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3 7 ?
N81J /Vc? ia&SS'S
OPERATORS ON CONTINUOUS FUNCTION SPACES
AND WEAK PRECOMPACTNESS
DISSERTATION
Presented to the Graduate Council of the
University of North Texas in Partial
Fulfillment of the Requirements
For the Degree of
DOCTOR OF PHILOSOPHY
By
Catherine Ann Abbott, B.S., M.A,
Denton, Texas
August, 1988
Q-i.
Abbott, Catherine Ann, Operators on Continuous Function
Spaces and Weak Precompactness Doctor of Philosophy
(Mathematics), August, 1988, 62 pp., bibliography, 36 titles.
If T:C(H,X)-+Y is a bounded linear operator then there
exists a unique weakly regular finitely additive set function
m:5fr(H)-«(X,Y**) so that
T(f) - J H f d m .
In this paper, bounded linear operators on C(H,X) are studied
in terms the measure given by this representation theorem.
The first chapter provides a brief history of representation
theorems of these classes of operators. In the second
chapter the represenation theorem used in the remainder of
the paper is presented.
If T is a weakly compact operator on C(H,X) with
representing measure m, then m(A) is a weakly compact
operator for every Borel set A. Furthermore, m is strongly
bounded. Analogous statements may be made for many
interesting classes of operators. In chapter III, two
classes of operators, weakly precompact and QSP, are studied.
Examples are provided to show that if T is weakly precompact
(QSP) then m(A) need not be weakly precompact (QSP), for
every Borel set A. In addition, it will be shown that weakly
precompact and GSP operators need not have strongly bounded
representing measures. Sufficient conditions are provided
which guarantee that a weakly precompact (QSP) operator has
weakly precompact (QSP) values. A sufficient condition for a
weakly precomact operator to be strongly bounded is given.
In chapter IV, weakly precompact subsets of L 1(fi ,X) are
examined. For a Banach space X whose dual has the
Radon-Nikodym property, it is shown that the weakly
precompact subsets of L1(p,X) are exactly the uniformly
integrable subsets of ,X). Furthermore, it is shown that
this characterization does not hold in Banach spaces X for
* which X does not have the weak Radon-Nikodym property.
TABLE OF CONTENTS
Ohapter P a g e
I. INTRODUCTION 1
II. REPRESENTATION OF OPERATORS ON C(H,X) 7
III. WEAKLY PRECOMPACT AND GSP OPERATORS
ON CONTINUOUS FUNCTION SPACES 24
IV. WEAKLY PRECOMACT SUBSETS OF L 47
BIBLIOGRAPHY 6 0
111
CHAPTER I
INTRODUCTION
This paper will concentrate on the study of bounded
linear operators on continuous function spaces. The vehicle
for this study is the Riesz Representation Theorem; that is,
if T: C (H,X)-4Y is an operator and m is a vector measure so
that T<f) = /jjfdm for each f € C(H,X), then we study the
relationship of T to m. It seems appropriate at this time to
give a brief overview of the Riesz Representation Theorem.
We will mention only selected highlights from its abundant
history and refer the reader to [DU, p. 180-183], [Di, p.
416], and [DS, p. 373,380-381] for additional references.
In a note in Gomptes Sendus in 1909 [R], Frederic Riesz
established the following theorem
Etant donnee I'operation lineaire A[f(x)], on peut
determiner la fonciion a variation bornee «(&), telle que
quel Ie que so i t la fonetion continue f(x), on ait
W ( * ) ] = ll0 f (x)da(x) .
The following expanded version of the theorem appears in [RN,
p. 110].
The Stieltjes integral
J b f(t)i,(x) 3.
formed with a fixed function of bounded variation ot(x),
defines a linear functional in the space C of continuous
functions f(x), and conversely, every linear functional can
be written in the integral form.
In 1913, Radon [Ra, p. 1333] extended the representation
theorem to Include functionals on C(Rn), and this time the
representation theorem was expressed in terms of measures.
In a note at the end of Saks" Theory of the Integral [Sa,
p. 322], Banach demonstrates an extension of the Riesz
Representation Theorem to C(H) for H compact and metric. An
extension for functionals on C(H) for H a compact Hausdorff
space was accomplished by Kakutani in 1941 [K, Theorem 9, p.
1009]. The theorem appeared as follows.
Let 0 be a compact iausdorff space, and let C(U) be
defined as usual. Then every bounded linear functional f(x)
defined on C(0) which satisfies the conditions
11/ II - '
x > 0 implies f(x) > 0
can be expressed in the form
f(x) = \^x( t)fi(dt)
where p>(E) is a completely additive non-negative set function
defined for all Borel sets B of ft such that p(U) - 1.
At this point the study branched in several directions.
Various authors, e.g. Halmos [Ha], Hewitt [He], Edwards [Ed],
and Bourbaki [Bour] have examined linear functionals on C(H)
for H locally compact. Gelfand [G] and Bartle, Dunford and
Schwartz [BDS] examined operators on C(H) which take their
ranges in a Banach space. The following theorem appears in
the paper of Bartle, Dunford, and Schwartz.
^ ® ® oper&tor on C (S) to X, there exists a unique
set function p:Yhl** such that
(a) fi(')x* 6 &(S) for each x* € /*
(b) the mapping x ^(*)x of I into &($) is continuous
with ike X and C(S) topologies in these spaces, respectivelyj
(c) x*7f = $ s f ( s ) p ( i s ) x * , f € C(S), x* € I*;
(*) |^| - semi-variation of )i over S.
Conversely, if (i is a set function on £ to X** satisfying (a)
and (b), then equation (c) defines an operator T:C(S)->X with
norm given by ( i ) , and such that t*x* = p(*)x*.
The studies of Singer took another direction. He chose
instead to examine the linear functionals on C(H,X). In a
1959 paper [Si, Theorem 2, p. 35] of his, we find the
following theorem.
Le dual de I'espace C^ est equivalent a I'espace de tous
les champs de formes f completement a d d i t i f s , a variation
bornee et regvliers, la norme etant
|/| * y*r f . e£Q e
I 'equivalence $ <—> / entre ces deux espaces est donnee par la
relation
<x, - ig<*(9),ifq> (z € CA).
The paths merge in the study of operators T:C(H,X)->Y, where X
and Y are Banach spaces. In Dinculeanu's book [Di, p.
398-399] we find the following theorem.
L e t Z C /' be a n o r m i n g s p a c e . I f U : % g ( T ) -» / i s a
l i n e a r n a p p i n g w i t h | | ^ | | < 03 f o r A € 56, t h e n t k e r e e x i s t s an
a d d i t i v e s e t f u n c t i o n m:% -» x (E,Z*) w i t h f i n i t e
s e m i - v a r i a t o n s u c h t h a t
9 ( f ) - - i f d m , f o r f € * g ( T ) .
M o r e o v e r , f o r e v e r y o p e n s e t G we h a v e
IM - ' ( ' ) •
Additional contributions to the representation theorem
(some quite technical in nature) were made by Brooks and
Lewis [BrL] and Batt and Berg [BB]. The theorem in [BrL] is
stated as follows.
I f L : B -4 F i s an o p e r a t o r , t h e n t h e r e i s a u n i q u e w e a k l y
r e g u l a r s e t f u n c t i o n -> B ( E , F * * ) s o t h a t
L ( f ) = f j [ f t * , f ^ 3 .
A representation theorem for operators more in keeping
the nature of the original representation theorem of
Riesz may be found in [Tu].
We conclude this chapter with a discussion of some of the
terminology and notation used in the remainder of this paper.
In the chapters which follow X , Yf and Z will represent real
Banach spaces. The dual of the Banach space X will be •
denoted by X . We identify X with its cannonical image in * *
X . The symbol H will denote a compact Hausdorff space;
$(H) will denote the <r-algebra of Borel subsets of H and S an
arbitrary <r-algebra. The set of all X! simple functions
taking their range in the Banach space X will be denoted by
SJJ(S) . We will use U^(£) to denote the uniform closure of
the X valued simple functions; that is, UX(E) is the closure
of SX(E) where the topology is given by the norm defined by
II'HOD = sup<||f(h)||x:h€H>.
The Banach space of continuous X valued functions on H will
be represented by C(H,X), where the norm on C(H,X) is again
given by the formula
ll fL - »uP<||«<h)||x:h€8>. In the case where X = R, we will use S(£), U(£), and C(H) in
place of SR(S), UR(S), and C(H,R).
will use ca(£,X), or ca(£) if X = IR, to denote the set
of all countably additive set functions x . We say that
an element /t of ca(S,X) is regular if for each E € E and e >
0 there exists a set F G S whose closure is contained in E
and a set G whose interior contains E so that J/t(C)j < e for
all C € S with C C G\F. The set of all such elements of
ca(E,X) will be denoted by rca(£,X).
We say that a function T:X -» Y is a bounded linear
operator if T is both continuous and linear. A function
x :X-*IR is said to be a continuous linear functional if it is
both continuous and linear. The term operator (functional)
will be understood to mean bounded linear operator
(continuous linear functional). We shall use £(X,Y**) to
denote the Banach space of bounded linear operators having X * *
for a domain and subsets of Y for a range. For T €
«(X,Y**)f ||T|| will be given by
11*11= ""pfllToolHMi*1*-$ j(c
Let m:E-*2(X,Y ) be an additive set function on a
*-algebra E. The semivariation m of m is defined by
m(A) = sup {II S m(B. ) (x.) It: II (A) , llx.lKl}.
