q(x)

= supx∈A,x 6=0

q[T (x

q(x)1

k

)] = supy∈A,q(y)=1

q[T (y)] = ‖T‖q .

(d) Let T ∈ CL(A). Using (c), clearly q(Tx) ≤ ‖T‖q .q(x) for eachx 6= 0 in A Clearly, this also holds for x = 0.

(e) Let S, T ∈ CL(A). Using (d),

‖ST‖q = supq(x)≤1

q[S(T (x))] ≤ supq(x)≤1

‖S‖q q(T (x))

≤ supq(x)≤1

‖S‖q ‖T‖q · q(x) = ‖S‖q . ‖T‖q .

(f) By Theorem A.6.25, (CL(A), ‖.‖q) is complete and so it is an

F -algebra. See also ([Bay03], p. 5; [Rol85], p. 41).(g) Let a ∈ A. Then

‖La‖q = supq(b)≤1

q(La(b)) = supq(b)≤1

q(ab)) ≤ supq(b)≤1

q(a).q(b) = q(a).

On the other hand, since q(eλ) ≤ 1 for all λ ∈ I,

‖La‖q = supq(b)≤1

q(ab)) ≥ q(aeλ) for all λ ∈ I;

so‖La‖q ≥ lim

λq(aeλ) = q(lim

λaeλ) = q(a).

Hence‖La‖q = q(a). Similarly, ‖Ra‖q = q(a).


Remark. In the definition of ‖T‖q given by (∗), we cannot make

the blank assumption that, for q an F -norm on E, ‖T‖q always exists

for each T ∈ CL(E). This assumption cannot be justified in view of thefollowing counter-examples.

(1) First, ||T ||q need not be finite for a general F-norm. For example,let E = R2; q(x1, x2) = |x1| + |x2|1/2; T (x1, x2) = (x2, x1). Then q is anF -norm on E, but ||T ||q =∞: for any n ∈ N,

||T ||q ≥q[T (n, n2)]

q(n, n2)=q(n2, n)

q(n, n2)=n2 + n1/2

2n→∞.

(2) Even when considering the subspace of those T for which ||T ||q <∞, then ||.||q need not always be an F-norm, since (F5) need not hold.For example, for a fixed sequence (pn) with 0 < pn ≤ 1, pn → 0, considerthe F-algebra E of sequences (xn) ⊆ R with |xn|pn → 0 and q((xn)) =supn≥1|xn|

pn. Then ||T ||q <∞ for all multipliers of E, but ||.||q is not anF -norm, i.e. it makes the space CL(E) into an additive topological groupbut not into a topological vector space (as it would lack the continuity ofscalar multiplication in the absence of (F5) (cf. [Rol85], Example 1.2.3,p. 8)).

In view of the above remark, we shall need to assume that (E, q) isa k-normed space (or a k-normed algebra) whenever ||T ||q is consideredfor T ∈ CL(E).

Theorem A.6.27. Let (A, q) be an k-normed algebra with q sub-multiplicative and A having an identity e. Then A is isometrically iso-morphic to a closed subalgebra of (CL(A), tu).

Proof. Define q/ : A −→ R by

q/(x) = supz∈A,z 6=0

q(xz)

q(z), x ∈ A

Then, for any x ∈ A,

q/(x) = supz∈Az 6=0

q(xz)

q(z)≤ sup

z∈A,z 6=0

q(x)q(z)

q(z)= q(x),

and also

q(x) = q(xe) = q(e).q(xe)

q(e)≤ q(e) sup

z 6=0

q(xz)

q(z)= q(e)q/(x).

Hence, q/ is equivalent to q.


Now for each x ∈ A, consider Lx : A −→ A given by Lx(y) = xy,y ∈ A. Then clearly Lx : x ∈ A is a subalgebra of CL(A). Further,

‖Lx‖q = supz∈A,z 6=0

q(Lx(z))

q(z)= sup

z∈A,z 6=0

q(xz)

q(z)= q/(x).

Hence (A, q/) is isometrically isomorphic to the subalgebra Lx : x ∈ A of CL(A) under the mapping x→ Lx.

Next Lx : x ∈ A is closed in (CL(E), tu), as follows. Let T ∈

Lx : x ∈ A‖.‖q . Then there exists a sequence Lxn ⊆ Lx : x ∈ A

such that Lxn‖.‖q−→ T . We show that T = Lxo for some xo ∈ A. Now, for

any y ∈ A,

T (y) = limn→∞

Lxn(y) = limn−→∞

xny = limn→∞

(xne)y

= limn→∞

Lxn(e)y = T (e)y = LT (e)(y);

i.e. T = Lxo , where xo = T (e) ∈ A. Hence T ∈ Lx : x ∈ A, and soLx : x ∈ A is closed in (CL(E), tu).

Definition. Let Y be a TVS and A a topological algebra, both overthe same field K (= R or C). Then Y is called a topological left A-moduleif it is a left A-module and the module multiplication (a, x)→ a.x fromA× Y into Y is separately continuous.

Definition: Let Y be a topological left A-module, and let b(A) (resp.b(Y )) denote the collection of all bounded sets in A (resp. in Y ). Then:

(a) The module multiplication (a, x)→ ax is called b(A)- (resp. b(Y )-) hypocontinuous if, given any neighborhood G of 0 in Y and any D ∈b(A) (resp. B ∈ b(Y )), there exists a neighborhood H of 0 in Y (resp.V of 0 in A) such that DH ⊆ G (resp. V B ⊆ G).

(b) The module multiplication (a, x)→ ax is called jointly continuousif, given any neighborhood G of 0 in Y , there exist neighborhoods U of0 in A and H of 0 in Y such that UH ⊆ G.

Clearly, joint continuity ⇒ hypocontinuity ⇒ separate continuity.The following theorem shows that, under some additional conditions,separate continuity ⇒ hypocontinuity.

Theorem A.6.28. Let (Y, τ) be a topological left A-module. If Y isultrabarrelled, then the module multiplication is b(A)-hypocontinuous. Inparticular, every ultrabarrelled topological algebra has a hypocontinuousmultiplication.


Proof. Let G be a neighborhood of 0 in Y and D ∈ b(A). For anya ∈ A, define La : Y → Y by La(x) = a.x, x ∈ Y . Clearly, eachLa is linear and also continuous (by separate continuity of the modulemultiplication). Further, La : a ∈ D is pointwise bounded in CL(Y, Y ).[Let x ∈ Y and G1 a neighborhood of 0 in Y . Since D in bounded in A,by separate continuity of the module multiplication, D.x is bounded inY and so there exists r > 0 such that D.x ⊆ rG1. So La(x) : a ∈ D =a.x : a ∈ D = D.x ⊆ rG1, showing that La : a ∈ D is pointwisebounded in CL(Y, Y ).] Since Y is ultrabarralled, by the principle ofuniform boundedness, La : a ∈ D is equicontinuous. Hence, given anyG ∈ WY , there exists a neighborhood H of 0 in Y such that La(H) ⊆ Gfor all a ∈ D; i.e. D.H ⊆ G.


