Approximation of Vector Valued Functions

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APPROXIMATION OF VECTOR VALUED FUNCTIONS

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NORTH-HOLLAND MATHEMATICS STUDIES

25

Notas de Matemhtica (61) Editor: Leopoldo Nachbin Universidade Federal do Rio de Janeiro and University of Rochester

Approximation of Vector Valued Functions JOaO B. PROLLA IMECC, Universidade Estadual de Campinas, Brazil

1977

NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM

NEW YORK

OXFORD

@ North-Holland Publishing Company - 1977 AN rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

North-Holland ISBN: 0 444 85030 9

PUBLISHERS :

NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM NEW YORK OXFORD SOLE DISTRIBUTORS €OR THE U.S.A. AND CANADA:

ELSEVIER NORTH-HOLLAND, INC. 52 VANDERBILT AVENUE, NEW YORK, N.Y. 10017 Library of Congress Cataloging in Publication Data

P r o l l a , Joao B Approximation of v e c t o r valued f u n c t i o n s . (Notas de matemitica ; 61) (North-Holland mathematics s t u d i e s ; 2 5 ) Bibliography: p . I n c l u d e s indexes. 1. Vector valued f u n c t i o n s . 2 . Approximation theor I. T i t l e . 11. S e r i e s . Q ~ l . N i 6 no. 6 1 [QA3201 5101.8s [ 515 . 7 l 77-22095 I S B N 0-444- 85 030-9

.

PRINTED IN THE NETHERLANDS

PREFACE

T h i s w o r k d e a l s w i t h t h e many v a r i . a t i o n s o f t h e Stonei l e i e r s t r a s s T h e o r e m f o r v e c t o r - v a l u e d f u n c t i o n s a n d some of i t s a;)?lications.

?or a more d e t a i l e d d e s c r i p t i o n o f

its

contents

s z e t h e I a t r o d u c t i o n a n 5 t h e Tab1.e of C o n t e n t s . The book is 1a:;re-

ly se1i'- c o n t a i n e d . T h e a m o u n t o f F u n c t i o n a l I n a l v s i s

required

i s m i n i m a l , e x c e 3 t f o r C h a p t e r 8 . B u t t h e r e s u l t s oE t h i s Chapter a r e n o t u s e d e l ~ e ~ ~ h e 'The r e . b o o k c a n be u s e d by r j r a d u a t e skudents who

Iia-Je t a k e n t h e u s u a l f i r s t - y e a r r e a l a n d corrplex a n a l v s i s

courses. T h c t r e a t m e n t o f t h e s u b j e c t h a s n o t a n D e a r e d i n boo!< f o r n p r e v i o u s l y . Z v e n t h e p r o o f of t h e S t o n e - W e i e r s t r a s s

ren i s new, a n d d u e t o S . ' l a c h a d o .

Theo-

7Je also g i v e r e s u l t s i n non-

a r c h i m e d e a n a p p r o x i m a t i o n t h e o r y t h a t a r e new a n d Dieudonne - KaTlanskv Theorem t o nonarchim.edean

extend

the

v e c t o r - valued

f u n c t i o n spaces. I t h a n k P r o f e s s o r S i l v i o :lachado,

cie F e d e r a l do 9 i o de J a n e i r o ,

from t h e Universich-

f o r h i s v a l u a b l e c o m m e n t s a n d re-

n a r k s o n t h e s u h j e c t . I i l i t h o u t h i s h e 1 3 t h i s v o u l d be a d i f f e r e n t a n d ?oorer b o o k .

I t h a n k also P r o - F e s s o r L z o ~ o l d o ! J a c l i b i n ,

t h e U n j v e r s i d a d e F e d e r a l do 710 d e J a n e i r o a n d

the

from

Vniversity

vi

of R o c h e s t e r , whose a d v i c e a n d e n c o u r a g e m e n t w a s n e v e r f a i l i n g . Finally,

I w i s h t o t h a n k A n g e l i c a h4arquez

M o r t a r i for t y p i n g t h i s m o n o g r a p h .

J O E 0 B.

PROLLA

Campinas, A p r i l 1 9 7 7

and

Elda

CONTENTS PREFACE

................................................... ............................................... 1 . THE COMPACT-OPEN TOPOLOGY

INTRODUCTION

CHAPTER

v ix

1

............................ 1 L o c a l i z a b i l i t y ............................... 3 § P r e l i m i n a r y lemmas ........................... 4 § S t o n e - W e i e r s t r a s s Theorem f o r modules ........ 7 § § 5 . The complex s e l f - a d j o i n t case ................ 1 0 13 § 6 . Submodules o f C ( X ; E ) ......................... 5 7 . An example: a theorem o f Rudin ............... 1 7 19 § 8 . B i s h o p ' s Theorem ............................. 25 § 9 . V e c t o r f i b r a t i o n s ............................ 21 5 10 . Extreme f u n c t i o n a l s .......................... 5 11 . R e p r e s e n t a t i o n o f v e c t o r f i b r a t i o n s .......... 35 p 12 . The a p p r o x i n a t i o n p r o p e r t y .................. 40 Appendix . Non-locally convex s p a c e s ................. 4 3 CHAPTER 2 . THE THEOREM OF DIEUDONNE ....................... 46 CHAPTER 3 . EXTENSION THEOREMS ............................. 52 CHAPTER 4 . POLYNOMIAL ALGEBRAS ............................ 57 §

. 2. 3. 4. 1

B a s i c definitions

. B a s i c d e f i n i t i o n s and lemmas ................. 57 2 . S t o n e - W e i e r s t r a s s s u b s p a c e s .................. 6 7 5 3 . C(x)-modules ................................. 72 5 4 . Approximation o f compact o p e r a t o r s ........... 7 4 CHAPTER 5 . WEIGHTED APPROXIMATION ......................... 79 5 1. D e f i n i t i o n o f Nachbin s p a c e s ................. 79 5 2 . The Bernstein-Nachbin a p p r o x i m a t i o n problem .. 8 0 3 3 . S u f f i c i e n t c o n d i t i o n s f o r s h a r p l o c a l i z a b i l i t y 88 5 4 . Completeness o f Nachbin s p a c e s ............... 9 0 5 5 . Dual s p a c e s o f Nachbin s p a c e s ................ 96 Appendix . Fundamental w e i g h t s ....................... 107 5

1

vii

viii

CONTENTS

. 7. 8.

CHAPTER 6

THE SPACE C o ( X ; E )

CHAPTER

THE SPACE C b ( X ; E )

CHAPTER

5

THE c-PRODUCT

.

1

. 5 3. 2

5 5 CHAPTER 9

§

4

.

S

.

.

1. 2

.... L . SCHWARTZ ..................

113 127 138

......................... S p a c e s of c o n t i n u o u s f u n c t i o n s .............. The a p p r o x i m a t i o n p r o p e r t y .................. M e r g e l y a n ' s Theorem ......................... L o c a l i z a t i o n o f t h e a p p r o x i m a t i o n p r o p e r t y ..

144

...........

153

General d e f i n i t i o n s

Valued f i e l d s

............................... .........................

K a p l a n s k y ' s Theorem

............................... f u n c t i o n s ..................... V e c t o r f i b r a t i o n s ........................... Some a p p l i c a t i o n s ........................... B i s h o p ' s Theorem ............................ T i e t z e E x t e n s i o n Theorem .................... The compact-open t o p o l o g y ................... The n o n a r c h i m e d e a n s t r i c t t o p o l o g y ..........

138

141 146 149

153 156

Normcd s p a c e s

162

Vector-valued

163

.............................................. SYMBOL I N D E X .............................................. I N D E X ..................................................... BIBLIOGRAPHY

...

w i t h t h e s t r i c t topology

NONARCHIMEDEAN APPROXIMATION THEORY

. § 3. § 4. § 5. § 6. § 7. § 8. § 9. 9 10 .

