Functional Analysis

And Infinite-Dimensional Geometry

Marian Fabian2

Petr Habala13

Petr Hajek12

Vicente Montesinos Santalucıa4

Jan Pelant2

Vaclav Zizler12

1 Department of Mathematics, University of Alberta, Edmonton2 Mathematical Institute, Czech Academy of Sciences, Prague

3 Department of Mathematics, Faculty of Electrical Engineering,Czech Technical University, Prague

4 Departamento de Matematica Aplicada,Universidad Politecnica de Valencia

Preface

PrefaceBanach spaces provide a framework for linear and nonlinear functional analysis,

operator theory, abstract analysis, probability, optimization and other branchesof mathematics. This book is intended as an introduction to linear functionalanalysis and to some parts of infinite-dimensional Banach space theory.The first seven chapters are directed mainly to undergraduate and graduate

students. We have strived to make the text easily readable and as self-containedas possible. In particular, we proved many basic facts that are considered “folk-lore”. An important part of the text is a large number of exercises with detailedhints for their solution. They complement the material in the chapters andcontain many important results.The last five chapters introduce the reader to selected topics in the theory

of Banach spaces related to smoothness and topology. This part of the bookis intended as an introduction to and a complement of existing books on thesubject ([BeLi], [DGZ3], [Dis1], [Dis2], [Fab], [JoL3], [LiT2], [Phe2], [Woj]).Some material is presented here for the first time in a monograph form.

The text is based on graduate courses taught at the University of Albertain Edmonton in the years 1984–1997. These courses were also taken by manysenior students in the Honors undergraduate program in Edmonton.As a prerequisite, basic courses in calculus and linear algebra should be enough.

For the most part, Royden’s book [Roy] should be sufficient.The chapters are best read consecutively. However:— Chapter 4 as well as the latter part of Chapter 3 (James boundaries) can be

omitted in the case of a more elementary functional analysis course. Chapter 4is used only marginally in Chapters 8–10.— The spectral theory (Chapter 7) can be approached after the first two

chapters and the beginning of Chapter 3 were covered; it is not needed in latterchapters.The book can serve as a textbook for the following types of courses in functional

analysis:1. Graduate two-semester course: Chapters 1–9.2. Graduate one-semester course: Chapters 1–3, 5, and 6 or 7.

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3. Graduate one-semester advanced course: Chapters 8–10 or 11, 12.4. Undergraduate first course in functional analysis: Chapters 1–3 and a part

of Chapter 7.5. Undegraduate second course in functional analysis: Chapters 4–6, Chapter 8

and 10.The first three chapters together with Chapter 7 can be used in service courses

for students of probability, physics, or engineering.

The principal part of the text was prepared at the Department of Mathemat-ics, University of Alberta in Edmonton. Each author spent some time at thisdepartment. Habala and Hajek obtained their PhD degrees there and Zizlerwas a faculty member there. We all thank this department for excellent work-ing conditions. We also thank our present home institutions for enabling us tofinalize the book. We are indebted to the grant agencies in Canada, the CzechRepublic, Germany, Spain and the U.S. for supporting our research in Banachspace theory over the years.We are grateful to our colleagues and students for many helpful discussions.

Our special thanks go to Jon Borwein, Gilles Godefroy, Jirı Jelınek, Kamil John,Lopez Pellicer, Jose Orihuela, Nicole Tomczak-Jaegermann, Jon Vanderwerff,and Dirk Werner. We also thank our colleagues that allowed us to include someof their recent unpublished results.We thank Marion Benedict for her excellent typing of the first version of the

manuscript and the staff of Springer-Verlag for their efficient work.Above all, we are deeply indebted to our wives for their support and encour-

agement.

We would be glad if this book inspired some young mathematicians to chooseBanach spaces as their field of interest, and hope that students and researchersin Banach space theory will find the text useful. We wish the readers a pleasanttime spent over this book.

Prague and ValenciaSummer 2000The authors

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Contents

Preface 5

1 Basic Concepts in Banach Spaces 1Holder and Minkowski inequalities, classical spaces C[0, 1], `p, c0,

Lp[0, 1], . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Operators, quotient spaces, finite-dimensional spaces, Riesz’s lemma,

separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Hilbert spaces, orthonormal bases, `2 . . . . . . . . . . . . . . . . . . . 17Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2 Hahn-Banach and Banach Open Mapping Theorems 38Hahn-Banach extension and separation theorems . . . . . . . . . . . . 39Duals of classical spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 45Banach open mapping theorem, closed graph theorem, dual operators 51Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3 Weak Topologies 66Weak and weak star topology, Banach-Steinhaus uniform boundedness

principle, Alaoglu’s and Goldstine’s theorem, reflexivity . . . . . 67Extreme points, Krein-Milman theorem, James boundary, Ekeland’s

variational principle, Bishop-Phelps theorem . . . . . . . . . . . . 79Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4 Locally Convex Spaces 112Local bases, bounded sets, metrizability and normability, finite-dimensional

spaces, distributions . . . . . . . . . . . . . . . . . . . . . . . . . 113Bipolar theorem, Mackey topology . . . . . . . . . . . . . . . . . . . . 122Representation and compactness: Caratheodory and Choquet repre-

sentation, Banach-Dieudonne, Eberlein-Smulian, Kaplansky the-orems, Banach-Stone theorem . . . . . . . . . . . . . . . . . . . . 127