"B IKA) 1 1 11 11 1,1
(We use 11(A) to denote a partition of A.) If m(A) < oo for
all A6£, then m is said to have finite semivariation. A
predecessor of this property, known as the Gowurin a-property, appears in [Go]. The reader should note that finite semivariation does not imply finite total variation.
rsj
If m(A^)-K) whenever (A^) is a pairwise disjoint sequence from
S, then m is said to be strongly bounded. Equivalent
formulations of this property may be found in [BrL, Lemma
3.1]. Examples in [Le] and [Dol] show that countable
additivity does not in general imply strong boundedness. The
measure m is said to be weakly regular provided m * (•) = (x,y ) '
* <m(•)x,y > is a finite regular Borel measure for each x € X
He *
and y €Y . For terms used but not defined in this paper, the
reader may consult [DU] and [DS].
CHAPTER II
REPRESENTATION OF OPERATORS ON C(H,X)
For the convenience of the reader, the representation
theorem for bounded linear operators on C(H,X) used in this
paper will be presented in this chapter. We will begin with
the Riesz Representation Theorem for linear functionals on a
compact Hausdorff space. The proof which is given here comes
from [DS]. Next, a representation theorem for bounded linear
operators on UX(E) due to Dinculeanu [Di] will be given.
Finally, the representation for bounded linear operators
T:C(H,X)-4Y found in [BrL] will be demonstrated. The proof of
the representation theorem for operators on C(H,X) will
employ the technique of embedding UX(»(H)) in C(H,X)**. Once
this objective is accomplished, the representation theorem
for bounded linear operators on UX(#(H)) may be put to work.
The uniqueness and weak regularity of the representing
measure will follow from properties of the representing
measure given in the Riesz Representation Theorem.
1. Theorem.[DS, p.258-265] If M is a compact Mausdorff
space, there is an isometric isomorphism between C(i}* and
rca(H), where E is the stgma algebra of Bore I subsets of H,
so thai corresponding elements of x* and p satisfy the
identity
x*f s tgfil*
£E2fif: First it will be shown that each /t € rca(E)
defines an element x* of C(H)* by the above formula and in *
this case ||x || = ||/t||. Let ft € rca(S) and f € C(H). It is
easily shown that f is p integrable. Define X*:C(H)-HR by
x (f) = JHfd/t
* Clearly x is linear. Furthermore,
5 ® o } l f | t , l H < H ) - tl'lUkll-
Thus x* defines an element of C(H)* for which ||x*|| < ||/*||.
1c
Let e > 0. Let b® a pairwise disjoint collection
of elements of £ such that
^ IkII - f -
Choose s i € {1, -1} so that s ^ f E j ) • | /t (E±) | . Since p is
regular, we may choose closed sets Cj C Ej so that |/»|(E1\C1)
< | for l<i<k and open sets G ± D C i so that |^|<Gi\Ci) K |
for l<i<k. Since H is a compact Hausdorff space we may
insist that the collection (G^) be pairwise disjoint. For
l<i<k, choose continuous functions so that c K f ^ h j O for h€H,
ffCj) - 0, and f(G*) = 0. Set f - E s ^ . Hence
* k k k II* II 2 / f d* ^ ) " S |f|(E \0 ) - £ U|(G.\C.)
i=l i=i a -1 i=i1 1 1 1
> - 2f > || |j - s,.
Thus ||x || > ||/»||. Therefore ||x*|| =
Next, it will be shown that every element x* of C(H)* may
be represented by an element ft of rca(E). Let x* € C(H)*.
Since C(H) is a subspace of U(£), x has a norm preserving
V* * * extension to U(L) . Let y denote one such. Define X by . * M E ) = y (*E) for each E € £. Clearly X is additive, p(E)|
$ •
< ||y ||, and y f = JHfdJ for f € S(£) . Since S(£) is dense in
U(E), it follows that x*f • JHfdJI for all f 6 U(E) .
Now X « X1 - X2, where X and X2 are non-negative. Thus
to complete the proof, it suffices to consider the case where
4 is non-negative and obtain /( € rca(E) so that /Hfd/i =
for f € C(H).
For a closed set F, define
/»1(F) - inf{X (G) :GDF,G open), and for ECH, define
P 2 ( E ) = sup{/t1(F) :FCE, F closed}.
Observe that /»1 and are non-negative and non-decreasing.
We note that /^(F) = /»2(F) for F a closed set.
Let Fj be closed and G1 be open. If G is an open set so
that GDFFJXGJ), then (GJUJGDF^ Hence j (G^UG) < JL (G ) +
-1(G). Thus
= inf (C) rCDFj ,c open)
< X (G1UG) < X (G1) + X (G).
Since the above statement is true for all open sets
G^fFjXGj), the following holds:
^(Gj) + inf{J(G):G3(F1\G)1, G open)
= X (G1)+/r1(F1\G1) .
10
Let P be an a r b i t r a r y c l o s e d s e t . The p r e v i o u s s t a t e m e n t
h o l d s f o r a l l open s e t s . In p a r t i c u l a r t h e s t a t e m e n t h o l d s
f o r a l l open s e t s G^DFflFj. Thus
/ ' i < F i ) ^ i n f { i (Gj )+ /<j (F^\Gj) : G j 3 ( F j f l F ) , Q± open}
< i n f U (G 1) : G 1 3 ( F 1 f l F ) , G1 open}
+ s u p { / t 1 ( F 1 \ G 1 ) :G1D(F1f lF) , Gj open}
< ^ 1 ( F i r i F ) + sup{/» 1 (C) :CC(F 1 \F) , C c l o s e d }
- ^ ( F j H F ) + / » 2 ( F 1 \ F ) .
I f ECH, t h e n
P 2 ( E ) » s u p { / t 1 ( F 1 ) : F 1 C E , FX c l o s e d }
< sup{/J 1 (FriF 1 ) iFJCe, F x c l o s e d }
+ s u p ^ f F ^ F ) ^ C e , F 1 c l o s e d }
< s u p { ^ 1 ( C ) :CC(EflF) , C c l o s e d }
+ s u p { / t 2 ( C ) :CC(E\F) , C c l o s e d }
- ^2(EflF) + / t 2 ( E \ F ) .
I t w i l l now be shown t h a t
^ / » 2( E n p ) + / , 2 ( E X F ) *
Let Ka and K2 be d i s j o i n t c l o s e d s e t s . S i n c e H i s a compact
H a u s d o r f f s p a c e , we may c h o o s e d i s j o i n t open n e i g h b o r h o o d s
and W2 o f K1 and K2 r e s p e c t i v e l y . I f G i s an open
n e i g h b o r h o o d o f KjUKg, t h e n } (G) > MGflWj) + ^(GflWg)
Thus
/ j 1 (K 1 UK 2 ) - i n f {J (G) iGDfKjflKg) , G open}
> i n f U (Gf1W1) jGDd^flKg) , G open}
+ i n f {J (GflW2) : GO (KjflKg } , G open}
> ^ ( K j ) + / ( ! (K 2) .
11
Hence
/»2(E) > 8up{/»1(K1)+/ll(K2):K1C(EnP), K2C(E\F), ^ ^ closed)
> fi 2 (EfiF) + /»2(E\F).
Thus
/ t2
( E ) 85 / t2
( E n F ^ + * 2 ( E \ p )
for ECH and F a closed set.
Thus every closed set is measurable. Define ft to be
the restriction of n 2 to the algebra generated by the closed
subsets of H. By [DS, III. 5.2, p. 133], it follows that is
additive. Observe that for a closed set F, /i (F) = /« (F) = 1 2
/MF). Hence by the definition of and it follows that
/» is regular. Thus by [DS, III.5.14,p. 138], /t has a regular
countably additive extension to the Borel subsets of H.
To complete the proof, it will be shown that
lBtu -
for f € C(H). Let f 6 C(H). Without loss of generality
assume that f is non-negative and that 0 < f(h) < l for all h
€ H. Let e > 0. Partition H into disjoint sets,
from the ^-algebra of Borel subsets of H so that
t-
£ a ji(E ) + e > J fd/t, i==l 1 1 H
where a^ = inf {f (h) :h€E^}. Since /J is regular, we may choose
closed sets F^CE^ SO that
n ^S^a1/t(F1) + 2e >
Since H is a compact Hausdorff space and f is continuous,
there exist disjoint open sets so that
12
b =inf<f(h):h€G.} > a, - -A 1 i - i n
for i - l,. . . n. Thus
n S + 3e >
Recall once again that if F is a closed set, then p^(F) =
= M * ) ' Now if G is an open set so that FCG, clearly
M F ) < /1(G). Since p( G) = sup {/j (F) :FCG} , /{(G) < /1(G). Thus
n n S b »(G ) < E b.i(G ) < JHfdJ. 1=1 i=i 1 H
Hence
<*> j H(a^>j Hfd,.
Now /»< H} = -1(H); thus
/(l-f)dJ < JH(l-f)d/».
Note that the function g defined by g = 1—f is a function for
which 0 < g(h) < 1 for all h € H. Thus the inequality (*)
must hold for g. Hence JHgd/» = JHgd/i and consequently JHfd/»
= /HfdJ.
Before proceeding with the proof of the representation
theorem on UX(E), we must make the following definition.
2- fiatinltlorj. Let m:E -+ 2(X,Y) be an additive set
function on a sigma algebra E. For f = E* x. € S (E), i X
define
J fdm = Em(A A) ( x ± ) .
For f € Uy(E), let (f ) be a sequence from S„(E) so that f A n X n
13
converges to f uniformly. Define
Jfdm = lim Jf dm. n-tao
The reader may consult [Di, p.106-142] for properties of this
integral.
3- lhfi2ZSm.[Di, Theorem 1, p. 145] for each bounded
linear operator T: V f there exists a unique finitely
additive set function m with finite seaivariat ion so
that
f ( f ) - tfd*
for f e flys;.
Furthermore if 7 corresponds to n, then a(S) *
PVQQf; I»et T:Ujj(S)-fY be a bounded linear operator.
Define m:S-»a»(X,Y) by
m(A)(x) = T(xAx) for A € £, x € X.