§

OF

w i t h t h e uniform topology

171 181 187 189 193 198 206

213 215

I N T RODUC T I ON

The t y p i c a l p r o b l e m c o n s i d e r e d i n t h i s book

is

the

f o l l o w i n g . One i s g i v e n a v e c t o r s u b s p a c e W o f a l o c a l l y convex space L of continuous vector-valued

f u n c t i o n s , w h i c h i s a modu-

l e o v e r an a l g e b r a A of continuous s c a l a r - v a l u e d f u n c t i o n s ,and t h e problem i s t o d e s c r i b e t h e c l o s u r e o f W i n t h e space L. I n c h a p t e r 1 w e s t a r t w i t h t h e c a s e i n whichL=C(X;E) w i t h t h e compact-open t o p o l o a y . Vhen t h e a l g e b r a A i s

self-ad-

j o i n t , t h e s o l u t i o n o f t h e above p r o b l e m i s g i v e n by t h e Stone-1Veierstrass theorem f o r modules. A very e l e g a n t and p r o o f d u e t o S . Machado ( s e e [ 3 8 ! )

i s p r e s e n t c . : h e r e . A s a co-

r o l l a r y one g e t s t h e c l a s s i c a l S t o n e - W e i e r s t r a s s self-adjoint self-adjoint,

direct

theorem

s u b a l g e b r a s of C ( X ; C ) . When t h e a l g e b r a A a s o l u t i o n o f t h e p r o b l e m i s g i v e n by

for

is not Bishop's

t h e o r e m . The p r o o f t h a t w e i n c l u d e h e r e i s a q a i n due t o S . Mac h a d o (see

[ 3 7 1 ) . The main i d e a

i s t o use a "strong"

Stone-

theorem f o r t h e r e a l c a s e p l u s a t r a n s f i n i t e a r -

-!!eierstrass

gument. T h i s i s done i n Machado's p a p e r v i a Z o r n ' s Lemma. Here

w e u s e t h e t r a n s f i n i t e i n d u c t i o n p r o c e s s found i n t h e o r i g i n a l p a p e r o f B i s h o p (see [ 8 ] ) . W e p r e f e r t h i s new method o v e r

de

B r a n g e ' s t e c h n i q u e , b e c a u s e i t can be a p p l i e d t o o t h e r s i t u a t i o n s i n weighted approximation theory

,

namely w h e r e

t h e o r e t i c t o o l s a r e e i t h e r p a i n f u l t o apply o r not a t all. In

5

available

9 o f t h i s C h a p t e r w e t r e a t a s p e c i a l c a s e o f vec-

t o r f i b r a t i o n s , and p r o v e i n t h i s c o n t e x t a " s t r o n g " -Weierstrass

measure

t h e o r e m d u e t o Cunningham a n d Roy (see

Stone-

[ 153 ) . This

r e s u l t i s u s e d i n t h e n e x t s e c t i o n t o c h a r a c t e r i z e extrgoe fun*

ix

I N T R O DUCT I 0N

X

t i o n a l s . A s corollaries, w e g e t t h e Arens-Kelley theorem scalar-valued

for

f u n c t i o n s , and S i n g e r ' s t h e o r e m ( v e c t o r - v a l u e d

c a s e ) . The r e s u l t s o f Buck [12]

and S t r o b e l e [63]

are a l s o ob-

t a i n e d . I n a n a p p e n d i x w e t r e a t t h e non l o c a l l y convex case. C h a p t e r 2 d e a l s w i t h v e c t o r - v a l u e d v e r s i o n s o f Dieud o n n g ' s t h e o r e m on t h e a p p r o x i m a t i o n o f f u n c t i o n s o f t w o v a r i a b l e s by means of f i n i t e sums o f p r o d u c t s o f f u n c t i o n s o f v a r i a b l e (see

[ 181 )

one

.

Chapter 3 i s devoted t o T i e t z e type extension

r e m s f o r vector-valued

theo-

f u n c t i o n s d e f i n e d on compact s u b s e t s o f

a completely r e g u l a r Hausdorff s p a c e. A t y p i c a l

result

says

t h a t , i f Y C X i s a compact s u b s e t o f a c o m p l e t e l y r e g u l a r s p a c e X , a n d E i s a F r g c h e t s p a c e , t h e n C b ( X ; E ) IY = C ( Y ; E ) . The s u b j e c t m a t t e r o f c h a p t e r 4 i s t h e n o t i o n o f pol y n o m i a l a l g e b r a s . T h i s n o t i o n was i n t r o d u c e d

[ 471 , and t h e name

in

PeaczGnski

i s d u e t o W u l b e r t ( c f . P r e n t e r 14911 1. I n h i s

d e f i n i t i o n PeXczyfisky u s e d m u l t i l i n e a r mappings, w h e r e a s Wulbert used polynomials. A t h i r d e q u i v a l e n t d e f i n i t i o n given i n B l a t t e r

[ 41

. We

is

p r e s e n t h e r e S t o n e - W e i e r s t r a s s theorem

f o r polynomial a l g e b r a s . A s a c o r o l l a r y w e g e t t h e i n f i n i t e dimensional version of t h e Weierstrass polynomial approximation t h e o r e m . Pe?czyfiski a t t r i b u t e s t h i s r e s u l t t o S . Mazur

(un-

p u b l i s h e d ) i n t h e case of Banach s p a c e s . A much s t r e n g t h e n e d form o f M a z u r ' s r e s u l t w a s p r o v e d i n t h e j o i n t p a p e r Machado, P r o l l a

Nachbin,

[ 461 , namely t h a t t h e p o l y n o m i a l s o f f i n i t e t y -

p e from a r e a l l o c a l l y convex s p a c e i n t o a n o t h e r are d e n s e

in

t h e s p a c e o f a l l c o n t i n u o u s f u n c t i o n w i t h t h e compact-open top o l o g y . P r e n t e r [ 481 e s t a b l i s h e d M a z u r ' s r e s u l t f o r s e p a r a b l e

I NT R O D U C T I 0 N

xi

H i l b e r t s p a c e s . I n t h i s c h a p t e r w e a l s o p r o v e B i s h o p ' s theorem f o r p o l y n o m i a l a l g e b r a s u s i n g t h e d e f i n i t i o n g i v e n by P e a c z y f i s k i . I t r e m a i n s a n open p r o b l e m f o r t h e m o r e g e n e r a l polynomial algebras. Chapter 4 ends with a study of t h e

approxi-

m a t i o n o f compact l i n e a r o p e r a t o r s by p o l y n o m i a l s o f f i n i t e t y pe * I n C h a p t e r 5 w e are c o n c e r n e d w i t h w e i g h t e d mation o f v e c t o r - v a l u e d f u n c t i o n s , i .e.

,

with the

approxi-

Bernstein-

Nachbin a p p r o x i m a t i o n problem. W e e x t e n d t h e f u n d a m e n t a l

of Nachbin (see f o r example [ 4 3 ] )

work

from t h e r e a l o r s e l f - a d j o i n t

complex c a s e t o t h e g e n e r a l complex case, i n t h e same way t h a t B i s h o p ' s t h e o r e m g e n e r a l i z e s t h e S t o n e - W e i e r s t r a s s theorem. I n t h e j o i n t p a p e r w i t h S . Machado

[ 401 , w e a c c o m p l i s h e d t h i s f o r

v e c t o r f i b r a t i o n s . Here, however, w e r e s t r i c t o u r s e l v e s t o t h e p a r t i c u l a r case of v e c t o r - v a l u e d

to

f u n c t i o n s . As a corollary

o u r s o l u t i o n o f t h e B e r n s t e i n - N a c h b i n a p p r o x i m a t i o n problem w e g e t a s t r e n g t h e n e d v e r s i o n o f K l e i n s t u c k ' s s o l u t i o n of t h e bounded case (see

[ 351 ) o f B e r n s t e i n - N a c h b i n p r o b l e m , as w e l l

as o f B i s h o p ' s t h e o r e m f o r w e i g h t e d s p a c e s p r o v e d by P r o l l a [51].

The r e s u l t of Summers [64]

f o r scalar-valued functions

is likewise generalized. I n t h e f i n a l t w o paragraphs

of Chapter 5 we

study

t h e problem o f c o m p l e t e n e s s of Nachbin s p a c e s and t h e c h a r a c t e r i z a t i o n o f t h e d u a l s p a c e o f a Nachbin s p a c e . I n an a p p e n d i x t o C h a p t e r 5 , w e p r e s e n t a v e r y s i m p l e p r o o f , due t o G.