7. Measure Theory

In this section, we present some measure theoretic terminology (see[Hal50, Var65, Whe83, Bog07]).

Definition. Let X be a given set. A collection A(X) of subsets ofX is called an algebra if it satisfies the following conditions:

(a) X ∈ A(X);(b) The union of finite number of sets in A(X) is also in A(X);(c) For any A,B ∈ A(X), the difference set A\B is also in A(X).

Clearly ∅ ∈ A(X). Further, A(X) is closed under finite intersections.An algebra A(X) is called a σ-algebra if it closed under countable unions(hence also under countable intersections).

Definition. If A(X) is an algebra of subsets of X, then a functionµ : A(X)→ R+ is called

(i) finitely additive if, for any A1, ..., An ∈ A(X) with Ai ∩ Aj = ∅for i 6= j,

µ(∪ni=1Ai) =n∑

i=1

µ(Ai);

(ii) countably additive if, for any sequence of sets A1, A2, ..... ∈ A(X)with

⋃∞i=1Ai ∈ A(X) and Ai ∩Aj = ∅ for i 6= j,

µ(∪∞i=1Ai) =

∞∑

i=1

µ(Ai).

Now, let X be a completely regular Hausdorff space, and let c(X)(resp. k(X), z(X), cz(X)) the collection of all closed (resp. compact,zero, cozero) subsets of X (see Section A.1). We refer to [GJ60] forgeneral facts about z(X) and cz(X).

The family of zero sets is preserved by finite unions and countableintersections. A compact Gδ-set is a zero set. Any zero set is closed anda Gδ-set; the converse is true in a normal space.

The family of cozero sets is preserved by finite intersections and count-able unions, and forms a base for the topology of X . Disjoint zero setsare contained in disjoint cozero sets.

Definition. Let Ba(X) denote the smallest σ-algebra of subsets of Xcontaining z(X), and let Bo(X) denote the smallest σ-algebra of subsetsof X containing c(X). Clearly, the σ-algebras Ba(X) and Bo(X) areclosed under countable unions and intersections The members of Ba(X)(resp. Bo(X)) are called Baire (resp. Borel) sets. Since every zero set

7. MEASURE THEORY 297

is closed, Ba(X) ⊆ Bo(X), but, in general, Ba(X) 6= Bo(X) even if X iscompact. For instance, in the compact Hausdorff space [0, 1]ω1, singletonsets are closed sets but are not zero sets. It is true that every compactset in the Baire σ-algebra Ba(X) is a zero set, but not every compactK ⊆ X is in Ba(X). See ([Whe83], p. 109). However, if X is a metricspace or, more generally, a perfectly normal space, then Ba(X) = Bo(X)([Whe83], p. 109; [Bog07], p. 13). (A topological space X is calledperfectly normal if every closed set F ⊆ X has the form Z = f−1(0) forsome f ∈ C(X), or equivalently, if it is normal and every closed subsetof X is a Gδ-set.)

Definition. If Σ is a subfamily of Ba(X), then we say that a finite,real-valued, finitely additive set function µ on Ba(X) is Σ-regular if, foreach A ∈ Ba(X) and ε > 0, there exists F ∈ Σ, F ⊆ A, such that

|µ(H)| < ε for all H ∈ Σ with H ⊆ A\F.

A positive Baire measure µ on X is a finite, real-valued, non-negative,finitely additive set function on the algebra Ba(X) which is z(X)-regular(i.e., equivalently, for any B ∈ Ba(X),

µ(B) = supµ(Z) : Z ⊆ B,Z ∈ z(X).

A (signed) Baire measure is the difference of two positive Baire measures.The spaces of all Baire measures and positive Baire measures are denotedby M(Ba(X)) and M+(Ba(X)), respectively.

If µ ∈M(Ba(X)), then the set functions µ+ and µ− defined on Ba(X)defined by

µ+(A) = supµ(B) : B ∈ Ba∗(X), B ⊆ A,

µ−(A) = − infµ(B) : B ∈ Ba∗(X), B ⊆ A,

are members of M+(Ba(X)) and µ = µ+ − µ−. The set function |µ| =µ+ + µ− is again in M+(Ba(X)) and is called the total variation of µ. IfA ∈ Ba(X),

|µ|(A) = supn∑

i=1

|µ(Bi)| : Bi ∈ Ba∗(X), A =

n⋃

i=1

Bi.

The inner and outer inner measures determined by µ ∈M+(Ba(X)) are

µ∗(A) = supµ(Z) : Z ∈ z(X), Z ⊆ A,

µ∗(A) = infµ(U) : U ∈ cz(X), A ⊆ U,

for each subset A of X .


M(Ba(X)) is a Banach space with the norm ||µ|| = |µ|(X) andM+(Ba(X)) is a closed subset of M(Ba(X)).

Definition. If µ ∈ M+(Ba(X)) and f ∈ Cb(X). If f ≥ 0, we definethe integral∫

X

fdµ = sup

∫

X

hdµ : 0 ≤ ϕ ≤ f, ϕ a simple function w.r.t. Bo(X).

For arbitrary f ∈ Cb(X), we can write f = f+ − f−, where f+(x) =maxf(x), 0 and f−(x) = max−f(x), 0. Then we define

∫

X

fdµ =

∫

X

f+dµ−

∫

X

f−dµ.

Now if µ ∈ M(Ba(X)), µ = µ+ − µ− with µ+, µ− ∈ M+(Ba(X)), andf ∈ Cb(X), we define

∫

X

fdµ =

∫

X

fdµ+ −

∫

X

fdµ−.

For a full discussion of integration with respect to a finitely additivemeasures, see [Hal50, Roy88, DS58].

The following version of the classical Riesz representation theorem iswell-known (see, [Var65, Whe83]).

Theorem A.7.1 (Riesz-Alexandroff) Let X be a completely regularHausdorff space. Then (Cb(X), u)∗ = M(Ba(X)) via the linear isomor-phism L←→ µ, where

L(f) =

∫

x

fdµ for all g ∈ Cb(X) and ||L|| = |µ|(X).

Remark. ([Whe83], p. 117) The Borel measures µ on Bo(X) arisein the usual Riesz Representation Theorem for compact spaces. For ashort and elegant proof of this result, see Garling [Gar73]. In the com-pletely regular space, several difficulties arise if we attempt to represent

members of (Cb(X), u)∗

by Borel measures. Each L ∈ (Cb(X), u)∗ can berepresented by a finitely additive c(X)-regular measure ν on Bo(X), butν need not be unique; unfortunately, there may be uncountably many νwhich work. Uniqueness of the representing Borel measure is restored ifX is normal.