For each A € E, the map m(A) :X-4Y is clearly linear. Now
||m(A) (x)j| - ||TCxjfA)|j < ||X||||T||
for all x € X. Thus m takes its range in X(K,Y). Let A and
B be in S so that AOb = ^. Then
m(AUB) (x) - T U A U Bx ) * T(XaX+XbX) = m(A) (x)+m(B) (x) .
Thus m is finitely additive. If f € SX(E), then it is clear
that T(f) = Jfdm. Let A € S; then
m(H) » sup{ ||E m(A1)(x1)||:Aieir(H) ,||xi||<l>
-SUp<||/*Hfd»||:(6Sx(£), IKIL-D
" IITH-
14
Thus m has finite semivariation. Let f € Uv(£). Choose a A
sequence {fn) from SX(E) so that converges uniformly to f.
Then
T(f) - lim T(f ) = lim Jf dm = ffdm. n-too
n n-ta>
The uniqueness of m is clear, as it must be the case that
t U a x ) » J*Axdm - m(A) (x) .
We shall now proceed with the task of embedding U (£) in ** x
C(H,X) . First we make the following definition.
4. D e f i n i t e . For f* € C(H,X)* and x € X define
<x,f*>:C(H) 4 R by
* * <x,f >(p) > f (p«x) for f € C(H).
* Now <x,f > defines a bounded linear functional on C{H). Let
^(x, f*) b e t h e u n i <J u e regular Borel measure in the Riesz
Representation Theorem so that
<x,f*>(F) - J H F ^ ( X f f * j .
For f € C(H,X) and f* € C(H,X)*, define <f,f*>:C(H) -4 R by
<f,f*>(?) = f*(pf),
where pf(h) » p(h)f(h). As before, note that <f,f*> defines
a bounded linear functional on C(H). Let p.. * be the (r / r )
unique regular Borel measure which represents it.
For A € :»(H) and x € X, define £ Ax on C(H,X)* by
XAx(f*) = A) for f* € C(H,X) .
Thus Sjj(se(H)) may be viewed as a subspace of C(H,X)**. For A
€ &(H) and f € C(H,X), may be defined analogously.
15
5. UiSfiCSm.[BrL, Lemma 2.1] If I and Y are Banack
sfaces . . i , X j ^ <= S j M D ) , Hen • ||-||M.
n Pfgpf: Let s = II y x. be a simple function In
1 * 1 A. 1 1
canonical form, I.e. A.fl A. • <f> and x t x for 1 ± j. i i
Without loss of generality assume J|x1 jj = JJsjJ . Choose x* €
Bx* so that ||xjJ| = x (Xj). Let t € Aj. Define ?:$(H) -» X* by f 0 if t (f A
?(A) = | x* if t € A
Now Jfdtf = <f(t),x > for f € C(H,X); thus y may be viewed as
an element of C(H,X)*. It is easily shown that y is
countably additive and Jyj is regular. In addition
9(H) - sup{||S?(B)xi|| : B € 11(A) , ||xi|| < 1}
- SU P < | x * ( x ) | : ||x|| < 1} = 1 « X*(X )
ll^llc(H,X) Now,
8(11) = )(7) " S r . (?) » #,(A.). 1 - 1 Ki l-l 1-1 (Xj/?) i i-r(xi,9)lrti
Let e > 0. For each i > l, choose a compact set C A^ SO
k i ? i ( x ^ , if) I (^i^i) < e. Next, choose a collection
pairwise disjoint open sets so that K. C v,, t ^ i i A A
k k
i =2 Vi ' a n d i52l/t(x4 ,») I ( Vi X Ki ) < e' Choose >k C C (H) so JL 1 B1
that p1(Ki)=l, ^i(V^)=0 and 0 < ^ < l. Then for i > l the
support of yi is contained in V1. Set G1 = {x€H:(x)=l).
oo Note that G1 is a Gg set as Gi = (x€H: p i( X)
>l~}. Now
16
But ,
Thus,
. S . I ' U j . f j ' V - 'H ' i * ( V f ) l
- , 5 , l ' « < e .
< r i a ' ( K 1 , f ) * < * 1 - T > < r i ) = K f j X j )
= / j j P i x i ^ ' = P ^ * ) = 0 .
i l 1 l ' ' ( x 1 . , ) ( G i , l * f -
Now
( A T A ^ K ? ) = / . ( X i , ¥ ) (*!> - < * ! . ? > U A i ) - ? ( * ! > * » )
= l a - » < V X 1 - **<*!> " | K | | .
Hence,
l i l / A / i ' " ! " I I N I + <Ai> I
- IKII " i 5 2 I'1 < X ; i , » ) ( A i ' I
* I M + j U ^ ^ ' V l ]
* IKII - , j l « W - £
> HxJI - 2 c .
T h e r e f o r e | H | c ( h . x ) - > | K | | > H ^ .
Now, f o r t h e r e v e r s e i n e q u a l i t y , l e t s be a s b e f o r e and (
^ B C ( H , X ) * ' c h o o s e ( K ^)» {G^}» { V l } , and {p^} a s b e f o r e ,
e x c e p t l e t i range from l t o k . Then
l < S - H " l i l / ( x . , ( ) < A i ' l
17
* li5iI'l(x1,o<Ai' " '(Xj.o ( 0i' ]I
+ 3 f
k 5 iii5,'ixiiic(H.x)+ 3 c
< IjxJI + 3c .
Therefore |H| c ( h > x,*- < Hs^.
6. IbfifljEfiffi.CBrL, Theorem 2.1] Let f € C(M,I). There
exists a sequence simultaneously
approximating f in the uniform and C(M,I)** topologies.
SXSSL: Let A„ = <x: ||f(x)||x > 1, = n <x: ||f(x)||x > i-|>.
Note that A n € ®(H) as A n is a compact . Let f € C(H,X).
Define to be f* . n
Note that
IIfn - f L - ll*An
f - f L < i t 0 -
Thus f n -• f uniformly.
By Definition 4, f^ may be viewed as an element of * *
C(H,X) . We will next show that f converges to f In
* * 2 C(H#X) . Let € > 0. Choose n so that - < e for n > N. Let *
f € BC(H,X)*' s l n c e X A is the pointwise limit of elements n
of C(H), by applying the Lebesgue Dominated Convergence
Theorem, we may choose f € C(H) so that
I ' u . f V V - '(f,f*j<"l < e-
18
Hence
- f*(f)| - k ( f, f*,(A n) " f*(f)|
- I ' u . f V V +
< € + |f*(pf - f)|
< € + ii' ii ii" - 'ii
< e + - < 2e . n
oo
Thus f n -t f in the C(H,X)** topology.
We will now proceed with the construction of the desired
simple functions. Partition An into sets <A .} n so that n ni i a-1
Ani € f o r e a c h n a n d 1 a n d ||f(x) " f(Y)|| < ^ for x and y
k ft
Now in A n l. Fix x n l € ((Ani) and set s n > .S x A x n l
n i
HSn ~ f L = Hi?, *a . xni " fll loo n i
k
S Hi?, *A xni " *A fllao + ll*A f " f L < n'
n i n n
Thus s n -» f uniformly.
Let e > 0. Choose N so that ^ < e for n > N, and fix n >
N. For convenience set Bi - a r 1, xa - X r 1 , and kn= k. Let
'* € BC(H,X) * ' H o t e t h a t
k s "n " «n)(f »l " I i?i *BjXi "
= l i l . ^ X j . f W " ' ( f . f W l l -
Choose compact sets Ki C b, so that
i ^ l ' i x ^ f ' i h w <<• an<s i l 1 i " ( f . f M < vc i > -
Next, choose a pairwise disjoint sequence {V.}k of open 1 i - 1
19
sets so that K± C ,
k
For each i choose pi € C(H) so that p± (K±) » l, ^(Vj) = 0,
and 0 < < l. Thus
lv f*> - V*>l - - Nf,,*)'8!)! 5 I ^ I X j . f V W l + R f f . f V W l
+ l^< X l,e*)< Ki' " ^ (f,f*)
( Ki'l
S 2 f + l ^ l x ^ f V i ' -
S 2 f + Sl'*(x1,f*)l<Vi\Ki'
+ |Sf*(x1p1 - f P l)|
+ s k((,f * ) l'V Ki» < 4f + Hf IIIIVl - f'illoo
< 4e + i < 5e. * *
Thus s n 4 f in C(H,X)
7. lhfi2Eeffl. [BrL, Theorem 2.2] If T:C(M,X) -» Y is a
bounded linear operator, then there is a unique weakly
regular set function n:m(H) -» z(X, Y**) so that
Uf) • for f € C(t,l).
furthermore, ||7|j * %(g)m
a|c g|c & ale
EEfiof: Now SX(»(H)) C C(H,X) . Hence T :Sx(® (H) ) -t
20
** * * Y . By Theorem 5, it follows that T has a continuous
extension T to all of U^(»(H)). Hence by Theorem 6, T(f) » * *
T (f) - T(f) for f € C(H,X). By Theorem 4, there exists
m:«8(H) -+ 2(X,Y ) representing T. Thus,
T < V > = T <*AX) = ^ H V d m = m< AM x)-
Let x 6 X and y* € Y*. Then
<m(A)(x),y*> = <T**(^ax),y*>
- A r A x ( T * ( y * ) ) = > ( x > T« ( y' ) ) (A)
where ^(x,T*(y*)) ***© regular Borel measure
guaranteed by the Riesz Representation Theorem. Hence the
choice of m is unique.
In addition, m(H) =||T|| = ||T|| .
If T:C(H,X)->Y is a bounded linear operator with
representing measure m, we may actually show that |m *J is
regular. To this end, consider the following.
8. Pefjpitjon. Let H be a compact Hausdorff space. An
operator T:C(H,X)~*Y is said to be dominated if there exists a
positive linear functional p € C(H)* so that ||T(f)|| <
<P.||f< ->||>.