Z a p a t a (see [68])

,

o f M e r g e l y a n ' s theorem

c h a r a c t e r i z i n g f u n d a m e n t a l w e i g h t s on t h e r e a l l i n e . T h i s s u l t w a s t h e n u s e d by Z a p a t a t o show t h a t Hadamard's

re-

problem

INTRODUCTION

xii

on t h e c h a r a c t e r i z a t i o n of q u a s i - a n a l y t i c classes o f f u n c t i o n s

i s e q u i v a l e n t t o B e r n s t e i n ' s problem on t h e c h a r a c t e r i z a t i o n of f u n d a m e n t a l w e i g h t s . The r e s u l t o f C h a p t e r 5 are a p p l i e d i n C h a p t e r 6 Co(X;E),

to

t h e s p a c e of a l l c o n t i n u o u s f u n c t i o n s t h a t are E-val-

ued and v a n i s h a t i n f i n i t y on a l o c a l l y compact s p a c e X I e q u i p ped w i t h t h e u n i f o r m c o n v e r g e n c e t o p o l o g y . W e a l s o p r e s e n t here B r o s o w s k i , D e u t s c h and Morris t h e o r e m (see [ 101 )

on

f u n c t i o n a l s of t h e u n i t b a l l of t h e d u a l of Co(X;E),

extreme generaliz-

ing it t o vector fibrations. Analogously, i n Chapter 7 w e apply t h e

results

of

C h a p t e r 5 t o t h e s p a c e Cb(X;E) o f a l l bounded c o n t i n u o u s funct i o n s , e q u i p p e d w i t h t h e s t r i c t t o p o l o g y o f Buck. W e g e t

both

Stone-Weierstrass and B i s h o p 's theorem f o r t h i s topology.

We

a l s o c h a r a c t e r i z e e x t r e m e f u n c t i o n a l s of p o l a r s e t o f n e i g h b o r hoods of t h e o r i g i n of C b ( X ; E ) . The eighth C h a p t e r d e a l s w i t h t h e € - p r o d u c t of L . S c h w a r t z and t h e a p p r o x i m a t i o n p r o p e r t y f o r c e r t a i n s p a c e s

of

f u n c t i o n s , e.9. Aron a n d S c h o t t e n l o h e r 1 3 1 r e s u l t on t h e e q u i v a l e n c e b e t w e e n t h e a p p r o x i m a t i o n p r o p e r t y f o r a complex B a n a d s p a c e E and t h e s a m e p r o p e r t y f o r t h e s p a c e of h o l o m o r p h i c mapp i n g s on E w i t h t h e compact-open t o p o l o g y . A l s o , t h e p r o o f d u e t o K.-D.

B i e r s t e d t 151 of t h e vector-valued version of

g e l y a n ' s t h e o r e m on a p p r o x i m a t i o n i n t h e complex p l a n e i s

Mer-

to

b e f o u n d i n t h i s C h a p t e r . I t e n d s w i t h some r e s u l t s o f B i e r s t e d t [ 6 ] on t h e " l o c a l i z a t i o n " o f t h e a p p r o x i m a t i o n p r o p e r t y v i a ma-

ximal anti-symmetric sets.

INTRODUCTION

xiii

Chapter 9 deals with nonarchimedean approximation Theory. The first results in this areawere proved by J. Dieudonng. He proved in [7O]

, for functions with values in the field of

p - adic numbers, the analogues of Weierstrass polynomial approximation theorem, and of Stone- Weierstrass Theorem on densityof separating subalgebras. To ?rove these Theorems he first established the analogues of Tietze's Extension Theorem and his Theorem on appoximation of functions on Cartesian products.

own In

1949, Kaplansky generalized Dieudonng's Stone- Weierstrass Theorem to the case of functions with values in any field with (rank one) valuation. (See Kaplansky [72

).

a

The case of arbi -

trary Krull valuations (or of archimedean valuations other than the usual absolute value of a ) was established Rasala and Waterhouse in

by

Chernoff,

[69].

We here treat only the case of rank one, i.e. real valued nonarchimedean valuations. We extend the Dieudonn6 -Kaplansky Theorem to vector valued functions, more precisely to functions with values in a nonarchimedean normed space over field

(F,

I

-

1).

some

valued

Our treatment cover the case of A-modules,wfiere

A is an algebra of F-valued functions, and in the case

E = F

extends Kaplansky's result in the sense that we compute the distance of a function from a module. As a corollary one gets description of the closure of a module and the density We also present Murphy's treatment of vector fibrations slightly modified version (see [74 ] )

. Results on

the

result. in

a

ideals are also

given, extending a result of I. Kaplansky on ideals of function algebras (see I. Kaplansky, TopoZogicaZ A Z g e b r a , Notas de Matemstica NP 16 (Ed. L. Nachbin) , Rio de Janeiro.)

C H A P T E R

1

THE COMPACT-OPEN TOPOLOGY

9 1

BASIC DEFINITIONS

Throughout this monograph X denotes a non-void Hausdorff space, and E denotes a non-zero locally convex space is over the field M (M= IR or C) The topoloqical dual of E denoted by E', and the set of all continuous seminorms on E is denoted by cs(E) The vector space over IK of all continuous functions taking X into E is denoted by C(X;E). For every non-void compact subset K C X and every continuous seminorm p € cs(E),

.

.

f + l defines a seminorm on C(X;E). The topology defined by all seminorms is called the c o m p a c t - o p e n t o p o l o g y . When E is a normed space, and t + I(tlI is norm, we write

such

for the corresponding seminorm on C(X;E). In particular, E = M , we write

when

its

and, if no confusion may arise, C(X) = C(X;M). The vector subspace of all f € C(X;E) such thatf(X) is a b o u n d e d subset of E, is denoted by Cb(X;E) and topoloqized by considering the family of all seminorms f

where p

€

-+

cs(E)

Ilfllp

=

sup {p(f(x));

. This topology

x

E

XI,

is referred to as the

topozogy

2

COMPACT

-

OPEN TOPOLOHY

of u n i f o r m c o n v e r g e n c e on X , or as the u n i f o r m t o p o l o g y .

When X is c o m p a c t , the two spaces C(X;E)and C b ( X ; E ) coincide, and the compact-open and the uniform topoloay are the same. I It1 I is itsnonn, When E is a normed space, and t we write +

for the correspondinq norm on C b ( X ; E ) . If E = K,and no confusion may arise, we write C b ( X ) = C b ( X ; M ) . Given a non-empty subset S c C ( X ; E ) , we define an equivalence relation on X , by settina, for all x, y E X , x 5 y (mod. S ) if, and only if, f(x) = f(y) for all f E S . Since the elements of S are continuous functions, the eauivalence classes (mod. s) of X are closed subsets. The set S c C ( X ; E ) is said to be s e p a r a t i n g o n X if the eauivalence classes (mod. S) of X are sets reduced to Doints. This is eauivalent to say that, for any such that pair x, y E X of distinct points, there is f E S f(x) # f(y). If S is separatina on X, we also say that S s e p a r a t e s t h e p o i n t s of X .

If K C X is a c Z o s e d non-empty subset, andScC(X;E), then SIK denotes the subset of C ( K ; E ) consistina of all gEC(K;E) such that there exists f E S with the property that q(x)= f(x), for all x E K . In particular, if K C X is compact and E = M, then C (K) = Cb (X) 1 K, bv the Tietze Extension Theorem, when X is completely reqular. It follows easily from the above definitions that for any closed subset K C X , if x,y E K then x E y (mod. S) if equivalence and only if x :y (mod. S I K ) . Moreover, aiven any class Y C K (mod. SIK) there is a u n i q u e equivalence class Z C X (mod. S ) such that Y = Z rl K . Suppose that E is a H a u s d o r f f space,and S C C ( X ; E ) Let A = {t$ o f; t$ E E', f E S } . Then for every x,y E X , x :y (mod. S ) if, and only if, x : y (mod. A ) . In fact, the "onlyif" part is true even when E is not Hausdorff.

.

COMPACT

5

2

-

3

OPEN TOPOLOGY

LOCALIZABILITY

L e t A b e a s u b a l q e b r a o f C ( X ; X). A vector W C C(X;E)

subspace

w i l l b e c a l l e d a module o v e r A , o r a n A-module, + a ( x ) f ( x ) belonqs t o W, €or e v e r y a E A

if

the function x f E

and

w. Notice t h a t , i f B d e n o t e s t h e s u b a l r r e b r a o f C ( X ; M )

g e n e r a t e d by A and t h e c o n s t a n t f u n c t i o n s , t h e n W i s a n

A-mo-

d u l e i f , a n d o n l y i f , W i s a B-module.

Moreover, t h e esuival e n c e r e l a t i o n x z !I (mod. A ) i s t h e same as x z v (mod. B).