In view of the Riesz-Aleksandrov theorem, any L ∈ (Cb(X), u)∗ can beidentified with a unique µ ∈ M(Ba(X)) and may write µ(f) for

∫xfdµ for

brevity. In his fundamental paper, Varadarajan [Var65] analyzed threesubspaces of M(Ba(X)), as follows:

7. MEASURE THEORY 299

Definition. (1) A measure µ ∈ M(Ba(X)) is called σ-additive if,for any sequence Zn of zero sets of X with Zn ↓ ∅, |µ(Zn)| → 0. Itis equivalent to that µ be countably additive on Ba(X), so that µ canbe extended uniquely to a countably additive set function on Ba(X). Weshall always assume that this extension has been made. Any countablyadditive measure on Ba(X) must be z(X)-regular (using the fact thatevery cozero set is a countable union of zero sets), hence is σ-additive.

(2) µ is called τ -additive if it is c(X)-regular and, for any net Zα ofzero sets of X with Zα ↓ ∅, |µ(Zα)| → 0. This behavior is not preservedby decreasing net of cozero sets. If X is locally compact, every τ -additivemeasure is compact zero set regular. Any τ -additive measure µ on Ba(X)can be extended to a unique c(X)-regular Borel measure ν such thatFα ↓ F , ν(Fα) → ν(F ). If every σ-additive measure in M(Ba(X)) isτ -additive, then X is called measure compact.

(3) µ is called tight if, given any ε > 0, there exits a compact setK ⊆ X such that |µ|(X) < |µ|∗(K) + ε. Any tight Baire measure canbe extended to a k(X)-regular Borel measure ν. We shall always assumethat this extension has been made.

Let Mσ(Ba(X)) (resp. Mτ (Bo(X)), Mt(Bo(X))) denote the space ofall bounded real-valued σ-additive (resp. τ -additive, tight) measures onBa(X) (resp. Bo(X), Bo(X)). Restricting the above measures to Ba(X),we clearly have Mt(Ba(X)) ⊆ Mτ (Ba(X)) ⊆ Mσ(Ba(X)) ⊆ M(Ba(X)).Note that Mt(Bo(X)) ⊆Mσ(Ba(X)) need not hold in general as an µ ∈Mτ (Bo(X)) is not necessarily z(X)-regular. Every tight µ ∈ M(Ba(X))is τ -additive and hence it has a unique extension µ′ ∈Mτ (Bo(X)).


8. Uniform Spaces and Topological Groups

Definition. [Will70, p. 238] Let Y be a non-empty set.(1) The subset ∆ = ∆(Y ) = (x, x) : x ∈ Y of Y × Y is called the

diagonal on Y .(2) For any A ⊆ Y × Y, we define we define A−1 by

A−1 = (y, x) : (x, y) ∈ A.

If A = A−1, then A is called symmetric.(3) For any A,B ⊆ Y × Y, we define A B by

A B = (x, y) ∈ Y × Y : ∃ z ∈ Y such that (x, z) ∈ A and (z, y) ∈ B.

If A = B, we shall write A A = A2. Note that,

(x, z) ∈ A, (z, y) ∈ B ⇒ (x, y) ∈ A B.

Definition. [Will70, p. 238] A subset H of subsets of Y ×Y is calleda uniformity (or a uniform structure) on Y if

(u1) (Y ) = (x, x) : x ∈ Y ⊆ A for all A ∈ H.(u2) A ∈ H =⇒ A−1 ∈ H, i.e. each A ∈ H is symmetric.(u3) If A ∈ H, ∃ some B ∈ H such that B2 ⊆ A.(u4) If A,B ∈ H, ∃ some C ∈ H such that. C ⊆ A ∩ B.(u5) If A ∈ H and A ⊆ B, then B ∈ H.In this case, the pair (Y,H) is called a uniform space and we shall

often denote it by H(Y ). The members of H are called vicinities of H.

Definition. [Will70, p. 238-239] (1) Let H1 and H2 be two unifor-mities on a set Y . If H1 ⊆ H2, then H1 is said to be

weaker than H2 and H2 is said to be stronger than H1.(2) The weakest uniformity on a set Y is I = Y × Y , called the

indiscrete uniformity.(3) The strongest uniformity on a set Y is D = A ⊆ Y ×Y : ∆ ⊆ A,

called the discrete uniformity.

Definition: Let H be a uniformity on a set Y . Then a subcollectionB ⊆ H is called a base ( or basis) for H if, for any A ∈ H, there existsB ∈ B such that B ⊆ A.

Theorem A.8.1. Let B be a base for a uniformity H on Y . Thenit has the following properties:

(b1) ⊆ B for all B ∈ B :(b2) If B ∈ B, ∃ some C ∈ B such that C−1 ⊆ B.(b3) If B ∈ B, ∃ some C ∈ B such that C2 ⊆ B.

8. UNIFORM SPACES AND TOPOLOGICAL GROUPS 301

(b4) If A,B ∈ B, ∃ some C ∈ B such that C ⊆ A ∩ B.Conversely, if a collection B of subsets of Y × Y satisfies the above

conditions (b1)-(b4), then the collection

H = A ⊆ Y × Y : A contains some B ∈ B

is a uniformity on Y whose base is B.We next define a topology on a set Y induced by a uniformity H on

Y .

Definition: [Edw65, p. 26] Let H be a uniformity on a set Y . Forany x ∈ Y and A ∈ H, let

NA(x) = y ∈ Y : (x, y) ∈ A.

Then, for each x ∈ Y , the collection NA(x) : A ∈ H satisfies all theconditions of a neighborhood system at x. Consequently, there existsa unique topology τH on Y such that, for each x ∈ Y , the collectionNA(x) : A ∈ H forms a local base at x. In this case, the topologyτH on Y is said to be the topology induced by (or derived from) theuniformity H.

Notation. For any K ⊆ Y and A ∈ H, let

NA(K) = ∪NA(x) : x ∈ K = y ∈ Y : (x, y) ∈ A for some x ∈ K.

As we shall see below, every semi-metric (in particular, metric) spaceis a uniform space and every uniform space is a topological space.

Theorem A.8.2. [Edw65, p. 30] Let (Y, d) be a semi-metric space(in particular, a metric space). For each r > 0, let

Dr = (x, y) ∈ Y × Y : d(x, y) < r.

Then the collection B = Dr : r > 0 is a base for a uniformity on Y .More generally, we have:

Theorem A.8.3. [Edw65, p. 27] Let D = dα : α ∈ I be a familyof semi-metrics on a set Y , and let

D+ = maxdα1, ..., dαn for all finite α1, ...αn ⊆ I.

For each r > 0, let

Dd,r = (x, y) ∈ Y × Y : d(x, y) < r, d ∈ D+.

Then the collection B = Dd,r : d ∈ D+, r > 0 is a base for a uniformityon Y . (Note that D ⊆ D+.)

Definition: [Edw65, p. 28]


A topological space (Y ,τ) is said to uniformizable if its topology canbe derived from a uniformity, i.e., there exists a uniformity H on Y suchthat τ = τH.

Theorem A.8.4. (a) [Will70, p. 256] A topological space (Y, τ) isuniformizable ⇔ it is completely regular.

(b) [Will70, p. 257] A uniformity H on Y is pseudometrizable ⇔ ithas a countable base.