9. Theorem. [BrL, Theorem 2.8] If T:C(H,X)-+Y is a
bounded linear operator with representing measure m, then T
is dominated if and only if J m I € rca(s&(H)).
21
£caa£: Suppose that |m| € rca(»(H)). Thus ||/Hfdm|| <
si**ce /H( )d|m| defines a positive functional, we
conclude that T is dominated.
Conversely suppose that T is dominated by a positive
functional p. Using the Riesz Representation Theorem we
obtain a positive regular Borel measure (i for which p(f) =
JHfd/i for all f € C(H) .
Let e > 0. Let A € E, x 6 X, and y* € Y*. Since m, * (x,y )
is a regular Borel measure, there exists a sequence (K ) of n
compact sets and a sequence (G ) of open set so that K C A C n n
Gn a n d lm(x,y*) H Qn^ Kn* < n* L e t {^n^ b e a se<3uence of
continuous functions so that Pn(Kn) = 1 and Pn(Gnc) • 0. By
the Lebesgue Dominated Donvergence Theorem, lim f » dm * jHB, J" P n <X'Y )
= m(x,y*)(A) a n d l l m J V n ^ = ^ ( A )* N o t e t h a t
n-to
IVn d m( X,y*)l = l<**'/H'nxd">l
< II/hVHI * ^Hlknxlld^ *
Thus m(Xfy*)(A) < /»(A) for all x € X, y* € Y*, and A € £.
Hence ||m(A)|| < /t (A) for all A € S. Thus |m| (A) < ^ (A) for
all A € S, and it follows that |mj € rca(E).
10. Defjnitipn. Let m:E-tt(X,Y**) be a finitely additive
set function. For y* € Y*, define my*:E-«* by
®y*(A)(x) = <m(A)(x),y*>.
11. Corollary. If T:C(H,X)-+Y is a bounded linear
22
operator with representing measure m, then | * | € rca($(H))
Pffppf: If m is the representing measure for a bounded
linear operator T, then for y* € Y* we have (y*oT)(f) •
Y (T(f)) = /Hfdmy*. Thus ||y*oT|| - m *(H) = |» *1* Hence $ - y
y OT is dominated. Therefore m^* 6 rca(36(H)).
We may now use Theorem 9 to establish a representation
theorem for continuous linear functionals on C(H,X).
12. Corollary. If x* € C(H,X)\ then there exists m €
rca(SB(H),X ) so that
x*(f) = /Hfdm.
* •
PrQQt• Let x € C(H,X) . By Theorem 11 there exists a
measure m:s®(H)->(ie(X,R)=X*) so that |mr| € rca(S) for all r €
R. Since nij • l«m, m € rca(»(H) ,X*) .
In this paper we will be primarily concerned with weakly
precompact and QSP operators on continuous function spaces
and how these operators influence their representing
measures. In addition we will find it convenient to use
several properties of unconditionally converging and weakly
compact operators. We close this chapter with the formal
definitions of these classes of maps.
23
13• A subset S of a Banach space X is said
to relatively weakly compact if the weak closure of S is
weakly compact.
An operator T:X-*Y is said to be weakly compact if T(B )
is relatively weakly compact.
A subset S of a Banach space X is said to be a GSP set if
for all finite measure spaces (fi,S,p), bounded linear
operators T:X-H^f/O, and e > o there exists B € £ such that
/»(fl\B) < e and T(S) | B • {T(x)*B:x€s> is totally bounded.
A bounded linear operator T:X-+Y is said to be GSP if
T(BJJ) is a G S P s e t .
A subset of S of a Banach space X is said to be weakly
precompact if every bounded sequence of elements of S has a
weak Cauchy subsequence.
A bounded linear operator T:X->Y is said to be weakly
precompact if T(B^) is weakly precompact.
A bounded linear operator T:X-+Y is said to be
unconditionally converging if ST(xn) is unconditionally
convergent whenever £xnis weakly unconditionally Cauchy.
CHAPTER III
WEAKLY PRECOMPACT AND GSP OPERATORS
ON CONTINUOUS FUNCTION SPACES
In this chapter GSP operators and weakly precompact
operators on C(H,X) will be examined. The focus of this
discussion will be on the relationship between these classes
of operators and their representing measures. For many
important classes of operators on C(H,X), properties of the
operator are reflected in the values of the representing
measure. For example, if T:C(H,X)-+Y is an operator with
representing measure m, then
(i) m(A) is weakly compact for each Borel set A,
whenever T is weakly compact,
(ii) m(A) is compact for each Borel set A whenever T is
compact,
(iii) m(A) is absolutely summing (p-summing) for each
Borel set A whenever T is absolutely summing (p-summing),
(iv) m(A) is unconditionally converging for each Borel
set A whenever T is unconditionally converging,
(v) m(A) is completely continuous for each Borel set A
whenever T is completely continuous,
(vi) m(A) is weakly completely continuous for each
Borel set A whenever T is weakly completely continuous.
24
25
The implications in (i) and (ii) are in Batt and Berg
[BB] and Brooks and Lewis [BrL]. The conclusions in (iii)
are from Swartz [Sw2] and Bilyeu and Lewis [BiL]. The result
in (iv) may be found in Swartz [Swl], Dobrakov [Do2] and
Bilyeu and Lewis [BiL], The assertion in (v) is obtained by
Bilyeu and Lewis[BiL], and (vi) is established by Bombal and
Cembranos in [BC]. (The reader may refer to these references
for the definitions of classes of operators not explicitly
discussed in this paper.)
In [Swl] Swartz showed that if T is unconditionally
converging, then m is strongly bounded. As each member of
the six classes of operators listed above is unconditionally
converging, it follows easily that each must have strongly
bounded representing measure.
In this chapter examples will be given to show that
neither of the above conclusions necessarily hold for the
representing measures of weakly precompact operators or QSP
operators. That is, if T is weakly precompact (GSP), then m
need not be strongly bounded and m(A) need not be weakly
precompact (GSP). However, sufficient conditions will be
given for a weakly precompact (GSP) operator to have a
representing measure with weakly precompact (GSP) values.
The following fundamental theorem of H. Rosenthal [Ro]
characterizes weakly precompact Banach spaces. His theorem
has proved important in the study of Banach spaces.
26
1« Theorem• A Banach space contains a subspace
isomorphic to if and only if it has a bounded sequence
with no weakly Cauchy subsequence. Consequently, a Banach
space is weakly precompact if and only if it contains no copy
of l t .
It is well known that If an operator T:X-»Y Is weakly
compact, then Its adjoint, T , Is weakly compact. However,
an analogous statement does not hold for weakly precompact
operators. In fact, T weakly precompact need not even Imply * *
that T is weakly precompact.
2. • If a bounded linear operator T:X-+Y is
weakly precompact then neither T* nor T** need be weakly
precompact.
Let I:cQ-+co be the identity operator. The operator I is
weakly precompact as cQcontains no copy of lt . However, I* 9fe *
and I are the identity operators on lt and / respectively .
Now, l^ contains an isomorphic copy of lt. Thus neither I* * *
or I can be weakly precompact.
It is the case, however, that the following holds.
3. Theorem. If T:i-*Y is a bounded linear operator so
that 7 is weakly precompact, then I is weakly precompact.
27
ECflfif: First, we observe that if a Banach space Z
contains a copy of lt then so must Z*. To see this, let
J:ij-*Z be an isomorphism. Then by [DS, VI.2.7] we see that * *
J : Z is onto. Since contains a copy of lt, J
contains a bounded sequence with no weak Cauchy subsequence.
This sequence must be the image of a bounded sequence in Z*
which has no weak Cauchy subsequence. Thus, by Theorem 1, Z*
contains a copy of lt .
Now, suppose that T:X-+Y is a bounded linear operator for
which T is not weakly precompact. Then X contains a copy, W,
of lt on which T acts as an isomorphism [Lo, Theorem 3], Let
J: Z i-+X be an isomorphism taking lt to W. Note that TJ is an
isomorphism. Hence (TJ) — J T is onto I . Since I 00 oo
contains a copy of It , I^ contains a bounded sequence with no
weak Cauchy subsequence. This sequence must be the image of
some bounded sequence in Y*. Thus J*T* cannot be weakly
precompact. Therefore T* is not weakly precompact.
A fundamental theorem of C. Stegall [St], proves to be
very useful in the discussion of GSP operators. Stegall's
Theorem is a consequence of the proof of the factorization
theorem of Davis, Figiel, Johnson and Pelczynski in [DFJP].
We state these two theorems and requisite definitions for
reference purposes.
4. Definition. Let (QX.it) be a finite measure space.
28
A function f:fl->X is said to be p-measurable if there exists a
sequence (fn) of /»-simple functions so that lim llf - fll = o n-x» n 11
H-almost everywhere.
A /t-measurable function f:fi-»X is said to be Bochner
integrable if there exists a sequence of simple functions
(fn) so that
/Oll'n " '11^ " °" n-to
In this case we define / fdft for E € £ by E
= lim / f c n-»oo
The collection of all ft -Bochner integrable functions f:fl-+X is
denoted by L^/^X). For a study of properties of the Bochner
integral the reader should consult [DU, II.2].
A Banach space X is said to have the Radon-Nikodya
property with respect to <0,E,/e) if for each ^-continuous
vector measure G:S-»X of bounded variation, there exists a g €
L1(/t,X) so that
G(E) = JEgd/»
for all E € S.
If X has the Radon-Nikodym property with respect to every
finite measure space then X is said to have the
Radon-Nikodym property RNP.
5. Theorem• A bounded linear operator 7;1HF is weakly
compact if and only if 7 factors through a refelexive space.
That is, T:X-*Y is weakly compact if and only if there exists
29
*
a reflexive space 2, and bounded linear operators A:X-+Z and
B:Z->Y so that I = 8A.