DEFINITION 1.1

Let W

C(X;E)

b e an A - m o d u l e . We s a y t h a t

W

i s l o c a l i z a b l e u n d e r A i n C ( X ; E ) if t h e c o m p a c t - o n e n c l o s u r e of W i n C ( X ; E ) is t h e s e t of a l l f E C ( X ; E ) s u c h t h a t f l y b e l o n g s t o t h e c o m p a c t - o p e n c l o s u r e of WIY i n C ( Y ; E ) for e a c h equivalence c l a s s Y

C X

(mod. A ) .

T h i s i s e c r u i v a l e n t t o s a y t h a t t h e compact-open clos u r e of ?%7 i n C ( X ; E ) i s t h e s e t of a l l f E C ( X ; E )

such that,aivPn

Y

C X an equivalence

E

> 0; and p E c s ( E ) , t h e r e i s u E W such t h a t p ( f ( x ) - u ( x ) )
1 - 6. W e now u s e ( 3 ) and o b t a i n

I n view of 0
0,there is

a c o n t i n u o u s l i n e a r mappinq u of f i n i t e rank from E t o E , an e l e m e n t u E E ' 8 E , such t h a t p ( x

5

8

-

u(x))
e x i s t s an o r d i n a l p E

G

T.

B i s h o p ' s arrrument t h a t

such t h a t each element S E P

P

berethere

is anti-

s-metric f o r A i s as follows. W e r e c a l l t h a t a s u b s e t S c X i s

COMPACT

-

23

OPEN TOPOLOGY

a n t i s y m m e t r i c f o r A if, for any f E A,

the restriction flS is real-valued implies that f I s is constant. Assume that P,+l is a proper refinement of P, for all u E Then Pa+l contains a set not in P, for all T < u + 1. Therefore the cardinal number of subsets of X is > I I . This contradicts the definition of Hence there

G .

G

.

G

, such that P p = P ~ + ~Iaence . pI, = *A, exists an ordinal p E where denotes the closed , pair-wise disjoint , partition of X into maximal antisymmetric sets for A.

yA

THEOREM 1.27 ( B i s h o p

[ 8 ] ; Glicksberg

L e t X be a

[26]1

com-

C(X;lK) b e a r e a l s u h a l g e b r a . L e t W c C(X;IK) b e a n A - m o d u l e . For e a c h f E C(X;M), f b e l o n g s t o t h e c l o s u r e of W if, and o n l y if, flS b e l o n g s t o t h e c l o s u r e of i n C ( S ; I K ) , for a l l s

p a c t H a u s d o r f f s p a c e and l e t A

C

WIS

This is an immediate corollary of the following ger for of Theorem 1.27. PROOF

stron-

THEOREM 1.28 (Machado [ H I ) L e t X b e a c o m p a c t H a u s d o r f f space and l e t E b e a serninormed s p a c e . L e t A c C(X;D() b e a r e a l s u b a l g e b r a and W C C(X;E) an A-module. L e t f E C(X;E). F o r each 0

E

G,

t h e r e is S ,

E Pu s u c h t h a t

(a) S , c S , f o r a l l T < u ,

T E

G

;

(b) inf IIf-q/'= inf !'flS,-qlS,~\. q EW

q EW

PROOF

bra

B v a real subalqebra P. C C(X;C) we mean that the

is an IR-alaebra. Let u E f Assume that, given f E C(X;E), a set S, properties (a) and (b) has been found for all T < u .

alae-

A,

Ist CASE there is S ,

.

= T

(J

E

+ 1, with

T E

G.

P, such that S, C S,

inf q EW

I If-ql 1

=

with

By the induction hypothesis, , 11 T and E

for all 11

inf ' l f ' s T - c - l1~. T l ff EIQ

E P,

COMPACT

24

-

OPEN TOPOLOGY

Let A T C A be the subalgebra of all h E A such that and hlST is real. By Theorem 1.26 applied to the alqebra A T I S , the module WIST (over A T I S T I there is a set Sa E Pa = PT+l such that

On the other hand Sa proves (a) and (b) in this case. 2nd

CASE.

ST by construction.

The ordinal a has no predecessor. Define Sa =

Then Sa E Pa and Su c S, for all sume by contradiction that

where

C

d = inf {\\f-gll:g

E

T < a,

T E

G. To prove

This

r)

, 1 be g i v e n . Let

belong t o A ,

ql,

...'9,

E W and T E x f ( " E ; E )

T E E ' €3 E ,

a l s o be g i v e n . I f

n = 1,

and T o q1 E W because W i s invariant under

compo-

POLYNOMIAL ALGEBRAS

63

s i t i o n w i t h e l e m e n t s of E ' 8 E. Suppose n > 1. S i n c e

..., ..

... ... $n-l(xn-l)vo

$ n ( x n ) v , where $i E El, xn) = $ l ( x l ) and v E E. A s s u m e (1) i s t r u e f o r n - 1. Then

T(xl,

(xl,.

* $1 (x,)

tXn-l)

and t h e r e f o r e t h e mapping x l o n g s t o W.

+

belongs t o

$l (ql ( X I 1

.. . $n-l

is

W

v e c t o r s p a c e , w e may assume t h a t T i s o f form

,... ,n,

i = 1

xf

(n-lE;E),

(qn-l(x)

vo be-

C a l l it h. L e t g = ($n o q n ) Q vo. Then g E W ,

t h e r e f o r e x * P ( h ( x ) , q ( x ) ) b e l o n g s t o W.

a

and

Choose $ E E l such that

$ ( P ( v o , v o ) ) = 1. Then $ 0 v b e l o n g s t o E' 8 E and

x+$(P(h(x),g(x)))v b e l o n g s t o W. However $ ( P ( h ( x ), q ( x ) ) ) v = $ (P ($1(9, ( X I

- .. $,,,

DEFINITION 4 . 7

(qn-l

(XI

) v o , Qn (qn ( X I )vo) 1 =

A v e c t o r subspace W C C(X;E)

n o m i a l a l g e b r a (of t h e 1 ' 2 kind) i f it p r o p e r t i e s ( 1 ) - ( 4 ) of Lemma 4.6. A vec t or nd c a l l e d a polynomial a l g e b r a of t h e 2 Lemma 4 . 6 i s t r u e f o r a l l T E X c n E ; E )

has any o f t h e equivalent subspace W C C(X;E) is k i n d i f p r o p e r t y ( 1 ) of and a l l n > 1.

A polynomial a l g e b r a W o f t h e

2"d k i n d ) i s c a l l e d s e l f - a d j o i n t and it i s c a l l e d e v e r y - w h e r e E

kind (resp.of

the

the algebra

d i f f e r e n t from zero i f ,

for

any

By "polynomial a l g e b r a " , w e mean a polynomial

al-

X, t h e r e i s g

CONVENTION.

5 ' 1

f E W) i s a s e l f - a d j o i n t s u b a l q e b r a of C(X);

A = {$ o f; $ E El,

x

if

i s called a poly-

E

W such t h a t g ( x )

# 0.

g e b r a o f t h e lSt k i n d . LEMMA 4 . 8

L e t E and F b e t w o n o n - z e r o

l o c a l l y c o n v e x Hausdorf3c

s p a c e s . Then (a)

(b)

The v e c t o r s u b s p a c e g e n e r a t e d b y t h e u n i o n of a l l 9 2 ( E ; F ) , w i t h n 2 1, i s a p o l y n o m i a l a l gebra. The v e c t o r subspace F f ( E ; F ) i s a algebra

.

polynomial

64

POLYNOMIAL ALGEBRAS

Let W c C ( E ; F ) be the vector subspace qenerated by union of all F f f ( E ; F ) , with n 2 1. Then W C F f ( E ; F ) = z ( E ) In fact,

the

PROOF

Q

F.

Therefore, Lemma 4 . 8 follows from Lemma 4 . 6 and the following LEMMA 4 . 9 F o r any n o n - z e r o l o c a l l y c o n v e x s p a c e E, s p a c e T f ( E ) i s an a l g e b r a .

the v e c t o r

PROOF It is enough to prove that any product ~I~.Q~....,@~ m linear forms Qi E E ' (i = 1,2,. ,m) can be written as

..

of a

linear combination of elements of F F ( E ) . By the "polarization formula" we have m 1 (1) xl. X = - c €1, Em(EIX1+ + Em Xm) , m m! 2 where the summation is extended over all possible combinations of E l = 2 1, E 2 = 2 I,..., E~ = 2 1, for all x1,x2, xm E M . Since E l Q1 + + E~ Qm E E l ,

....