(b) [Edw65, p. 27-28]

A topological space (Y, τ) is uniformizable with uniformity H⇔ thereexists a family dα : α ∈ I of semi-metrics on Y , which generates theuniformity H.

Note. In particular, every metric space and every completely regularspace is a uniform space.

Definition: [Edw65, p. 28]

Let (Y,H) be uniform space.(1) A net xα : α ∈ I in Y is said to be convergent to x ∈ Y if, given

any A ∈ H, there exists an α0 = α0(A) ∈ I such that

(xα, x) ∈ A for all α ≥ α0;

in this case, we write xα → x.(2) A net xα : α ∈ I in Y is said to be a Cauchy net if, given any

A ∈ H, there exists an α0 = α0(A) ∈ I such that

(xα, xβ) ∈ A for all α, β ≥ α0.

Definition: A uniform space (Y,H) is said to be complete if everyCauchy net in Y is convergent to a point in Y.

Definition: [Edw65, p. 30] Let (Y,H) be a uniform space. The setS ⊆ Y is said to be totally bounded if, given any A ∈ H, there exists afinite set

x1, x2, ...., xn ⊆ Y

such that

S ⊆ ∪ni=1NA(xi) = ∪ni=1y ∈ Y : (y, xi) ∈ A.

Equivalently, [Will70, p. 262], a set S ⊆ Y is totally bounded if, givenany A ∈ H, there exists a finite cover U1, U2, ...., Un of S such that

∪ni=1Ui × Ui ⊆ A.


Theorem A.8.5 [Will70, p. 262] A subset S of a uniform space(Y,H) is compact ⇔ it is complete and totally bounded.

Definition. [Will70, p. 243] (a) Let (Y,HY ) and (Z,HZ) be uniformspaces and xo ⊆ Y. A function f : Y → Z is said to be continuous at xoif, for any B ∈ HZ , there exists an A ∈ HY such that

(f(x), f(xo)) ∈ B whenever x ∈ Y with (x, xo) ∈ A.

(b) Let (Y,HY ) and (Z,HZ) be uniform spaces and S ⊆ Y. A functionf : Y → Z is said to be uniformly continuous on S if, for any B ∈ HZ ,there exists an A ∈ HY such that

(f(x′), f(x′′)) ∈ B whenever x′, x′′ ∈ Y with (x′, x′′) ∈ A.

Note. If f is uniformly continuous on S ⊆ Y, then clearly f iscontinuous on S; the converse holds if S is compact.

We next recall the notion of a ”topological group” and show thesespaces are uniform spaces.

Notations: Let G = (G, ·) be a group.

For any V,W ⊆ G, let

V ·W = a · b : a ∈ V, b ∈ W,

V −1 = a−1 : a ∈ V .

Definition: Let G = (G, ·) be a group, written multiplicatively, andτ a topology on G. Then the G is called a topological group if

(TG1) The mapping P : (x, y) → x · y of G × G → G is jointlycontinuous (i.e. given any x, y ∈ G and any neighborhood U of xy,there exist neighborhoods V and W of x and y respectively such thatP (V,W ) ⊆ U or V ·W ⊆ U).

(TG2) The inverse mapping I : x → x−1 of G → G is continuous(i.e. given any x ∈ G and any neighborhood U of x−1, there exists aneighborhood V of x such that I(V ) ⊆ U or V −1 ⊆ U).

In this case, we denote G = (G, ·, τ).

Note. The two conditions (TG1 & TG2) are equivalent to the singlecondition:

(TG) The map (x, y)→ xy−1 of G×G→ G is jointly continuous.

Examples:(1) (R,+) with the usual topology is a TG.(2) (R\0, ·) with the usual topology is a TG.


(3) (Q,+), (Q\0, ·) with the induced topologies are topologicalgroups.

(4) (C,+), (C\0, ·) are TG.(5) Every TVS (X, τ) is a TG under the operation of +; in fact,

by (TVS1), (x, y) → x + y is continuous and, by (TVS2), the inversionx → x−1 = −x is continuous. In particular, every normed vector space(X, ‖·‖) is a TG under the operation of +.

(6) T = λ ∈ C: |λ| = 1 is a TG under multiplication; here λ−1 = λ.It is called a torus group [Pa78, p. 88].

Remark. By definition, any TVS (E, τ) is topological group under+.

Lemma A.8.6. Any group G = (G, ·) with the discrete topology τIis a TG.

Proof. (TG1) We verify that the map (x, y) → x · y is continuous.Let x, y ∈ G. Let U be any neighborhood of xy in (G, τI). Since (G, τI)is a discrete space V = x and W = y are τI-open neighborhoods ofx and y, respectively. Clearly

V ·W = x · y ⊆ U.

Hence the map (x, y)→ x · y is continuous.(TG2) We show that the map I : x → x−1 is continuous. Let x ∈ G

and U any neighborhood of x−1 in (G, τ1). Now V = x ∈ τ , and

I(V ) = V −1 = x−1 ⊆ U.

Thus I = x→ x−1 is contains on G. Hence (G, ·, τI) is a TG.

Lemma A.8.7. Any group G = (G, ·) with the indiscrete topologyτD is a TG.

Proof. (TG1) We show that the map (x, y)→ x · y is continuous. Letx, y ∈ G. Since (G, τD) is an indiscrete space, G is the only non-emptyopen set in G and so U = G is the only τD-neighborhood of G. TakeV = G and W = G as τD-neighborhoods of x and y respectively. Then

V ·W = G ·G = G = U.

Hence (x, y)→ x · y is continuous.(TG2) We show that I = x → x−1 is continuous. Let x ∈ G. Then

U = G is the only τD-open neighborhood of x−1. Take V = G as τD-neighborhood of x. Then

I(V ) = V −1 = G−1 = G = U.

So I : x→ x−1 is continuous.


Thus (G, ·, τD) is a TG.

Theorem A.8.8. Let G = (G, ·, τ) be a TG. Then, for any fixeda ∈ G, the maps La : x→ a · x and Ra : x→ x · a are homeomorphismsof G to G.

Proof. We prove the theorem for La.La is 1-1: Let x, y ∈ G with La(x) = La(y). Then

a · x = a · y =⇒ a−1 · (a · x) = a−1 · (a · y)

=⇒ e · x = e · y =⇒ x = y.

La is onto: Let y ∈ G (domain) and

La(a−1 · y) = a · (a−1 · y) = e · y = y.

This shows that La is onto.La is continuous: Let xo ∈ G. By (TG1), the map (a, x) → a · x is

continuous at xo or, equivalently, La : x→ ax is continuous at xo.(La)

−1 is continuous: Now La(x) = a · x = z (say), and so

L−1a (z) = x = a−1 · z = La−1(z).

Thus L−1a = La−1 . By above part, La−1 is continuous. Hence (La)−1 :

G→ G is continuous.Thus La : x→ a · x is a homeomorphism. Similarly Ra : x→ x · a is

a homeomorphism.