6. ThQQCQifl. A bounded linear operator T:I-*¥ is QSP if
and only if T factors through a space 2 so that 2* has the
Hadon-Nikodnt property.
Theorem 6 was proved by demonstrating that if T:X-+Y is a
QSP operator, then the same construction in [DFJP] of a
space through which T factors must yield a space Z so that Z
has the Radon Nikodym property.
It should be observed that if I:X-« is the identity
operator on X, then the construction in [DFJP] of the space
through which T factors yields a space isomorphic to X.
Stegall's proof together with this observation makes possible
the following easy example.
Esssmls* If T:X-4Y is a bounded linear operator which
is QSP then neither T nor T** need be QSP.
Let I:cQ->co be the identity operator. As CQ* « L{ has
the Radon-Nikodym property, it follows from Theorem 6 that I
is a GSP operator. Now, I i s the identity on lt. In
this case the space Z obtained via the construction in
[DFJP1] is isomorphic to lt. However, It* « J does not have
the Radon-Nikodym property. Hence, I* is not a QSP operator. * *
Similarly, I is the identity operator on I and the space
30
* *
constructed in [DFJP] is isomorphic to I . Now, I * does 00 00
not have the Radon-Nikodym property. As a consequence I
cannot be a GSP operator.
The following example shows that the analog of Theorem 3
does not hold for GSP operators.
8. Example• If T:X-+Y is a bounded linear operator so *
that T is GSP, then T need not be GSP.
Let I:JT-*JT be the identity operator on the James Tree
space [Jam]. Now, all even duals of JT have the
Radon-Nikodym property and all odd duals do not have the
Radon-Nikodym property.[LSt] In view of Theorem 6, all odd
adjoints of I are GSP operators. Now, as every adjoint of
the identity operator is an identity operator, in light of
our earlier observation, it is clear that I and all its even
adjoints fail to be GSP operators.
The following theorem of Riddle, Saab, and Uhl [RSU] is
an additional consequence of the factorization theorem in
[DFJP]. Their theorem provides a nice characterization of
weakly precompact operators.
9. Theorem. The operator T:I-*Y is weakly precompact if
and only if T factors through a space which does not contain
ii-
31
The next theorem, due to S. Musial [Mu] in the case where
each separable subspace of X is contained in a separable and
complemented subspace of X and L. Janika [Jan] in the general
case, aids in the further understanding of weakly precompact
operators. We first make an essential definition.
PsfAnitAon- Let (fl,E,/i) be a finite measure space.
Suppose that f:fl-*X is a function so that x*f € L (/t) for all * * ** *
x € X . For E € E, there exists x_ € X for which £
) 38 fEx*fd/t
for all x* € X* [DU, II.3.1]. If x £ € X for all E € E, then
we say that f is Pettis integrable. We say that x is the E
Pettis integral of f over E and we write
* E = P .
A Banach space X is said to have the weak Radon-Nikodym
property with respect to (Q,E,/») if for every /i-continuous
vector measure G:E-« of bounded variation, there exists a
Pettis integrable function g so that
G(E) - P - JEfd/i
for all E € E.
If X has the weak Radon-Nikodym property with respect to
every finite measure space (ft,E,/i), then X is said to have
the weak Radon-Nikodym property.
Theorem *[Jan, Mu] A Banack space 2 does not contain
a copy of if and only if Z* has the weak iaiou-Nikoiya
32
property.
We may now use the conclusions of [DFJP], [RSU], [St],
and [Mu], [J] to establish a relationship between weakly
compact, GSP, and weakly precompact operators.
12. Uisacfil. Suppose that T:I->Y is a bounded linear
operator. If T is weakly compact, then T is GSP. I f T:X-*¥
is GSP, then f is weakly pre compact .
Ecfifil: Suppose T:X->Y is weakly compact. By Theorem 5, T
must factor through a reflexive space. However, every
reflexive space has the Radon-Nikodym property. Hence, from
Theorem 6, one must conclude that T is a GSP operator.
Suppose T:X-*Y is GSP. By Theorem 6, T factors through a
space Z whose dual has the Radon-Nikodym property. Since any
space having the Radon-Nikodym property must have the weak
Radon-Nikodym Property, by Theorem 10, Z cannot contain a
copy of ij. Hence, by Theorem 9, T is weakly precompact.
At this point examples may be easily produced to show
that weakly compact, GSP, and weakly precompact are distinct
classes of operators.
13. Example• GSP operators need not be weakly compact.
Earlier, we observed that the identity operator on c is o
33
a GSP operator. However, this operator is riot weakly
compact.
14. Example. Weakly precompact operators need not be
GSP operators.
Consider again the identity operator I:JTMJT on the James
Tree space. Recall (Example 8) that all odd adjoints are
GSP, but the operator I and all its even adjoints fail to be
GSP. Neither JT nor any of its duals contain lt [LSt,
Corollary 1], Consequently, the operator I and all its
adjoints are weakly precompact.
We now direct our attention to operators on C(H). Here
we find a much closer relationship among the three classes of
operators under scrutiny. To assist us in our investigation
we shall use the following theorem [DU, VI.1.3, p. 149].
15. ThQQrem. // £ a „ - f i e \ i 0f subsets of S, and
T:0(E)-*Y is a bounded linear operator that is not weakly
compact, then there is a linear subspace of V(Yt) that is an
isomorphic copy of I^ on which T acts as an isomorphism.
16. Theorem. If T:C(B)-*¥ is a bounded linear operator,
then the following statements are equivalent.
( i ) I is weakly compact.
( i i ) T** is GSP.
34
(iii) 7** is weakly precompact.
( * v ) 7 \fJ(Yi) 1 3 w e a k l 9 precompact
Prpof: (i)-f(ii) if T is weakly compact, then T** is
weakly compact. Thus, by Theorem 12, T** is GSP. 9|C 3§6
If T is GSP, by Theorem 10, T is weakly
precompact.
(iii)-»(iv) This implication is clear.
(iv)-»(i) Suppose T is not weakly compact. Then • *
T |U(£)ls n o t w e a k lY compact. By Theorem 15, there exists a
linear subspace of U(£) that contains an isometric copy of t GO
* * on which T a c t s a s a n isomorphism. Now I^ contains a
copy of I a n d thus T j u(S) a c* s a s a n isomorphism on a * *
copy of I j. Hence T |u(£) canno"t be weakly precompact.
Of course, the representing measure of any operator on
C(H) will have weakly precompact values. In fact, it should
be observed that if X contains no copy of lt, then every
bounded linear operator T on C(H,X) will have a weakly
precompact valued representing measure. However, the next
example will show that even if an operator on an arbitrary
C(H,X) space is weakly precompact, then the representing
measure need not have weakly precompact values.
17. Example. If T:C(H,X)-*Y is a weakly precompact
operator, then m(A) need not be weakly precompact.
35
Let H = {0} U n = 1,2,...}. Let £ denote the Borel
subsets of H. Define an operator T on C(H,J ) as follows-00
T(f) = (f(0)n - f(i) n)™ = 1
for f € C(H»^00) • Here f(0)n and f(^)n denote the nth
coordinates of f(0) and f(^) respectively. Let f € C(H,i ) " 00
Then
l , < 0 ,n " '<5'nl S " "n'lli ? «• 00 11
Thus T(f) € cq. Hence TtCfH,!^) *4 cq, and clearly T is
linear. To show that T is bounded, note that
||T(f)||c - sup|f(0) -f(i)| o n
< suplf(O) I + supIf(~) I n n
< ||f(0)||, + ||,(i)||( < 2||,L. 00 00
Thus T is bounded.
consider the operator T on Uj(£)=U defined in the Now _ _ . — -
00
same manner; that is
*<«> - (f(0)n -
for f € U. The same arguments used for T show that T is
bounded and linear. Furthermore, T takes its range in I as 00
sup|5(f)n| - sup|f(0)n - f(i,n| < ,||i;||ro.
Note that T restricted to C(H,1 ®) is T. Define m:£-*e(Z J ) 00 00
by
m(A) (x) - T(^Ax) = ((ATAx(0))n - <*Ax(i) )®al.
Clearly, m is the measure guaranteed by Theorem II.3 for
36
which
T(f) - JHfdm
for all f € U. To show that m is the representing measure
for T, it suffices to show that m is weakly regular. Let x =
( xn ) € a n d y = € c0* = Without loss of
generality assume ||x|| = 1. Let e > 0. Choose N so that
GO £ |Yn| < e. Note that for A € E, n»N
m<x,y)<A> " <•<*>*.*> = J [<***«»>„ " ( V n " n l y n -
Furthermore note that any subset of H is either open or
closed. In fact, any set containing 0 must be closed and any
set not containing 0 must be open. It suffices to consider
two cases.
First, suppose A C h and 0 A. Then A is open. Set F =
A fl 1 < n < N-l}, and notice that F is certainly compact.
In addition
lm(*.y)l(AVP)
"" S U^{ Bg| im(x,y) I partitions A\p|
" ne?\F | U A^ X ,° , ,'> " ( W R " n l | v „ |
< £ I x I I y I < e . n€A\F nl
Now suppose A C H and 0 € A. Then A is closed and in fact,
compact. Let Q = [A D <i:i < n < N-l>] U > N> U {0>.
The set Q is open and
lB(x,y)l<°U> * ^ A k a \ * " l » l n " W ' R ' n l
37
ao
- n?„lynl < f"
n=N
Thus m is weakly regular and consequently m must be the
representing measure for T.
Now T must be weakly precompact as Cq contains no copy of
I!. However,
m(0)(x) - T(*{0)x) = (xn - 0 ^ - x
for all x = (XR) € l^. Thus m(0) Is the Identity operator on
• Hence m(0) is not weakly precompact.