...

...,

...

... x

belongs to cj)f;(E) (i = 1,2,...,m) (2)

Ql(X)

for all x

-+

,

[El

Q1(X) +

... +

Em

m Qm(x)]

and therefore substituting Qi(x)

for

xi

(1) yields

1 c ... Q m W= m7 !2

... Ern(E14+X)+

El

... +

EmQm(X)P

E E.

As

another example of a polynomial algebra

c C ( X ; E ) let us consider the following situation. Let

be a real finite-dimensional non-associative (i.e. not necessarily finiteassociative) linear algebra. This means that E is a dimensional vector space over IR in which a bilinear multiplication

W

(u,v)

E E x E

-+

u v

E

E E

is defined. Since E is finite-dimensional there is only one locally convex and Hausdorff topology on E, and we shall always multiplication consider this topology for E . Notice that the

65

POLYNOMIAL ALGEBRAS

being bilinear is then continuous. By defining operations pointwise, C(X;E) becomes a non-associative algebra over IR too, as well as a b i m o d u l e o v e r E : if u E E and f E C(X;E) the mappings x + u f(x) and vector subspace x + f (x)u belong to C(X;E) We shall call a W C C(X;E) a s u b m o d u l e o v e r E if it is a bimodule over E, i.e. if it is invariant under right and left multiplication by elements of E.

.

LEMMA 4.10

L e t E b e a r e a l f i n i t e - d i m e n s i o n a l c e n t r a l and s i m -

p l e n o n - a s s o c i a t i v e l i n e a r a l g e b r a . L e t W C C(X;E) b e a g e b r a o v e r IR w h i c h i s a s u b m o d u l e o v e r E. T h e n W i s a nomial a l g e b r a .

subalpoly-

Before proving Lemma 4.11 let us explain the terminology. All definitions are taken from Schafer [581. An algebra E is called a z e r o - a l g e b r a if uv = 0 €or all u,v E E. The subspaces of E which are invariant relative to the right and left E is multiplications are called the i d e a l s of E. The algebra and called s i m p l e if E has no (two-sided) ideals # 0 and # E, be the enveloping moreover E is not a zero-algebra. Let &(E) algebra of all right and left multiplications. &(E) is called are the m u l t i p l i c a t i o n a l g e b r a of E. Clearly the ideals of E the subspaces which are invariant relative to the multiplication algebra C / C ( E ) . It follows that a non-zero alqebra is simlinear ple if and only if &(E) is an irreducible alqebra of transformations. We define the c e n t r o i d of E to be the centratransformalizer of &(El in the alqebra &(E) of all linear centroid tions on E. It follows that T E d(E) belongs to the of E if and only if T(uv) = T(u) .v = u.T(v) for all u,v E E. Clearly, all T of the form T = h.idE,for X E R, belong to the centroid. We say that E is c e n t r a l if its centrold coincides with IR.idE. We have then the followinq fundamental result LEMMA 4.11

Let E be a r e a l f i n i t e - d i m e n s i o n a l

p l e n o n - a s s o c i a t i v e a l g e b r a . T h e n &(E)

= &(El.

c e n t r a l and sim-

POLYNOMIAL ALGEBRAS

66

PROOF Let r be the centroid of E. Then r is isomorphic to IR. The result follows from Theorem 4, Chapter X, Jacobson [3lj. PROOF OF LEMMA 4.10 By Lemma 4.11, &E) = &(E). Therefore, any W c C(X;E) which is a submodule over the algebra E is invariant under composition with any linear transformation TE&E). Since E is not a zero-algebra, choose a pair u,v C E such that u v # 0. Let @ E E' be a linear functional such that @(uv) = 1. Define A = {$(q); $ E E', 9 E W). By Lemma 4.1, A is a vector subspace of C ( X ; I R ) such that A Q E c W. It remains to Then prove that A is a subalgebra. Let $(q) and q(h) be in A . x + $(g(x))u and x + q(h(x))v belong to W, since A Q E c W. By hypothesis, W is a subalgebra of C(X;E) under pointwise operations. Thus the mapping x + [$ (g(x))u! [q (h (x))v] = $(g(x))rl(h(x))uv belongs to W. Call it f. Then +(f) E A. Clearly, @(f(x)) = $(g(x))q(h(x)) for all x E X, since $(uv) = 1. Thus W is a polynomial algebra. REMARK

The above proof of Lemma 4.10 can be applied to

any

In his Thesis r161, non-zero algebra such that d ( E ) = &(E). De La Fuente proved that &(El = g ( E ) for the followinq classes of algebras : (1) E a Clifford algebra of a real vector space of even dimension: (2) E a Cayley-Dickson algebra Dn, with n > 2. In his monograph 141, Blatter assumes E to have a non-zero square, i.e. assumes the existence of an element v E E such that v2 # 0. Thus his result cannot be applied to Lie algebras. A non-associative algebra E is said to be a L i e a l g e b r a if its multiplication satisfies the two conditions L

(i) v = o (ii) (uv)w + (vw)u + (wu)v = 0 for all u,v,w E E. From (i) and (ii) (known as the J a c o b i i d e n t i t y ) it follows that for all u,v E E.

(iii) uv = -w Conversely, if the field over which E is a space is of characteristic # 2, then (iii) implies (i).

vector

67

POLYNOMIAL ALGEBRAS

5 2 STONE-WEIERSTRASS SUBSPACES Motivated by the Stone-Weierstrass Theorem lary 1.9, 5 5 , Chapter 1) we state the followinq.

(Corol-

L e t W CC(X;E) be a v e c t o r s u b s p a c e . S t o n e - W e i e r s t r a s s h u l l o f W i n C(X;E), d e n o t e d b y A(W), i s

DEFINITION 4.12

The the

s e t o f a l l f u n c t i o n s f E C(X;E) s u c h t h a t

(1) f o r any x E X such t h a t f(x) # 0 , there i s g E W such t h a t g(x) # 0 ; (2) f o r any x,y E X such t h a t f (x) # f (y), t h e r e is g E W s u c h t h a t g(x) #.g(y).

Obviously, A(W) c C(X;E) is a vector subspace, containing W. Moreover, if E is a Hausdorff space, ii c A (W)

.

DEFINITION 4.13 L e t W C C(X;E) be a v e c t o r s u b s p a c e . We say t h a t W i s a S t o n e - W e i e r s t r a s s s u b s p a c e i f A(W) C Before proceeding, let us show that A(W) is in fact a self-adjoint closed polynomial algebra containing W. To do this let us introduce the following function 6w: R + {0,1) (see Blatter [4]) : a) R C X x X is the set of all pairs (x,y)suchthat x y (mod. W). b) Gw(x,y) = 0, if f(x) = 0 for all f E W.

w.

Gw(x,y) = 1, if f(x) # 0 for some f

c)

E

W.

It is clear that the following property holds: (x,y) E R * f(x) = GW(x,y)f(y) for all f E W. Let Al(W) be the set of all g E C(X;E) such that (x,.y) E R * g(x) = 6w(x,y)q(y). Clearly, w c A1(W). PROPOSITION 4.14 A1(W)

-

PROOF

Let f

E

For e v e r y v e c t o r s u b s p a c e W

A1(W). Let x

E

C

C(X;E), A(W) =

X be such that f(x) # 0.

If

g(x) = 0 for all q E W then 6w(x,x) = 0, and f(x)= GW(x,x)f(x) = 0, a contradiciton. This proves (1) of Definition 4.12. Let

P 0L Y N 0Pl I A L A L G E B R A S

68

x,y E q E

w.

x

.

b e such t h a t f ( x ) # f ( y ) A s s u m e q ( x ) = q ( y ) f o r all Then ( x , y ) E R. S i n c e f ( x ) # f ( y ) , w e may assume f (x)#O.

By (1) j u s t p r o v e d , t h e r e i s q E W w i t h q ( x ) # 0 . Gw(x,y) = 1. T h e r e f o r e f ( x ) = G W ( x , y ) f( y ) = f ( y ) , a t i o n . T h i s p r o v e s (2) o f D e f i n i t i o n 4.12, and so A1(W)

Hence contradic-

c

A(W)

.

C o n v e r s e l y , assume f E A ( W ) . L e t ( x , y ) E R. Suppose Since t h a t Gw(x,y) = 0 . Then g ( x ) = q ( y ) = 0 f o r a l l q E W. f E A(W),

f ( x ) = f ( y ) = 0 . Suppose now t h a t Gw(x,y) = 1.