Corollary A.8.9. Let G = (G, ·, τ) be a TG and U an open set inG. Then, for any a ∈ G, a · U and U · a are also open. Moreover, ifA ⊆ G, A · U and U · A are open.

Proof. Since the mapping La : x → ax is homeomorphism, it is,in particular, an open map. Hence La(U) or a · U is open. SimilarlyRa : x→ x · a is open and so U · a is an open set.

Next, if A ⊆ G, A·U = ∪x∈Aa·U which is open; also U ·A = ∪x∈AU ·ais open.

Lemma A.8.10. Let G = (G, ·, τ) be a TG. Then:(i) The map x→ x−1 is a homeomorphism.(ii) If a, b ∈ G, the map x→ a · x · b is a homeomorphism.(iii) If a ∈ G, the map x→ x · a · x−1 is continuous.(iv) If Z is a base of neighborhoods at e, then, for any x ∈ G,

x ·U : U ∈ Z is a base of neighborhoods at x, and so is U ·x : U ∈ U.

Theorem A.8.11. Let G = (G, ·, τ) be a TG with identity e and letU be a base of neighborhoods at e. Then:

(i) e ∈ U for each U ∈ U .


(ii) For each U ∈ Z, there exists some V ∈ Z such that V 2 = V ·V ⊆U .

(iii) For each U ∈ Z, there exists some W ∈ Z such that W−1 ⊆ U .(iv) For each U ∈ Z and a ∈ U , there exists some V ∈ Z such that

a · V ⊆ U .(v) For each U ∈ Z and x ∈ G, there exists some V ∈ Z such that

x · V −1x ⊆ U .(vi) For each U, V ∈ Z, there exists some W ∈ Z such that W ⊆

U ∩ V .(vii) ∩

U∈UU = e.

Proof. (i) Since each U ∈ U is a neighborhood of e and so e ∈ U foreach U ∈ U

(ii) Let U ∈ U . Since the mapping (x, y) → x · y is continuous at(e, e) and (e, e) → e · e = e, there exist neighborhoods V1 and V2 of esuch that V1 · V2 ⊆ U . Put V = V1 ∩ V2. Then V is a neighborhood ofe, V ⊆ V1 and V ⊆ V2. Hence

V 2 = V · V ⊆ V1 ⊆ U.

(iii) Let U ∈ U . Choose an open neighborhood W1 of e with W ⊆ U .Since x → x−1 is a homeomorphisms, W−1

1 is an open neighborhood ofe−1 = e. Let W = W1 ∩W

−11 . Then W ∈ U and

W−1 = W ⊆ U.

(iv) Let U ∈ U and a ∈ U . Since the map (x, y)→ x · y is continuousat (a, e) and (a, e)→ a · e = a, ∃ a neighborhood V of e in X such thata · V ⊆ U.

The remaining proofs are omitted.

Theorem A.8.12. [Ber74, p.17] Let G = (G, ·, τ) be a topologicalgroup with a base U of neighborhoods of e in G. Then the following areequivalent.

(a) G is Hausdorff.(b) e is a closed subset of G.(c) ∩

U∈UU = e.

Theorem A.8.13. (a) Let G be a TG. Then, for any neighborhoodU of e in G, there exists a neighborhood V of 0 such that V ⊆ U .Consequently, every topological group is regular.

(2) Every Hausdorff TG is completely regular.

Definition. [Ber74, p. 26] A metric d on a group (G, .) is called:


(i) left invariant

if d(a · x, a · y) = d(x, y) for all x, y, a ∈ G.(ii) right invariant if d(x · a, y · a) = d(x, y) for all x, y, a ∈ G.

Metrization Theorem A.8.14. (Birkhoff-Kakutani

) ([Ber74, p. 28], [Hus66, p. 49]) (a) Let G = (G, ·, τ) be a TG withidentity element e. Then G is metrizable iff it is Hausdorff and has acountable base of τ -neighborhoods of e.

(b) A mertizable TG admits a left invariant (or right invariant) com-patible metric.

Theorem A.8.15. (a) Every topological group (G, τ) is a uniformspace.

(b) Every topological vector space (E, τ) is a uniform space.

Proof. (a) Let B be any base of τneighborhoods of the identity e inG. For each V ∈ B, let

AV = (x, y) ∈ G×G : xy−1 ∈ V .

Then H = AV : V ∈ B is a uniformity on G, as follows.(u1) (G) = (x, x) : x ∈ G ⊆ AV for all AV ∈ H: [Let AV ∈ H

with V ∈ B. For any x ∈ G,

xx−1 = e ∈ V and so (x, x) ∈ AV .

Hence ∆ ⊆ AV for all AV ∈ H.](u2) AV ∈ H =⇒ (AV )

−1 ∈ H: [Let AV ∈ H with V ∈ B. We notethat (AV )

−1 = AV −1 :

(AV )−1 = (x, y) : (y, x) ∈ AV = (x, y) : yx

−1 ∈ V

= (x, y) : xy−1 = (yx−1)−1 ∈ V −1 = AV −1 .

Hence, if AV ∈ H, then (AV )−1 = AV −1 ∈ H since V −1 ∈ B.]

(u3) If AV ∈ H, ∃ some AW ∈ H such that (AW )2 ⊆ AV : [LetAV ∈ H with V ∈ B. Choose W ∈ B such that WW ⊆ V . We note that(AW )2 ⊆ AWW , as follows

(AW )2 = AW AW

= (x, y) : ∃ some z ∈ G such that (x, z) ∈ AW

and (z, y) ∈ AW

= (x, y) : ∃ some z ∈ G such that xz−1 ∈ W

and zy−1 ∈ W

⊆ (x, y) : ∃ some z ∈ G such that (xz−1)(zy−1) ∈ WW


= (x, y) : (x, y) : xy−1 ∈ WW = AWW .

Hence A2W ⊆ AWW ⊆ AV .]

(u4) If AV , AW ∈ H, ∃ some AU ∈ H such that AU ⊆ AV ∩ AW : [Let AV , AW ∈ H with V,W ∈ B Choose U ∈ B such that U ⊆ V ∩W .We note that AV ∩AW = AV ∩W , as follows:

AV ∩AW = (x, y) : xy−1 ∈ V ∩ (x, y) : xy−1 ∈ W

= (x, y) : xy−1 ∈ V ∩W = AV ∩W .

Hence AU ⊆ AV ∩W ⊆ AV ∩ AW .](u5) If AV ∈ H and AV ⊆ AW , then AW ∈ H: [Let AV ∈ H and

AV ⊆ AW with V,W ∈ B. We note that AV ⊆ AW implies V ⊆ W , asfollows. Let x ∈ V . Then xe−1 = xe = x ∈ V , and so (x, e) ∈ AV . SinceAV ⊆ AW , (x, e) ∈ AW . Hence xe−1 ∈ W or x ∈ W . So V ⊆ W . SinceW ∈ B, we have AW ∈ H.]