Example 17 also provides us with a GSP operator whose
representing measure has some values which are not GSP
operators. To see this, observe that T factors through c , a o
space whose dual has the Radon Nikodym property. In view of
Theorem 6, T must be GSP. Since m(0) is not weakly
precompact, by Theorem 12, m(0) is not GSP. Furthermore,
since the representing measure takes its range in 2(/ I ) 00 00
rather than in by [Do2] the representing measure is
not countably additive and hence not strongly bounded. This
example leads to the following theorem.
18. Theo^m. The Banach space I contains a copy of lt
if and only if there exists a Banack space Yf a compact
Mausdorff space M, a Borel subset A of M, and an operator
L:G(B,I)-*Y so that L is weakly precompact, but m(A) is not.
38
PfQflf * Suppose X contains an isomorphic copy of 2 j. Let
Y be the subspace of X isomorphic to Let J:Y -> J be an oo
embedding. As is injective, J may be extended
isometrically to all of X. Continue to call this extension
J. Let H = {0} U {~in € N>. Let £ denote the ^-algebra of
Borel subsets of H. Define J:on (5j) by J(f)(h) * J(f(h))
for h € H. Note that if f = then J(f) = S^A f(Xl).
Thus J maps Ux(£) to Uj (E ) . Furtermore, for f € C(H,X), J(f) 00
€ C(H, l^). Define L : C ( H , X H C q by L - ToJ. The reader may
observe that
t(f) = (J(f(0))n - J(«(i))n).
Note that L = TOJ defines an extension of L to U (S) having A
representing measure n:£-te(X, defined by n(A)(x) =
m(A)(J(x)). As n is weakly regular, n must be the
representing measure for L. Hence, n(0)(x) = m(0)(J(x)) «
J(x). Thus n(0) = J, which is not weakly precompact.
We now turn our attention to establishing some sufficient
conditions for m(A) to be weakly precompact for every Borel
set A. These conditions were motivated somewhat by the proof
in [BiL] of statement (v) in the introductory paragraph of
this chapter. We will begin by making the following
definition.
19- Definition. Let T:C(H ,XHY be a bounded l i n e a r
39
operator. For x € X, define T :C(H)-»Y by
Tx(
f) • T(fx) for f € C(H).
Observe that T x has representing measure m (IR,Y** )=Y**)
defined by n»x(A) = m(A)(x). (In Chapter II, m * was defined * * ^
for y € Y . Although the same notation has been used for
two different meanings, the intent will always be clear from
context.)
20. Theorem. If T :C(H ,X)-*Y is weakly precompact, then
the following statements are equivalent.
(*) I j , s precompac t .
( H ) is weakly precompact for all x in X.
( H i ) weakly compact for all x in I .
(*v) I j . ( B f i f f l j ) 15 dense in T^ ( ^ f f ( t y ) f o r x € / .
'Sr(Z)' (*) f (1C(l)!)! " '«»»« <» (%))•
P£oq£: (iH(ii) Suppose that T (£)is weakly * X
# * precompact. Hence T x .y ^ is weakly precompact for all x €
X. Thus by Theorem 16 T x is weakly precompact for all x€ X.
(ii )-*( iii) This implication follows from Theorem 16.
(iiiH(iv) Suppose that T is weakly compact for all x €
X. Then m is countably additive for all x. Hence W = * x
{m(x,y*) :llY ||-1} l s uniformly countably additive for all x.
Thus W x is uniformly regular. Let A € E. Choose an open set
Gi and a compact set so that K1CaCg1 and m ( x y* )(G 1\K 1) <
40
1 i' ^ o r Y ^ ®y*' i € II. Let be a continuous function
so that = 1 and = 0. Note that
<V*. V l » = * /„*»"»•(*. Y*) « < y \ T " < * a »
uniformly for y* G By*. Therefore T**(B C ( H )) is dense in
Tx (BU(S))*
(iv)-4( v) Suppose that T**(B C ( H )) is dense in T**(B U ( S ))
for all x. Hence n»x takes its values in Y. Thus by [Do2] m
is countably additive for all x. Consequently the set
{m(x,y*) :IIY II-1* i s uniformly countably additive and hence
uniformly regular. Let A 6 S and x € X. We may now proceed
as in (iii)-»(iv) to obtain a sequence (f.) in B_,„. so that * $ $ 4c
<y <Y »TX (JtA)> uniformly for y*€By*. Thus
<Y fT(xf>1)> -4 <y*,T**(xjtA)> uniformly for y* € By*. Since bSx(£)
1 s d e n s e l n bUx(£)' t (bC(H,X)) 1 s d e n s e l n
T (BUx(E))*
* * (v)->(i) Suppose that T (BC(H X ) ) is dense in • * '
T *BU (E)* a n d t h a t T i s w e a kly precompact. Let <f ) be a X ^
bounded sequence from "X(E) • Choose a bounded sequence (p )
from C(H,X) so that ||T**(p1) - T**f 1|| 0. Since T is weakly
precompact we may choose a subsequence f of f. so that * * * *
T ) is weakly Cauchy. Then T (f. ) must be weakly ii
* * Cauchy also. Thus T ^ is weakly precompact.
1 X
It should be noted that a stronger implication than
41
(iii)-+(iv) actually holds. It is actually the case that the
statement In (ill) implies that T x (B C ( H )) is dense in
Tx ' This implication holds for the following very
fundamental reason.
21. Theorem- A bounded linear operator T:I-*Y is weakly
compact if and only if is dense in
Proof: Suppose T is weakly compact. Let (xff) be a net
from Bx which converges to a point x in the w* topology. By
* * * * [VI.4.7,DS,p. 484] T (xtf) converges to T (x) in the weak
s|c 4c $ topology. Thus T (w -closure(Bx)) C w-closure(T**(B )).
* X
Recall that w -closure(Bx) = B **. Thus T**(B **) = «V X
• * *
T (w -closure(Bx) C w-closure(T**(Bx)) =* T**(BX). Thus
T (Bx) is dense in T**(BX**).
For the converse, suppose that T**(BV) is dense in A
T**(BX**). Thus T**(BX**) C T**(Bx) C Y. Hence T is weakly
compact by [DS, VI.4.2, p. 482].
22. Corollary. If T:C(M,X)-*Y is weakly precompact and
any of conditions (i)-(v) in Theorem 26 hold, then m(A) is
weakly precompact for every Borel set A.
Propf• Let T:C(H,X) -» Y be weakly precompact and suppose
that any of conditions (i)-(v) in Theorem 20 hold. Thus
42
* * T |U (£) 1 8 w e a k l Y precompact. Let A 6 S and (x ) be a
I X 1
bounded sequence from X. Note that (tf.x.) is a bounded A JL
sequence from Ux<£). Thus ( T * * ( V l ) ) . (.(AKxp) has a weak
Cauchy subsequence.
A theorem similar to 20 may be demonstrated for GSP
operators. First consider the following lemma.
23. Lemma. If ACJCI and B is a GSP set, then A is a GSP
set.
Pr99t: Let (fi,Etft) be a finite measure space and
b e a b o u n d e d linear operator. Let € > 0. Since B
is a GSP set, we may choose K € E so that /t(ft\K) < e and
{L(X)*K:X€B} is totally bounded. Hence {L(X)*„:X€B} is
compact. But {L(X)*K:X€A> C {L(X)ATK:X€B> . Thus,
<L(X)*k:X€A} is totally bounded. Therefore A is a GSP set.
24. Theorem. If f : 0(H, X)-*Y is a GSP operator, then the
following are equivalent:
(*) f I/? /vi *3 SSP. f j ( s;
( i i ) 7** is GSP for all x € I.
(***) Tx is weakly compact for all x € J.
(iv) fs**(*C(M)) is dense in
'e.(E) (v) i» de.se in t"(Mf - . ) .
43
( v * ) T \y (Y,) *s weakly precompact. I /
£ESSl: (iH(ii) Suppose that T**.^0 j> is QSP. Then • X
|U (S) l s w e a k l y precompact. Thus from Theorem 20 we see < X
* •
that T x is weakly precompact for all x € X. Hence by
Theorem 16, T** is QSP for all x € X.
This implication follows from Theorem 16.
The implications (iii)-f(iv) and (iv)-4(v) follow from
Theorem 20.
(v)-4(i) Suppose that T**(B C ( H X ) ) is dense in
T (BUx(E))* s i n c e T (BC(H,X)) i s a Q S P s e t a n d T**<Bu ( S ) )
^ 3lc sic
^ Lemma 23, T (B^ (£)) *s a set. Thus X
T lu x (S) 1 3 GSP-
Since every GSP operator is weakly precompact, it follows
from Theorem 20 that (vi) is equivalent to (i)-(v) under the
hypothesis that T is GSP.
** T
25. Coronary. If T :C(i ,X)-*Y is a GSP operator, n
represents T, and any of conditions ( i ) - ( v i ) of Theorem 24
hold, then n(A) is a GSP operator for every Bore I set A.
PrffOf: Let T: C (H, X) -4Y be a GSP operator and suppose that
any of conditions (i)-(vi) in Theorem 24 hold. Thus * * * *
T |ux(£) l s a Q S P °P e r a t o r• Let A € S. Now T (By s ) is
X
44
a GSP set as is any of its subsets. Since m(A)(B ) C * * ^
T ^BU ( S ) ^ * s a 6 S P s e t- Thus m(A) is a GSP X
operator.
26, Lemma. If I C cabv(%,X) and £ is weakly precompact,
then {|/t|.-/i € /} is uniformly covntably additive.