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

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5 Structure of Banach Spaces 143Projections and complementability, Auerbach bases . . . . . . . . . . . 143Separable spaces as subspaces of C[0, 1] and quotients of `1, Sobczyk’s

theorem, Schur’s property of `1 . . . . . . . . . . . . . . . . . . . 147Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

6 Schauder Bases 167Shrinking and boundedly complete bases, reflexivity, Mazur’s basic

sequence theorem, small perturbation lemma . . . . . . . . . . . 171Bases in classical spaces: block basis sequences, PeÃlczynski’s decompo-

sition method and subspaces of `p, Pitt’s theorem, Khintchine’sinequality and subspaces of Lp . . . . . . . . . . . . . . . . . . . 178

Unconditional bases, James’s theorem on containment of `1 and c0,James’s space J , Bessaga-PeÃlczynski theorem . . . . . . . . . . . 186

Markushevich bases: existence for separable spaces, extension prop-erty, Johnson’s and Plichko’s result on `∞ . . . . . . . . . . . . . 194

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

7 Compact Operators on Banach Spaces 209Compact operators and finite rank operators, Fredholm operators Fred-

holm alternative . . . . . . . . . . . . . . . . . . . . . . . . . . . 209Spectral theory: eigenvalues, spectrum, resolvent, eigenspaces . . . . . 216Self-adjoint operators, spectral theory of compact self-adjoint and com-

pact normal operators . . . . . . . . . . . . . . . . . . . . . . . . 223Fixed points: Banach’s contraction principle, non-expansive mappings,

Ryll-Nardzewski theorem, Brouwer’s and Schauder’s theorems,invariant subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . 234

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

8 Differentiability of Norms 249Smulian’s dual test, Kadec’s Frechet-smooth renorming of spaces with

separable dual, Frechet differentiability of convex functions . . . 251Extremal structure, Lindenstrauss’ result on strongly exposed points

and norm attaining operators . . . . . . . . . . . . . . . . . . . . 264Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

9 Uniform Convexity 294Uniform convexity and uniform smoothness, `p spaces . . . . . . . . . 294Finite representability, local reflexivity, superreflexive spaces and En-

flo’s renorming, Kadec’s and Gurarii-Gurarii-James theorems . . 300Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

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10 Smoothness and Structure 323Variational principles (smooth and compact), subdifferential, Stegall’s

variational principle . . . . . . . . . . . . . . . . . . . . . . . . . 324Smooth approximation: partitions of unity . . . . . . . . . . . . . . . 338Lipschitz homeomorphisms, Aharoni’s embeddings into c0, Heinrich-

Mankiewicz results on linearization of Lipschitz maps . . . . . . 341Homeomorphisms: Mazur’s theorem on `p, Kadec’s theorem . . . . . . 346Smoothness in `p, Hilbert spaces . . . . . . . . . . . . . . . . . . . . . 350Countable James boundary and saturation by c0 . . . . . . . . . . . . 353Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

11 Weakly Compactly Generated Spaces 368Projectional resolutions, injections into c0(Γ), Eberlein compacts, em-

bedding into a reflexive space, locally uniformly rotund and smoothrenormings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369

Weakly compact operators, Davis-Figiel-Johnson-PeÃlczynski factoriza-tion, absolutely summing operators, Pietsch factorization, Dunford-Pettis property . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382

Quasicomplements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390

12 Topics in Weak Topology 398Eberlein compacts, metrizable subspaces . . . . . . . . . . . . . . . . . 399Uniform Eberlein compacts, scattered compacts . . . . . . . . . . . . . 405Weakly Lindelof spaces, property C . . . . . . . . . . . . . . . . . . . . 414Corson compacts, weak pseudocompactness in Banach spaces, (BX , w)

Polish . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428

9

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Index

annihilator yol and Yol, 40, 55, 58,93, 148, 149

Banach limit, 62basis

algebraic, 34, 191Auerbach, 139, 164bimonotone, 191block, 172boundedly complete, 166-168, 182,

192

constant, 163, 169, 182equivalent, 169-171Hamel (see basis: algebraic)Markushevich, 188-190, 382,

410-412 )~'Itshrinking, 188, 197, 369, 370weakly compact, 364:1>10)weakly LindelOf, 411

monotone, 163, 191, 192normalized, 163orthonormal, 18, 20, 222Schauder, 161, 163, 165, 303, 307seminormalized, 303shrinking, 166-168, 184, 192, 260summing, 165, 181uncondition al , 180, 181, 196, 197

bump. See function(Bx, 1\·11),2,14(Bx, w), 73, 75, 414, 415(Bx, w*), 71-73, 319, 365, 395,

409-412

cardinality card(A), 23, 403~.~-closure M , M ,M, 64compact

Corson, 409-412, 427, 428countable, 345, 399, 420Eberlein, 365, 367, 388, 390-393,

409, 417, 419, 420scattered, 398-401, 419, 420uniform Eberlein, 394, 395, 418,

419 \Ja.\d.(vio.l\i2.~

complement, 137, 138, 147-149algebraic, 137, 147orthogonal Fol, 17, 18, 138quasieomplement, 377

constantbasis, 163, 169unconditional basis, 182

conv(M), 2,22, 85, 92, 104convergence

in norm -+, 65pointwise, 66, 68, 86