(b) This follows as in (a). In fact, let (E, τ) be a TVS, and let W bea base of neighborhoods of 0 in X . In this case, for each V ∈ W, let

AV = (x, y) ∈ E ×E : x− y ∈ V .

Then H = AV : V ∈ W is a uniformity on E, as follows:(u1) ∆ = (x, x) : x ∈ E ⊆ AV for all AV ∈ H (since, for any

x ∈ E,x− x = 0 ∈ V and so (x, x) ∈ AV ).

(u2) For any balanced V ∈ W,

A−1V = (x, y) : (y, x) ∈ AV = (x, y) : y − x ∈ V

= (x, y) : x− y ∈ −V = A−V .

Hence, if AV ∈ H , then A−1V = A−V ∈ H since −V ∈ W.](u3). Let AV ∈ H. Choose a balancedW ∈ W such thatW+W ⊆ V .

Then

A2W = AW AW

= (x, y) : ∃ some z ∈ E such that (x, z) ∈ AW

and (z, y) ∈ AW,

= (x, y) : ∃ some z ∈ E such that x− z ∈ W and z − y ∈ W

⊆ (x, y) : x− y = (x− z) + (z − y) ∈ W +W = AW+W .

Hence A2W ⊆ AW+W ⊆ AV .]

(u4). Let AV , AW ∈ H. Choose G ∈ W such that G ⊆ V ∩W . Then

AV ∩ AW = (x, y) : x− y ∈ V ∩ (x, y) : x− y ∈ W

= (x, y) : x− y ∈ V ∩W = AV ∩W .


Hence AG ⊆ AV ∩W ⊆ AV ∩AW .](u5).Let AV ∈ H and AV ⊆ AW . This implies V ⊆ W , as follows.

[Let x ∈ V . Then x − 0 = x ∈ V , and so (x, 0) ∈ AV . Since AV ⊆AW , (x, 0) ∈ AW . Hence x − 0 ∈ W or x ∈ W . So V ⊆ W .] SinceW ∈ W, we have AW ∈ H.]



Section A.1. This contains some basic definitions and results fromGeneral Topology. A full discussion of the results can be found in stan-dard text books. See, for example, Kelley [Kel55], Gillman and Jerison[GJ60], Dugundji [Dug66], Willard [Will70] and Husain [Hus77].

Section A.2. This is a collection of basic definitions and results fromtopological vector spaces. See, for example, Dunford-Schwartz [DS58],Edwards [Edw65], Horvath [Hor66], Husain [Hus65], Kelley and Namioka[KN63], Kothe [Ko69], A. Robertson & W. Robertson [RR64], Rudin[Rud91], and Schaefer [Scha71].

Section A.3. Here some linear topologies on CL(E, F ), known as theG-topologies are defined. The most important cases of the G-topologyare: the topology tu of uniform convergence on bounded sets, the topol-ogy tpc of precompact convergence, the topology tc of compact conver-gence and the topology tp of pointwise convergence.

This section also includes some fundamental theorems of FunctionalAnalysis such as the principle of uniform boundedness, Banach-Steinhaus

, open mapping theorem, and closed graph theorem. All these aregiven in the non-locally convex setting.

Section A.4. The notion of a shrinkable neighborhood of 0 in a TVSE is presented. It had first appeared in the Ph.D. thesis of R. T. Ives in1957 (see [Kl60a]), and further studied by Klee ([Kl60a, Kl60b]).

Important facts about shrinkable neighborhoods included in this sec-tion are:

(A) Every Hausdorff TVS E has a base of shrinkable neighborhoodsof 0.

(B) If W is a shrinkable neighborhood of 0 in a TVS E, then itsMinkowski functional ρW is continuous and positively homogeneous.

The proofs of these important results, due to Klee [Kl60a], may notbe found in the text books and therefore we have included them here.

Section A.5. This includes the notions of non-Archimedean val-ued fields, non-Archimedean TVSs and locally F-convex TVSs. Most ofthese results are collected from the monograph of Prolla [Pro82] (see also[Pro77, NBB71, vR78]).

We mention that an NA TVS over F or equivalently, a locally F-convex TVS (for non-trivially valued field F) may be considered as ananalogue of the usual locally convex TVS over K.


Section A.6. This section contains basic definitions and results fromtopological algebras and modules. The proofs of these results except fora few selected theorems are omitted; a full discussion of the results can befound in standard text books. See, for example, [Mal86, Mic52, Hus83,Dal00, Fra05, Zel71, Zel73].

Section A.7. Some measure theoretic terminology is considered. Fora full discussion, see Halmos [Hal50], Dunford-Schwartz [DS58], Royden[Roy88]. We also include some material on the classes of tight, τ -additiveand σ-additive measures on a topological spaces X. The detail may befound in the survey papers of LeCam [Lec57], Varadarajan [Var65] andWheeler [Whe83].

Section A.8. Uniform spaces are the carriers for the notions ofuniform convergence, uniform continuity, equicontinuity and like this.These notions are easily defined in metric spaces; the important qualityof metric spaces for this purpose being that the distance is a notionwhich can be applied uniformly to pair of points without regard to theirlocation. This quality is not possessed by topological spaces, where theneighborhoods of a point depend on the location of the point. Thereforeuniform spaces will require somewhat more structure than a topologyprovides. The general notion of a general uniform space was introducedby A. Weil in 193; another approach is due to W.J. Tuckey (1940). Seealso J.R. Isbell [Isb64]. For detail of the definitions and results of thissection, see [Edw65, Will70].

In this section, we have also introduced here the notion of a ”topolog-ical group” which is more general than the notion of a ”topological vectorspace. It is shown that every topological group is a uniform spaces. Fordetail see, [Hus66, Ber74].

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APPENDIX B

List of Symbols

A or cl(A), the closure of a subset A of a topological space, A.1, 1.1A0 or int A, the interior of a subset A of a topological space, A.1, 1.1A(x) = f(x) : f ∈ A, A.3, 4.1B(X), the algebra of all bounded functions ϕ : X → K ( = R or C), 1.1Bf(X) = ϕ ∈ B(X) : ϕ has finite support, 1.1Bσ(X) = ϕ ∈ B(X) : ϕ has σ-compact support, 1.1Bσo(X) = ϕ ∈ B(X) : ϕ has countable support, 1.1Bo(X) = ϕ ∈ B(X) : ϕ vanishes at infinity, 1.1Boo(X) = ϕ ∈ B(X) : ϕ has compact support, 1.1B(X)⊗E = ϕ⊗a : ϕ ∈ B(X), a ∈ E, where (ϕ⊗a)(x) = ϕ(x)a, x ∈ X ,1.1Ba(X), the σ-algebra of all Baire subsets of X,, A.7Bo(X), the σ-algebra of all Borel subsets of X, A.7β, the strict topology on Cb(X,E), 1.1βX, the Stone-Cech compactification of X, A.1, 1.1C, the field of complex numbers A.2CMr(A, Y ), the vector apace of all continuous linear right multipliersfrom A to Y .C(X), the algebra of all continuous functions ϕ : X → K, 1.1Cb(X) = ϕ ∈ C(X) : ϕ is bounded on X, 1.1Co(X) = ϕ ∈ C(X) : ϕ vanishes at infinity, 1.1Coo(X) = ϕ ∈ C(X) : ϕ has compact support, 1.1CVb(X) = f ∈ C(X) : vf is bounded for each v ∈ V , 1.2CVo(X) = f ∈ C(X) : vf vanishes at infinity for each v ∈ V , 1.2C(X,E), the vector space of all continuous functions f : X → E, 1.1Cb(X,E) = f ∈ C(X,E) : f is bounded on X, 1.1Co(X,E) = f ∈ C(X,E) : f vanishes at infinityCoo(X,E) = f ∈ C(X,E) : f has compact supportCrc(X,E) = f ∈ C(X,E) : f(X) is relatively compact in ECpc(X,E) = f ∈ C(X,E) : f(X) is precompact in E, 4.2CVb(X,E) = f ∈ C(X,E) : vf is bounded on X for each v ∈ V , 1.2