Efaof: Suppose K is weakly precompact and {J/»|:/e € K} is
not uniformly countably additive. Choose e > 0, a sequence
(/*j) from K, and a pairwise disjoint sequence (A^) from £ so
that > e for i = l, 2, . . .. Pick partitions
/n >n(i) _ m ^ _ { ij'j=l o f Ai a n d xij € B
x* s o that
n(i) .
for i = l, 2, . . . Define F:cabv(£,X) -4 Zj by F { f t ) =
^Xij^ (Bi j)) j=l, i=l = xi2^^Bi2^' * * ''
xln( 1 )P Bln( 1) ' x2l',(B21) X2n(2)^t(B2n(2))' * '
Note that ||F(/i)|| < ||i||. Thus F is bounded and clearly F is
linear. Thus F(K) must be weakly precompact. Let (x )°° = ij j=l
oo
Kk,ii < *• k2
F(/t1). Let kj = l. There exists k so that 2 |x..| < -j=k^ V 2
Thus ||F(/i1) - F(/ik )|| > |. Pick k so that S |x. I < §. 2 j=kg 2 2
Then ||F(/,1) ~ P(/»k )|| > | and ||F(*k ) - F(/»k )|| > |. This 2 3
process may be continued to obtain a sequence (/j. ) so that i
l l P ( ' k . ) " F ('k } l l - I f o r T h u s >)* has no norm 1 3 Ki 1
45
convergent subsequence. Since weak convergence and norm
convergence in l^ coincide, this sequence has no weak
convergent subsequence. Furthermore, I is weakly
sequentially complete. Hence (F(/» )) has no weak Cauchy i
subsequence. Consequently F(K) is not weakly precompact.
27. Theorem. If T:C(Jf,X)-tY is a bounded linear operator
so that J is weakly precompact, then a is strongly bounded.
E£S2£2l: Suppose T is weakly precompact. Then T ( B * ) = * *
{my*:y €By*} is weakly precompact. Thus {|m *|:y €By*> is
uniformly countably additive by Lemma 26. Hence m is
strongly bounded.
28. Corollary. If T:C(M,I)-*Y is a bounded linear
operator so that T* is weakly precompact, then T** 'X
weakly precompact and-, consequently, n(A) is weakly
precompact for every Borel set A.
h & ) t s
£ £ 0 0 f : Suppose T:C(H,X) -» Y is a bounded linear operator •
so that T is weakly precompact. Thus by the previous
theorem, m is strongly bounded. Thus m is countably
additive. This implies m is countably additive for all x €
X. Hence T x is weakly compact for all x € X. Since T* is
weakly precompact, from Theorem 3 we know that T is weakly
46
precompact. Thus by Theorem 20, T j0 j, is weakly • X
precompact. By Corollary 22, m(A) is weakly precompact for
every A € S.
It should be observed that for T to be weakly precompact
it is not sufficient for T to be weakly precompact. * X
The following example may be given.
29. Example. If T:C(H,X)-»Y is a bounded linear operator # • #
so that T |u (J]) 1 8 weakly precompact, then T need not be I X
weakly precompact.
Let {p} denote a set containing one point. Note that
C({p>,co) = co. Then U j £ ) - C q. Let I:C<{p},co) -4 C q be
the identity map. Observe that I and I**|u are weakly
co precompact, but I is not.
CHAPTER IV
WEAKLY PRECOMPACT SUBSETS OP L^/^X)
In [GS] N. Ghoussoub and P. Saab characterize Banach
spaces X for which weakly compact subsets K of 1^(0,X) are
defined by the following three properties:
(i) K is bounded.
(ii) K is uniformly integrable.
(Hi) for each E € S, the set {/Efd/t:f€K> is relatively
weakly compact.
This characterization may be stated as follows.
1- Tijeofeig. [GS, Corollary 3] A Banach space I and its
dual X have the ladon-Nikodym property if and only if for
every finite measure space any hounded and uniformly
integrable subset I of Lj(n,I) is relatively weakly compact
whenever for every B G E, the set is relatively
weakly compact in /.
In this chapter we will consider weakly precompact
subsets K C ,X) in terms of conditions analogous to (i),
(ii), and (iii). We begin by considering the following
lemma. The arguments for Lemma 2 and Theorem 3 are similar
to those for Lemma 3.2 and Proposition 3.1 in [BrL]. Let
47
48
f:0-4X and g:fl-+x . in the discussion which follows, <f,g>
will be the function mapping 0 to R defined by <f,g>(W) =
<g(w),f(w)> for w € 0.
2. IiSfflia. Suppose that I is a Ban&ch space so that X*
has ike iadon-jfikoiym property with respect to ( U I f
( f j is a sequence from L j ( n , I ) so that (*) < °°>
19
(**) {fB:»€ll} is uniformly integrable,
and
is weakly Cauchy for all A € E,
then ( f n ) is weakly Cauchy in L j f p , ! ) .
P r o o f ; Suppose that X has the Radon-Nikodym property,
(Q,£,/t) is a finite measure space, and (f ) is a sequence
from L1(/j,X) satisfying (i), (ii), and (iii). Let (fn) be a
sequence from K. Since X* has the RNP, L^p.X)* « L (p.X*)
[DS, IV. l.l]. Let g € L^ (/i, X ). Let sn be a sequence of
simple functions so that sn->g a. e. p and ||sn(t)|| < 2||g(t)||
for each n [DS, III.3.8].
Let e > 0. Since {f^} is uniformly integrable, there
exists a S > 0 so that if p (A) <6 for A € E, then JJfJd/j <
e. By Egoroff's Theorem, there exists A € £ so that ft (A) < 6
and sn->g uniformly on 0\A. Let A be one such element of £.
Pick N 6 0 so that for all n > N,
llsn(t) " 9(t)|| < <r
49
for all t€fi\A. Since (fR) satisfies ill, pick I G IN so that
if i and j > I,
< e*
Let i and j > I; then
|/0<f1,g>d/i - S a < t y g > f y \
- Un < fi , s N" g > d''l + + Uo < fj ' s N " 9 > d' (l
* I W ^ r V 3 ^ ! + |/*<fi'8H-B><V| + (
+ IW<frV0>d',l + < cl!filli+ »«IWL + £ + 'H'jB,+ "IWL-
Thus (fn) is weakly Cauchy in L1(^,X).
We may now characterize weakly precompact subsets of
L1(p,X) for Banach spaces X whose duals have the
Radon-Nikodym property.
3. Theoyqm. Let I be a Banach space so that I* has the
laion-N ikoiym property. If I C Lj(n,X) then I is uniformly
iategr&ble if and only if [ is weakly precompact .
Proof: Let (f^) be a bounded sequence from K. Define
</*n) bY
VA> - /Afn*'
Since K is uniformly integrable, it follows that (a ) is r n
uniformly countably additive. Construct a countable
subalgebra x - {^j of E so that ^JfO) - J ,0) for ail i j w4
n € N. Let £ be the *-algebra generated by A. Define
50
I I: Span{/t n:n€NHca ( E \ x ) by D(y) - Clearly II Is an
isometry. Since X has the Radon-Nikodym property, and hence
the weak Radon-Nikodym property, by Theorem III.l and Theorem
III.11 we see that for A € £, {/*n(A) :n€N} is weakly
precompact. Thus (/t ) has a subsequence (/t ) so that nk k
^Hfc^i^k l s w e a k l Y Cauchy for all i € N. Let x* € X*.
Hence lim x*(ft (A)) exists for all A € A. Hence by k-ta> nk *
[DS,IV.8.8], lim x (a (A)) exists for all A G E1 . Therefore k-to k
*^n = ^ A f n l s w e a k l Y Cauchy for all A € S* . Thus by ic k
Lemma 2, (f^) is weakly Cauchy in L 1(p^ l,X). Since II is an
isometry, (fR ) is weakly Cauchy in L (p,X). k 1
For the converse, let K be a weakly precompact subset of
Lj (/*, X). For f € L 1(/t,X), define vf € ca(£,X) by v (A) =
^ ' S e* K' = Since K is weakly precompact, the
same may be said of K1. Applying Lemma III.26, we see that
K1 is uniformly countably additive. The uniform countable
additivity of K1 implies the uniform integrability of K.
The conclusion in Theorem 3 may not be obtained for
Banach spaces X for which X lacks weak Radon-Nikodym
property, for if every weakly precompact subset of L1((i ,X) is
uniformly integrable, then X* must have the weak
Radon-Nikodym property. To see this, suppose that X is a
Banach space so that uniformly integrable subsets of L1(/t,X)
51
are weakly precompact. Note that if (xn) is a bounded
sequence from X, then (*Q*xn) is uniformly integrable in
L1(/t,X). Thus (XQ'X^) has a weak Cauchy subsequence
(*ft'xn }* However, this produces the subsequence (x ) which nk
is weakly Cauchy in X. Hence we conclude that X* possesses
the weak Radon-Nikodym property. Thus if X* does not have
the weak Radon-Nikodym property, then L± <t ,X) must have a
subset which is uniformly integrable and not weakly
precompact. In fact, by extending Example IV.2.2 in [DU] we
may make a stronger statement about those spaces X for which *
X does not have the weak Radon-Nikodym property.
4. Theorem. Suppose that X* does not have the weak
iadon-Kikodym Property. There exists a finite measure space
and a set I C £j(p,X) so that
( t ) i is bounded,
( i i ) I is uniformly integrable,
( H i ) { f £ f d p : f € [ } is relatively compact for all B € E,
and
(iv) I is not weakly precompact.
P r Q o f : Suppose X is a Banach space so that X* does not
have the weak Radon-Nikodym Property. Thus by Theorem
III.11, X contains a copy of I v Let (xn) be a copy of the
l± basis in X. Without loss of generality, assume ||xn|| < l
for all n € N. Since (xn) is a copy of the l% basis, we may
52
choose ^ € R so that
n n
* '£1-11-for every n € II and choice of scalars (a } n
i i=l*
Let 0 = [0,1], E = #([0,1]), and p, = Lebesgue measure.