If

which f ( x ) # f ( y ) , t h e r e would e x i s t g E W w i t h g(x) # g ( y ) , c o n t r a d i c t s ( x , y ) E R . Hence f ( x ) = f ( y ) . I n b o t h cases, f ( x ) = GW(x,y) f ( y ) , and t h e r e f o r e f E A1 (W)

F o r e v e r y v e c t o r s u b s p a c e W C C(X;E),

PROPOSITION 4.15

i s a closed self-adjoint PROOF

.

S i n c e Gw(x,y) E I0,l) f o r a l l ( x , y ) E R , A l ( W )

i s obvi-

o u s l y a polynomial a l g e b r a , c o n t a i n i n q W , such t h a t { $ o g ; $ E El, CJ E A1(W) 1 i s s e l f - a d j o i n t . L e t q E A 1 ( W ) ,

l e t { f a } be a n e t , f a

+

A(W)

p o l y n o m i a l a l g e b r a c o n t a i n i n g W.

9, f a E A l ( W ) .

and Since

L e t ( x , y ) E R.

K = ( x , y ) i s compact, and f o r e v e r y a , f a ( x ) = G W ( x , y ) f a ( y ) , w e

see t h a t q

E A1(W).

I t remains t o n o t i c e A l ( W )

= A(W)

by

the

preceding Proposition 4 . 1 4 . LEMMA 4.16

L e t W C C(X;E) b e a v e c t o r s u h s p a c e w h i c h i s

in-

v a r i a n t u n d e r c o m p o s i t i o n w i t h a n y e l e m e n t u E E' 8 E, and l e t A = ( 4 o f ; 6 E E', f E W ) . S u p p o s e t h a t E i s a H a u s d o r f f spaae. Then A(W)

= L A ( A Q E) = L A ( W ) .

. Let

Y C X b e an e q u i v a l e n c e c l a s s (m0d.A). L e t x , y E Y. I f f ( x ) # f ( y ) , t h e r e i s 9 E W s u c h t h a t a ( x ) # q ( y ) . By t h e Hahn-Banach Theorem, t h e r e i s $ E E' s u c h t h a t @ ( g ( x ) )# @ ( q ( y)) S i n c e $ o q E A , t h i s i s i m p o s s i b l e . Hence f is const a n t o v e r Y. L e t v E E b e t h i s c o n s t a n t v a l u e . I f v = 0 , t h e n such f a g r e e s w i t h 0 E A Q E o v e r Y. I f v # 0 , choose q E W

PROOF

L e t f E A (W)

.

over t h a t q ( x ) # 0 , f o r some x E Y. Notice t h a t q i s c o n s t a n t Y, s i n c e A and W d e f i n e t h e same e q u i v a l e n c e r e l a t i o n over X.

69

POLYNOMIAL ALGEBRAS

Let u E E , u # 0, be this constant value. Choose @ E E ' with @(u) = 1. Then h = ( @ o 9) 8 v belongs to A 8 E and aqrees with f over Y. Hence f E LA(A 63 E )

.

By Lemma 4.1, 51, A

Q E

c W. Therefore LA(A

8 E)

c

LAW). Finally, let f E LA(W). Let x E X be such that f(x) # 0, Suppose u(x) = 0 for all q E W. Let For Y C X be the equivalence class (mod. A) that contains x . every E > 0 and p E cs(E) there is g E W such that Hausdorff, p(f(x) - g(x)) < E . Hence p(f(x)) < E . Since E is f(x) = 0. This contradiction shows that f satisfies (1) of Defin i t i o n 4-12.similarly, one proves that f satisfies condition ( 2 ) of D e f i n i t i c m 4.12. So f E A(W). This completes the proof of Lemma 4.16. THEOREM 4.17

( S t o n e - W e i e r s t r a s s Theorem for p o l y n o m i a l

al-

g e b r a s ) . Suppose E i s a H a u s d o r f f s p a c e . E v e r y s e l f - a d j o i n t pol y n o m i a l a l g e b r a W C C(X;E) i s a S t o n e - W e i e r s t r a s s s u b s p a c e .

.

PROOF By Lemma 4.16, A(W) = LA(W) = LA(A 8 E) By Theorem 1.8, 5 5 , Chapter 1, applied to the A-module A Q E , we have LA(A Q E) = A Q E . Since W is a polynomial algebra, A Q E C W. Hence A Q E C ii. Putting all this together, A(W) c ii, i.e. W is a Stone-Weierstrass subspace. COROLLARY 4.18

Suppose E i s a H a u s d o r f f s p a c e . L e t W C C(X;E)

be a s e l f - a d j o i n t p o l y n o m i a l a l g e b r a . Then W i s d e n s e i f and only i f W i s s e p a r a t i n g and e v e r y - w h e r e d i f f e r e n t f r o m z e r o .

PROOF Just notice that if W is Separating and everywhere difdense ferent from zero, then A(W) = C(X;E). Conversely, every from subset of C(X;E) is separating and everywhere different zero, since E is Hausdorff. COROLLARY 4.19 (Nachbin, Machado, Prolla [46!)

(Infinite

di-

mensional W e i e r s t r a s s polynomial approximation Theorem). L e t E and F b e two n o n - z e r o r e a l l o c a l l y c o n v e x H a u s d o r f f s p a c e s . T h e n (E;F) i s d e n s e i n C(E;F) Moreover t h e v e c t o r s u b s p a c e g e n e -

Tf

.

70

POLYNOMIAL ALGEBRAS

r a t e d b y a l l pi(E;F), w i t h n > 1, i s d e n s e i n t h e

polynomial

a l g e b r a {f E C(E;F); f(0) = 0).

8,

PROOF By Lemma 4 . 8 , 5 1, (E;F) is a polynomial algebra.Since E and F are real, A = ( @ o q; @ E F', q E Ff(E;F)l is a subalgebra of C(E;IR). Since (E;F) contains the constants and is separating over E (because both E and F are non-zero),Corollary 4.18 above shows that (E;F) is dense in C(E;F)

Ff

.

pf

Let W be the vector subspace of C(E;F) qenerated by > 1. By Lemma 4 . 8 , 5 1, W is a the union of all f)i(E;F) with n polynomial algebra. Let A = {@I o q; @ E F', q E Wl. Since both E and F are real, A C C(E;IR). Let f E C(E;F) be such that f(0) = 0 . Let x E E be such that f(x) # 0 . Hence x # O.Let @EE' = 1 and let v E F with v # 0 . Then q = 0 v belongs with $(XI to W and g(x) = v # 0. Let x,y E E be such that f (x) # f (y). Hence x # y. Choose @ E E' with @(x) # $ ( y ) and v E F with v # 0 . Then q = $ Q v belongs to W and q(x) # q(y) This shows that f E A ( W ) . By Theorem 4 . 1 7 , f E as desired.

w,

.

REMARK Corollary 4.19 has an analogue for c o m p l e x spaces, if n Q F as the vector subspace qenerated by the we redefinepf(E) !"v, where v E F and set of all maps of the form x + [@(XI @ : E -c 4: is either a linear or an antilinear continuous form. Let T;(E;F) be the vector subspace generated by all $(El 0 F, > 1, defined as above. Then A = { $ o g; @ E F', q Epf(E)@F)= n Tg(E) is a self-adjoint subalgebra of C(E;Q). Suppose E i s a H a u s d o r f f s p a c e . For e v e r y v e c t o r s u b s p a c e W C C(X;E), A ( W ) i s t h e s m a l l e s t c l o s e d self-

COROLLARY 4.20

a d j o i n t polynomial algebra containing W.

PROOF By Proposition 4 . 1 5 , A ( W ) is a closed self-adjoint polyselfnomial algebra containing W. Let V C C(X;E) be a closed adjoint polynomial algebra containing W. Hence A ( W ) C A ( V ) . By = V. Therefore A ( W ) C V, as desired. Theorem 4 . 1 7 , A ( V ) c

v

COROLLARY 4 . 2 1

f B Z a t t e r [ 4 ] I Let E be a f i n i t e - d i m e n s i o n a l

t r a l and s i m p l e n o n - a s s o c i a t i v e

r e a l a l g e b r a . Every r e a l

censub-

P 0 L Y N0NI A L A L G E B RA S

71

a l g e b r a W C C(X;E) w h i c h i s a s u b m o d u l e o v e r E i s a e r s t r a s s subspace.