329

330 B. LIST OF SYMBOLS

CVpc(X,E) = f ∈ C(X,E) : vf(X) is precompact in E for all v ∈ V ,4.2CVo(X,E) = f ∈ C(X,E) : vf vanishes at infinity for each v ∈ V , 4.2C(X)⊗ E = ϕ⊗ a : ϕ ∈ C(X), a ∈ E, 1.1Cφ, the composition operator on CVo(X,E), Cφ(f) = f φ with φ : X →X , 10.2c(X), the collection of all closed subsets of X , A.7cz(X), the collection of all cozero subsets of X , A.7E ′, the algebraic dual of a vector space E, A.2E∗, the topological (or continuous) dual of a topological vector space E,A.2F, a field, A.4F (X,E), the vector space of all functions X → E, 1.1, 1.2, 9.1, 9.2FVo(X,E), the vector space of all functions f : X → E with vf vanishesat infinity for each v ∈ V, 1.2Fσ, countable union of closed sets, A.1f ≺ U means that there exists a Z ∈ z(X) such that Z ⊆ U andsupp(f) ⊆ Z, 7.1, 7.2‖f‖W = ||ρW f || = sup

x∈X|ρW (f(x))|, 8.1, 8.2

Gδ, countable intersection of open sets, A.1G(µ,W, ε) = f ∈ Cb(X,E) : µ(x : f(x) /∈ W) ≤ ε, 7.1, 7.2γ, the weak topology on (Cb(X,E), β), 7.1γV , the weak topology on (CVo(X,E), ωV ), 7.2HomA(A, Y ), the vector space of all continuous left A-module homomor-phisms of A into Y, 10.3∫xϕdµ, the integral of ϕ ∈ Cb(X) with respect to a scalar measure µ :

X → K, 1.7, 8., 8.2∫xdmf, the integral of f ∈ Cb(X,E) with respect to a vector measure

m : X → E∗, 8.1, 8.2k, the compact-open topology on C(X,E) or Cb(X,E), 1.1K, the field of scalars, always either the field R or the complex field C,1.1, A.1, A.2k(X), the collection of all compact subsets of X, A.7K+(X), the set of all non-negative constant functions on X, 1.2χA, the characteristic function of a subset A, 1.1, 1.2, A.1χc(X) = λχK : λ > 0 and K ⊆ X, K compact, 1.2χf(X) = λχF : λ > 0 and F ⊆ X, F finite, 1.2ℓp = ℓp(N) = x = (xn) ⊆ K :

∑∞n=1 |xn|

p <∞, 0 < p <∞, A.2

Lp = Lp[a, b] = f : [a, b]→ K :b∫a

|f(t)|pdt <∞, 0 < p <∞, A.2

B. LIST OF SYMBOLS 331

M(A), the set of all maximal ideals of codimension 1 in A,A.6, 5.2Mc(A)), the set of all maximal closed ideals of codimension 1 in A,A.6,5.2Mθ, the multiplication operator from CVo(X,E) into F (X,E) definedby Mθ(f) = θf with θ : X → C, 9.1, 9.2Mϕ, the multiplication operator from CVo(X,E) into F (X,E) definedby Mϕ(f) = ϕf with ϕ : X → E, 9.1, 9.2Md(A), the algebra of all double multipliers (S, T ) on an algebra A, 10.1Mσ(Ba(X)), the vector space of all bounded real-valued σ-additive mea-sures on Ba(X), A.7, 8.1Mτ (Bo(X)), the vector space of all bounded real-valued τ -additive mea-sures on Bo(X), A.7, 8.1Mt(Bo(X)), the vector space of all bounded real-valued tight measureson Bo(X), A.7, 8.1Mσ(Ba(X), E∗), the vector space of all bounded E∗-valued σ-additivemeasures on Ba(X), 8.1, 8.2Mτ (Bo(X), E∗), the vector space of all bounded E∗-valued τ -additivemeasures on Bo(X), 8.1, 8.2Mt(Bo(X), E∗), the vector space of all bounded E∗-valued tight measureson Bo(X), 8.1, 8.2M(µ, v,W, ε) = f ∈ CVo(X,E) : µ∗(x ∈ X : vf(x) /∈ W) ≤ ε, 7.2M(D,G) = x ∈ Y : D.x ⊆ G, 10.2N(ϕ,W ) = f ∈ Cb(X,E) : (ϕf)(X) ⊆W, 1.1N(A,W ) := N(χA,W ) = f ∈ Cb(X,E) : f(A) ⊆W, 1.1p, the pointwise topology on C(X,E), 1.1pf , the seminorm on E defined by pf(x) = |f(x)|, x ∈ E, A.2qx, the seminorm on E∗ defined by qx(f) = |f(x)|, f ∈ E∗, A.2R, the field of real numbers, A.2ρA, the Minkowski functional of A ⊆ E defined by ρA(x) = infλ > 0 :x ∈ λA, A.2ϕ⊗ a, the function, 1.1∏α∈I

Xα, the Cartesian product of a family Xα : α ∈ I of topological

spaces with the product topology, A.1σ,the σ-compact-open topology on Cb(X,E), 1.1σo, the countable-open topology on Cb(X,E), 1.1S+o (X), the set of all non-negative upper semi-continuous functions onX which vanish at infinity, 1.1S(X,E), a vector subspace of F (X,E), 9.1, 9.1SM(X), the collection of all submeasures µ : cz(X)→ [0,∞], 7.1, 7.2