Use r n to denote the nth Rademacher function on [0,1]; that
i s' r n ^ = s9n(sin(2n*t)). (Take sgn(0) = l.) For n € W,
define fn:[0,l]4X by f R - r ^ . Let K « <fn:n€N).
Observe that ||fj = J|fn|<^ < 1. In addition, lim llZ-fd^l
n-ta» '
0 for all E € S. Thus the set K = {fn:n€N) satisfies
conditions (i) and (Hi). Furthermore, as |j^n(^)J| ^ 1 for
all n € N and t € [0,1], K is uniformly integrable. Let
C IR. Observe that
> f/l £ |a1r1|d/» • ^ S |aA i=l i=l
Thus (fR) is a copy of the basis. Therefore K is not
weakly precompact.
Theorem 3 and Theorem 4 leave several questions
unanswered. First, may Banach spaces whose duals have the
Radon-Nikodym property be characterized as those Banach
spaces X for which weakly precompact subsets of L 1(/<,X) are
exactly those subsets of L^(^i,X) which are uniformly
integrable? If not, how may one characterize such spaces?
In addition, the question as of whether or not Banach spaces
53
having the weak Radon-Nikodym property may be characterized
as those spaces X for which subsets K of LjL(/(,X) satisfying
<i), (ii) and (iii) of Theorem 4 are weakly precompact
remains.
In [GS], N. Ghoussoub and P. Saab showed by construction *
that if X fails to have the Radon-Nikodym Property, then
there exists a finite measure space (fl,E,ft) and a subset K of
L1(/t,X) so that K is bounded, uniformly integrable, and
{ J E f d / t : f € K } is weakly compact for all E € E, but K is not
weakly compact. The question as of whether or not their set
K is weakly precompact arises. Suppose that X* has the
Radon-Nikodym Property. The following is a sketch of the
construction in [GS]. Let A denote the Cantor group with
Haar measure /, and let {A^ . s l<i<2n> denote the standard nth
partition of A. Set A = A. Thus A . = A U °'1 n,i n+1,2i-l
An+l,2i 1 8 c l°P e n' and p(An ..) = 2 n. We use x n d to denote
*a . The identity operator i:x-« factors the Haar operator
H. I ) which takes the basis of I into the usual Haar
basis of C(A) as a subspace of LJp). That is, there exist
bounded linear operators and V:X-»Lm(ft) so that H =
VOIOU. If {enl:n>0,l<i<2n} is an enumeration of the usual I
1 basis H may be defined as follows:
H ( e . ) = y — y — K ni' *n+l,2i-l *n+l,2i ~ hni
Set
f n ( t ) a n hji ( t ) eji' t € j=l i=l
54
In [QS] the sequence (fn) was shown to be bounded and
uniformly integrable. Furthermore, it was shown that
{/Afnd/»:n € IN) is weakly compact for all A € E.
For an operator T:X->Y, define T:!^ (/»,X)-+L1 (ft, Y) by
<T(f)(t)) = T(f(t)). Then
H(f )(t) - i S S L J t l h , , , t € A. j=l i=l 3 1 3 1
Note that for t € C{A), H(fn)(t) C c(A). Ghoussoub and Saab
show that {I(fn):n€ll} is not weakly compact by demonstrating
that {H(f n):n€ll} is not weakly compact. Unfortunately, the
same technique may not be used to determine whether or not
{fn:n € N} is weakly precompact, as <H(fn):n € N} is weakly
precompact. To see this recall that a bounded sequence (xn)
in a Banach space is equivalent to the basis if and only
if there exists a real number $ > 0 so that for any fc c 1' 2'
. . .,cn> C R
m m
'n=l|Cnl " l^Vnll Let {c1, . . c } C R. Let t € A. Note that t 6 A D
0,1
l.Pj"5 A2,p 3 * ' * f o r s o m e sequence of positive integers A, ^ 3 . 2
(Pk). Thus
Im f n 2^
s c [i S E h (t)h,,(«)] I wta n=l j=i i=i J1 31 1
55
I m j- n 2
2 ^ 1 C " > 1 i»i i X3 + 1. 21-1' *' -*J+1.21'l) 1
I m r n = sup S c I - S w€A n=l n S = l
(X. _ i (t) -AT. j = 1 j+l,2p.-1*~' *j+l,2p. (t))
Let q be the smallest non-negative integer so that w € A
and w £ A q + , then
q + 1
q»p ' q
(X j+l , 2 pi-i ( t )" J f j + 1' 2 pi ( t ) )' Uj+i,2p. ^ " ^ j + i ^ p . ( w ) )
1 if j+l < q+i
-1 if j+l = q+l
0 if j+l > q+i
Thus
S c S(f )(t) n=l " n
k m S c
n + (k-1) E c3>? n=l i=k+l l l
for some k G {0, 1, . . ., m). Let (fn ) be a subsequence of
(fR)- Set c - (-l)s and for all n £ n for s s some s set c =
n
0. Thus
£ c i?(f ) s=l ns V * / I L V » 8 ( f n 'H* - 2-
S=1 s s
56
S. However 2LJ|g | = r. Hence (H(fn )) is not equivalent to the
s-i s s Jj basis. Thus (H(fn)) is weakly precompact.
An essential part of the proof of Theorem 1 is the
following characterization of Banach spaces X whose duals
have the Radon-Nikodym property.
5. Theorem. TDU, IV.1.1] let be a finite measure
space, 1 < p < oo, and I a Banach space. Then I (pfX) -
£ (H,I ), where p * q = 1, if and only if / has the
iadon-ff ikodym property with respect to p.
A similar characterization may be made for Banach spaces *
X for which X has the weak Radon-Nikodym property.
6. Theorem, let I be a Banach space and a
finite measure space. The space I* has the weak
iadon-Jf ikodym property with respect to p if and only if for
each I € l^(p,X) , 1 < p < oo, there exists a Pettis
integrable function so that
L(f) * for f € l^(/t,I),
where <f,g>(m) = (9 (<*))( f (<»)) for w G ft .
Proof: Let X be a Banach space and (ft ,£,/<) a finite
* measure space. Suppose that X has the weak Radon-Nikodym
57
property with respect to p. Let L € L Define G on £ P
by
G(E)(x) = L(*eX) for E € S.
Thus
|G(E)(X)| - |L U EX,| < ||L||.||V||p - ||L||-||,e||p-||x||,
• and consequently G:E-»X .
To see that G is ft continuous and of bounded variation,
let E€S, {Ej, . . E n) a partition of E, and {xJf . .
x } C B . Thus n j\
n n I S G(E.){x.)| < IL( £ G(E,)(x.)J i-1 1 1 1 ' i=l 1 1 1
* IIHI• IIj/E^illp - W'/nlF^/if'5
*
Since X has the weak Radon-Nikodym property with respect to
there exist a Pettis integrable function g-.TrtX* so
that G(E) =* P - J gd/» for all E € E. If f is a simple
function, then L(f) - Jft<f,g>d/i. Let f € L (/i,X) and let tr (f ) be a sequence of simple functions so that f -+ f a. e. 11 n
on 0 and so that |fnU)| < 2|f(«)| for all u € 0. Note that
fft<fn'g> = L ( fn ) L ( f ) * n
Define fR(A) = JA<fn,g>d/t. Observe that
*n(A) = L(^Afn) -f L(f) .
58
In addition fn«/» for all n. Thus by [DS,p.l58],
lim ^„(E) = 0 /J( E)->0
uniformly in n. Let c > 0, and choose S > 0 so that if p(A)
< S, then fn(A) < e for all n € N. Choose N so that
M<t:||f(t)||>N) < s . Let B = < t: ||f (t) ||<N>. Hence
/fl|<fn.9>|^ » /B'=l<fn-9>ld' + /Bl
<fn'9>ld''
< £ + ' BI< S*A <
xni' 9 >l d' ni
< € + sup{JB £ <^Exi,g>:||x1||<2N, 11(B) partitions B} EGII ( B )
< t + suP^ ||/g<^^'(B^Afgxjd/'||: H ^ i l l — H ( B > >
• e + 2N|6|(B).
Since f n -» f a. e. , <fn>9> <f,g> a.e. . Certainly <fn»g>
is measurable for all n as f is measurable for all n.
Furthermore | | < 2|<f,g>|. Hence by Fatou's lemma,
/ | <f»g> | d/t < JJji J| <fn,g>|d/t.
Now applying the Lebesgue convergence theorem we obtain
lim/<f ,g>d/t = J<f,g>d/t. n-to
Therefore L(f) = J < f , g > d / t .
For the converse, suppose that for every L € L (p,X) P
there exists a Pettis integrable function g:£->X* so that L(f)
Let G:£-« be a /» continuous vector measure of bounded
variation. Let Eq € E so that ^(E ) > 0. It will be shown
that G has a Pettis integrable weak Radon-Nikodym derivative
on some B€S, bCEq, with /t(B)>0. A version of the Exhaustion
lemma will complete the proof [DU,III.2.5]. Choose k € IN so
59
that JG| (EQ) < K/T(EQ). Apply the Hahn decomposition theorem
to Kp - |G| to obtain BGE so that BCEq, /t(B) > 0, and |G|(E)
< k/j (E) for E € E, E C B. For f = x,, where x. € X, E. G i
E, and E^^Ej = t for i£j, define
L(f) - EG(EinB)(x1) .
Thus
XG(E FLB)
(i (E.flB)x. /MEJlB)
1
< Ek||^{(EinB)(xi)|| < ^jf^ < k/»(fl)q||f||p.
Thus since L is linear on simple functions, L defines a
continuous linear functional on the simple functions of
Lp(/t,X). Hence L has an extension to all of L (£,X). By
hypothesis there exists a Pettis integrable function G so
that L(f) » J<f,g>dm for all f € L (/t,X). Now
G(EflB) (x) = " JEnB<x,g-d/»>
for all x € X. Therefore G(EOB) = P - /EpB9<3/».
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