Stone-Wei-

PROOF By Lemma 4.10, 9 1, W is a polynomial algebra. Hence we may apply Theorem 4.17. COROLLARY 4.22 (De La F u e n t e [16] ) L e t E b e a C l i f f o r d algebra of a r e a l v e c t o r s p a c e of e v e n d i m e n s i o n o r a C a y l e y - D i c k s o n a l g e b r a Dn, w i t h n > 2 E v e r y r e a l s u b a l g e b r a W C C(X;E) w h i c h is a submodule o v e r E i s a S t o n e - W e i e r s t r a s s s u b s p a c e .

.

PROOF As noticed in the final Remark of 9 1, we can aPP1Y Lemma 4.10, 9 1. Therefore W is a polynomial algebra, and by Theorem 4.17 above, W is a Stone-Weierstrass subspace. THEOREM 4.23 Suppose E i s a non-zero Hausdorff space. Let W C C(X;E) b e a v e c t o r s u b s p a c e w h i c h i s i n v a r i a n t u n d e r comp o s i t i o n w i t h e l e m e n t s of E' Q E, and l e t A={$of; 4 E E',fEW). The f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t : (1) W i s l o c a l i z a b l e u n d e r A i n C(X;E). (2) W i s a S t o n e - W e i e r s t r a s s s u b s p a c e . (3) A i s a S t o n e - W e i e r s t r a s s s u b s p a c e . PROOF By Lemma 4.16, A (W) = LA(W). Hence (1) and (2) are equivalent. of Assume (2), and let f E C(X;K) be an element A(A). Let K C X compact and E > 0 be given. Choose $ E E', with $ # 0, and choose v E E with $(v) = 1. Let g = f Q v. Obviously g E A(W). By hypothesis q E Let p E cs(E) be such that !$(t)! 5 p(t) for all t E E. Let h E W be chosen so that p(g(x) - h(x)) < E for all x E K. Hence If (x) (6 o h) ( X I1 < E for all x € K. But $ o h E A, so € E ii and A is a StoneWeierstrass subspace. Finally, assume (3). Since ii = A(A), it followsfmm Proposition 4.15, that B = is a closed self-adjoint subalgebra of C(X;M). By Theorem 4.17 applied to the polynomial algebra B Q E, we have LB(B Q E) = B Q E. Hence LA(W) = LA(A 0 E) C

w.

-

L (B 0 E) = L (B A B

€4

- Q E,by

E) = B 0 E C A

Lemma 4.16

and

the

72

POLYNOMIAL ALGEBRAS

fact that €9 E c A Q E. By Lemma 4.1, 5 1, A €9 E is contained in W; hence LA(W) C i , which proves (1). We come now to Bishop's Theorem for polynomial alqebras of the 2nd - kind. L e t X be a compact H a u s d o r f f s p a c e and l e t E b e THEOREM 4 . 2 4 a semi-normed s p a c e . L e t W C C(X;E) be a p o l y n o m i a l a l g e b r a of t h e 2"d k i n d and l e t A = { $ o f; $ E El, f E W). For every f E C(X;E), f b e l o n g s t o t h e c l o s u r e of W, i f and o n l y if, f(S b e l o n g s t o t h e c l o s u r e of WIS i n C(S;E), f o r e a c h maximal A - a n t i s y m m e t r i c s u b s e t S c X.

PROOF By Lemma 4 . 6 , 5 1, A c C(X;C) is a subalgebra.For every f,g E W and $ E El, the function x * $(f (x))q(x) belongP to W, 2E;E). Hence, W is an since (u,v) * $(u)v belongs to A-module. It remains to apply Theorem 1.27 (5 8 , Chapter 1).

z(

5

3

C (X)-MODULES

In this section we shall suppose throughout that E is a locally convex Hausdorff space. Let S c C(X;E) be an arbitrary subset and let us define Z(S) = {x

E

X; q ( x ) = 0 for all g

E S}.

Obviously, Z(S) is a closed subset of X. On the other hand, Z C X is any closed subset let I(Z) = {f

E

C(X;E);f(x) = 0 for all x

E

if

Z}.

It is easy to check that, for any subset S C C(X;E) ,W = I(Z (S)) is a closed polynomial algebra, containing S, which is a C(X)module. Moreover, A = {I$ o f; $ E E l , f E W }is self-adjoint. Indeed, let q E A, say q = 4 o f, with $ E El, f E W. Choose a pair $ E E' and v E E with +(v) = 1. Let h = 9 €9 v. Let xEZ(S). Then g(x) = 0, and h(x) = q(x)v = 0, i.e. h E W. Since q=$ o h, E A, i.e. A is self-adjoint. it follows that Let V be a closed polynomial algebra, containing S,

73

PCLYNOMIAL ALGEBRAS

and such that (1) V is a C(X)-module; (2) { a o f; a E E ' , f E V} is self-adjoint. We claim that W = I ( Z ( S ) ) C V. Indeed, let f E W. Let x E X be such that f (x) # 0. Then x d Z ( S ) , i.e. there exist g E S c V such that g(x) # 0. Let x,y E X be such that f (XI # f (y) Then x and y do not belong simultaneously to Z ( S ) . Suppose x $ Z ( S ) . Since X is a completely regular Hausdorff space, there is such h E C(X) such that h(x) = 1, h(y) = 0. Let g E S c V be and that g(x) # 0. Then hq E V, since V is a C(X)-module, = V. h(x)g(x) = g(x) # 0 = h(y)g(y). By Theorem 1, 9 2 , f E If S = C(X;E) , then Z ( S ) = B . Conversely, if W is a closed polynomial algebra satisfying (1) and ( 2 ) and suchthat Z ( W ) = 8 , then W = I(Z (W)) = I ( B ) = C(X;E) This proves the following.

.

v

.

L e t S C C(X;E) b e an a r b i t r a r y s u b s e t , THEOREM 4 . 2 5 W = I(Z(S)). T h e n W i s t h e s m a l l e s t c l o s e d p o l y n o m i a l

and algebra

c o n t a i n i n g S and s u c h t h a t

(1) W i s a C ( X ) - m o d u l e ; ( 2 ) { $ o f; E E', f E W} i s s e l f - a d j o i n t . Moreover, W i s a c l o s e d polynomial algebra s a t i s f y i n g

(1) and ( 2 ) i f , and o n l y i f , W = I ( Z ( W ) ) . A c l o s e d p o l y n o m i a l a l g e b r a W s a t i s f y i n g (1) and ( 2 ) i s c h a r a c t e r i z e d b y t h e a s s o c i a t e d c b w d s e t Z(W). I n p a r t i c u l a r , W = C(X;E) i f , and o n l y i f , Z ( W ) = fl

.

COROLLARY 4 . 2 6

T h e maximal p r o p e r c l o s e d s e l f - a d j o i n t

poly-

n o m i a l a l g e b r a s w h i c h a r e C ( X ) - m o d u l e s a r e of t h e form W =

{f

E

C(X;E); f(x) = 0 f o r some x

E

XI.

Every proper c l o s e d s e l f - a d j o i n t polynomial COROLLARY 4 . 2 7 a l g e b r a W, w h i c h i s a C ( X ) - m o d u l e , i s c o n t a i n e d i n some maximal p r o p e r c l o s e d s e l f - a d j o i n t p o l y n o m i a l a l g e b r a w h i c h i s a C(X)m o d u l e ; i n f a c t , W i s t h e i n t e r s e c t i o n o f a l l t h e maximal prcper c l o s e d s e l f - a d j o i n t p o l y n o m i a l a l g e b r a s w h i c h a r e C(X) -modules and c o n t a i n i t .

74

POLY!!OMIAL A L G E B R A S

9 4 APPROXIMATION OF COMPACT OPERATORS If E and F are Banach spaces, let Lc(E;F) be the uniform closure in the space of bounded l i n e a r operators from E to F of the set E' 0 F of continuous linear operators of finite rank from E to F. The space Lc(E;F) is the space of compact linear operators from E to F if either E ' or F has the approximation property. In this case if u : E + F is a compact linear l i n e a r map operator, then given E > 0 there is a continuous 1 w E E' 8 F = pf(E;F) such that I (u(x) - w(x) I < E for all x E E, with IIxII 5 1. What happens if neither E' nor F has the approximation property? We will prove that the above approximation is always possible if we allow the finite-rank map w to be a poly(E;F). In [39] it was assumed tfiat nomial, i.e. an element of the space E is reflexive. We thank Prof. Charles Stegall for calling our attention to the factorization theorem of T. Figiel and W.B. Johnson that makes unnecessary the reflexivity of E.