332 B. LIST OF SYMBOLS

supp(f) = closure of the set x ∈ X : f(x) 6= 0, where f : X → E orK, A.1, 1.1TA, topological algebra, A.6TV S, topological vector space, 1.1, A.2u, the uniform topology on Cb(X,E), 1.1U(D,W ) = T ∈ CL(E, F ) : T (D) ⊆W, A.3, A.6, 4.2, 9.1, 10.1U(D,G) = T ∈ HomA(A, Y ) : T (D) ⊆ G, 10.3Ud(D,W ) = (S, T ) ∈Md(A) : S(D) ⊆W and T (D) ⊆W, 10.1,V > 0, a Nachbin family such that, for each x ∈ X , there is a v ∈ Vwith v(x) 6= 0, 1.2VMt(Bo(X), E∗) = vm : v ∈ V,m ∈Mt(Bo(X), E∗), 8.2w(E,E∗) or σ(E,E∗), the weak topology on a TVS E, A.2w(E∗, E) or σ(E∗, E), the weak∗-topology on the dual space E∗, A.2ω, the first uncountable ordinal, 1.1, 6.1, 6.2ωV , the weighted topology on CVb(X,E) or CVo(X,E), 1.2W =WE , a base of neighbourhoods of 0 in a topological vector space E,A.2, A.3Wπ,ϕ, the weighted composition operator on CVo(X,E),Wπ,ϕ(f) = π ·f ϕ with ϕ : X → X and π : X → L(E), 9.2z(ϕ) = ϕ−1(0) = x ∈ X : ϕ(x) = 0, a zero set of a function ϕ : X → R,A.1, A.7z(X), the collection of all zero sets in X, A.7

APPENDIX C

Index

absorbing set, A.2algebra, A.6

Banach algebra, A.6, 10.1F - (complete metrizable), A.6, 10.1, 10.2faithful (or without order), A.6, 10.1, 10.2hypo-topological, A.6, 9.2locally bounded, A.6, 10.2locally C∗-, A.6, 1A.1locally convex, A.6locally idempotent, A.6, 9.1, 10.2locally m-convex, A.6Q-, A.6, 5.1, 5.2semisimple, simple, A.6, 5.1, 5.2topological, A.6, 5.1, 5.2, 9.1, 9.2, 10.1, 10.2, 10.3

algebraic dual, 1.2approximate identity (left, right, two-sided), A.6, 10.1, 10.2

uniformly bounded, A.6approximation property, A.2, 4.1, 4.3, 4.5balanced set, A.2Baire set, A.7, 8.1base of neighbourhoods, 1.1, A.1, A.2Borel set, A.7, 8.1bounded function, A.2, 1.1bounded linear mapping, A.3bounded set, A.2Cauchy net, A.2characteristic function χA, A.1, 1.1compact support, A.1, 1.1continuous extension, A.1continuous function, A.1, 1.1, 1.2continuous linear mapping, A.3,convex set, A.2

333

334 C. INDEX

covering dimension, A.1, 4.1cozero set, A.1, A.7, 8.1dominate, 1.1equicontinuous, A.3, 4.1essential module, 1A.2, 10.3finite rank operator, A.2, 4.3, 5.6F -norm, A.2F -seminorm, A.2Fσ-set,F -space, A.2fundamental TVS, 10.1, 10.3Gδ-set, A.1Grothendieck approximation property (GAP), 4.3Hewitt (or real) compactification υX , 5.2homeomorphism, A.2homomorphism, A.6, 3.2, 5.2, 1A.1,10.3hyperplane, 1.2ideal (left, right, two-sided, modular), A.6, 5.1, 5.2idempotent set, 1.6, 9.3integral, 1.7, 8.1, 8.2isomorphism, A.6, 3.2, 5.2, 10.1, 10.3k-space, A.1, 1.1, 1.1, 3.1kR-space, A.1, 3.2locally convex TVS, A.2locally equicontinuous, 5.2maximal ideal space, A.6, 5.2measure, A.7, 7.1, 8.1

Baire, A.7, 7.1, 8.1Borel, A.7, 8.1E∗-valued, 8.1, 8.2σ-additive, A.7, 8.1Σ-regular, A.7τ -addivite, A.7, 8.1tight, A.7, 8.1, 8.2

measure compact, A.7, 8.1Minkowski functional, A.2, A.3, 6.1multiplicative linear functional, A.6, 5.2multiplier (left, right, double), 10.1Nachbin family, 1.2, 3.3, 4.2, 7.2, 9.1, 9.2non-Archimedean field, A.4, 4.6

C. INDEX 335

non-Archimedean TVS, A.4operator, 9.1, 9.2

multiplication, 9.1,composition, weighted composition, 9.2

P -space, A.1partition of unity, 4.1, 4.2precompact set, A.1, A.2, 3.2property V, 4.1quasi-complete TVS, A.2, 1.1quasi-norm, A.2quasi-ultrabarralled, A.3, 1.1quotient map, A.2, A.3quotient space, A.2, A.3radical, A.6semicontinuous (lower, upper), A.1, 1.1seminorm, A.2, 6.2separable, 6.1separation property, 4.4sequentially complete TVS, A.2, 3.1shrinkable neighbourhood, A.4Stone-Cech compactification βX , A.1, 5.2, 6.2submeasure, 7.1, 7.2submetrizable space, 6.1

separably, 6.1support, compact, finite, σ-compact, 1.1support of a function, A.1, 1.1theorem,

Arzela-Ascoli, 3.1, 3.2Banach-Steinhaus, A.3closed graph, A.3Hahn-Banach, A.2M. Krein and S. Krien, 6.1open mapping, A.3Riesz representation, A.7, 9.2Stone-Cech, A.1Stone-Weierstrass, 4.1, 4.2, 4.6Tietze extension, A.1,

topological basis (orthogonal, Schauder), A.6topological dual, A.2topological group, A.8

336 C. INDEX

topological module, A.4, 10.2, 10.3topological space, A.1

completely regular, A.1, 1.1hemi-compact, A.1, 1.1locally compact, A.1, 1.1sequentially compact, A.1, A.2σ-compact, A.1, 1.1, 6.1

topological vector space (TVS), A.2, 1.1admissible, A.2, 4.1complete, quasi-complete, sequentially complete, A.2F- (complete metrizable), A.2fundamental, A.2, 10.1locally bounded, A.2, 4.2, 6.1, 9.1locally convex, A.2metrizable, 1.1, A.2ultrabarrelled, quasi-ultrabarrelled, A.3, 1.1ultrabornological, A.3, 1.1, 10.1

topology,compact-open (k-), 1.1, 10.1, 10.2countable-open (σo-)Gelfand, A.6, 5.2pointwise (p-), 1.1strict (β-), 1.1, 10.1, 10.2σ-compact-open (σ-), 1.1uniform (u-), 1.1, 10.1, 10.2weak (w(E,E∗)-), A.2, 7.1, 7.2weak (γt,- mt-), 7.1weak (γv-, mv-), 7.2weighted (ωV -), 1.2

totally bounded (= precompact), A.2uniform boundedness principle, A.3, 10.1, 10.2, 10.3uniform space, A.8, 3.1VR-space, 3.2vanish at infinity, 1.1, 1.2weak approximation, 7.1, 7.2zer0-dimensional, A.1, 4.6zero set, A.1, 7.1, 8.1

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LINEAR TOPOLOGICAL SPACES OF CONTINUOUS ...linear topological spaces of continuous bounded vector-valued functions endowed with the uniform, strict, compact-open and weighted topologies