Tf

LEMMA 4.28 (Fiqiel [23!, Johnson [33]) L e t E and F be two real There Banach s p a c e s , and u : E -+ F a compact l i n e a r o p e r a t o r . e x i s t s a r e f l e x i v e r e a l Banach s p a c e G and compact l i n e a r oper a t o r s v : E + G and g : G + F s u c h t h a t g o v = u. THEOREM 4.29 L e t E and F he two r e a l Banach s p a c e s , and u : E -P F a compact l i n e a r map. T h e n , g i v e n E > 0 , t h e r e i s a = 0 c o n t i n u o u s p o l y n o m i a l of f i n i t e t y p e w E Tf(E:F) w i t h ~ ( 0 ) and s u c h t h a t

1 lu(x) -

w(x)

II

< E

f o r a l l x E E, w i t h 11x1 I < 1.

PROOF By the theorem of Figiel-Johnson there is a reflexive Banach space G and compact linear operators v : E + G and g : G + F such that g o v = u. Let X be a closed ball of G such reflexive, that v(x) E X for'all x E E, \ ! X I 1 5 1. Since G is X equipped with the a(G,G')-topology is compact. Let W be the vector subspace of C(X;F) generated by F:(G;F)] for all

75

POLYNOMIAL ALGEBRAS

n > 1. Then W is a polynomial alqebra, separatinq over X, and such that, qiven t E X, t # 0, there is q E W with g(t) # 0. Since q : G * F is a compact linear map the restriction qIx is in C(X;F). By Corollary 4.18, 9 2, a belonas to the closure of W in C(X;F). Given E > 0, let h E W be such that < 1, then I\g(t) h(t)ll < E for all t E X. If x E E, 1(x1( v(x) = t E X. Hence I I (q o v) (XI - (h o v) (x) 1 1 < E . Let w=hov; then w E Tf(E;F) and I lu(x) - w(x) 1 I < E for all 11x1 I 5 1.

-

f : E * F b e t w e e n two Banach s p a c e s is s a i d t o b e w e a k l y c o n t i n u o u s i f f is c o n t i n u o u s f r o m t h e weak t o p o l o g y u(E;E') i n E t o t h e norm t o p o l o g y i n F. All 41 E E' are weakly continuous, and as a corollmy Q F are weakly continuous too. all p E $(E) We shall denote by C(Ew;F) the vector space of all weakly continuous maps from E into F, equipped with the topology defined by the family of seminorms DEFINITION 4.30

A mapping

f

+

sup {IIf(x)I); x

E

K)

where K c E is a weakly compact subset. If we denote by X the space (E,a(E,E')), then C(Ew;F) with the above topoloqy is just C(X;F) with the compact-open topoloqy. L e t E and F b e two r e a l Banach s p a c e s . THEOREM 4.31 (El 8 F i s d e n s e i n C(Ew;F)

Tf

Then

.

PROOF Let X = (E,a(E,E')). By the remarks made after Defi(E) Q F is contained in C(X;F) Since $(El 8 F nition 4.30, is a polynomial algebra, which is separating and everywhere different from zero, we can apply Corollary 4.18, 9 2,with W = is dense in Ci)f(E) Q F C C(X;F), to conclude that (El @ F C (X;F) = C (Ew;F) in the compact-open topoloqy.

Tf

.

Tf

L e t E and F b e t w o r e a l Banack s p a c e s and s u p COROLLARY 4.32 p o s e t h a t E is r e f l e x i v e . L e t g : E + F b e a w e a k l y c o n t i n u o u s map and l e t r > 0 . G i v e n E > 0 , t h e r e is a c o n t i n u o u s polynomial of f i n i t e type h E (El Q F s u c h t h a t 1 \q(x) h(x) I I < E , f o r a l l x E E w i t h 11x11( r.

Tf

-

76

POLYNOMIAL ALGEBRAS

PROOF When E is a r e f l e x i v e Banach space, any closed ball of (x E E; 11x1 I 2 r} is weakly compact, and the topoloqy C(Ew;F) can be defined by the family of seminorms f

+

sup {IIf(x)jI; IIxlI < rl

where r > 0 . DEFINITION 4 . 3 3 A m a p p i n g f : E + F b e t w e e n two Banach s p a c e s i s s a i d t o b e w e a k l y c o n t i n u o u s on bounded s e t s i f t h e r e s t r i c t i o n of f t o a n y b o u n d e d s u b s e t X of i s continuous from the r e l a t i v e weak t o p o l o g y u(E,E') o n X t o t h e norm t o p o l o g y i n F .

Ix

Any weakly continuous mappinq f : E + F is weakly continuous on bounded sets, but the converse is false in qene(See ral, even in the case of a Hilbert space E and F = El. Restrepo [ S l j , pg. 194). When E is a r e f l e x i v e Banach space, we shall denote by C(Ewcb;F) the vector space of all f : E + F which are weakly continous on bounded sets, equipped with the topology definedby the seminorms f

+

sup ([lf(x)lj; x E XI

where X C E is bounded. Since every bounded set X C E is contained in some closed ball centered at the origin, thisbpolocry is also defined by the family of seminorms f

+

sup E ! If(x)I

1;

11x1 I 5 rl

where r > 0. The following result generalizes Theorem Restrepo [53]. THEOREM 4 . 3 4

3

L e t E and F b e t w o r e a l Banach s p a c e s and

p o s e t h a t E i s r e f l e x i v e . Then g : E

+

F i s weakly

on b o u n d e d s e t s , i f and o n l y if, t h e r e i s a s e q u e n c e p o l y n o m i a l s pn E Tf(E) Q F s u c h t h a t pn

+

of sup-

continuous

(p,)

of

g u n i f o r m l y o n bound-

ed s e t s .

PROOF Let q : E + F be such that there exists a sequence {pn} of polynomials pn E pf(E) 63 F such that pn + g uniformly on such bounded sets. Let X c E be a bounded set. Let r > 0 be Banach that X C (x E E; I Ix[[5 r} = Ur. Let Cb(Ur;F) be the

77

POLYNOMIAL ALGEBRAS

space of all bounded continuous mappings from Ur (equipped with the relative weak topology a(E,E') IUr into the Banach space F. A Since pnlUr + glUr uniformly, it follows that 9 E Cb(Ur;F). fortiori, glX is continuous from the relative weak

top01oqy

a(E,E') ! X on X to the norm topology of F. Conversely, assume that g : E + F is weakly continuous on bounded sets. Since every bounded set X c E is contained the topoin some closed ball {x E E;llx!! 5 n}, n = 1,2,3, logy of C(Ewcb;F) is metrizable and the result follows from the following.

...,

L e t E and F be two r e a l Banach s p a c e s and THEOREM 4.35 p o s e t h a t E i s r e f l e x i v e . Then (E) Q F i s d e n s e i n

Ff

sup-

C (EwcbiF) let Let 9 E C(Ewcb;F) be given. For each n = 1,2,3,..., PROOF 1 2 n}, equipped with the relative weak topoUn = { x E E; 1 logy a(E,E') IUn. Then q ( U n E C(Un;F). Let Wn = ($(El Q F) IUn.

!XI

Then Wn is a polynomial algebra contained in C(Un;F), which

is

separating and everywhere different from zero. By Corollary 4.18, 5 2, W is dense in C(Un;F). Hence, given E > 0, there is a con-

n

tinuous polynomial of finite type p E ( P f ( E ; F )

(Ip(x) - g(x)lI
0 and u E V such that v(x) < hu(x) , w(x) < Xu(x) , for all x E X. Any element of a directed family of upper semicontinuous positive functions on X is called a w e i g h t o n X. E Let E be a locally convex space. A function h:X v a n i s h e s a t i n f i n i t y if, given E > 0 and p E cs(E) , the set Ix E X; p(h(x)) 2 E } is compact. Hence p o h is upper semicontinuous, and therefore bounded on X. +

DEFINITION 5.1

L e t V b e a d i r e c t e d s e t of w e i g h t s o n X.

The

N a c h b i n s p a c e CV-(X;E) i s t h e v e c t o r s u b s p a c e of a l l f E C(X;E) s u c h t h a t vf v a n i s h e s a t i n f i n i t y , f o r e a c h v E V, t o p o l o g y z e d by t h e f a m i l y o f s e m i n o r m s

f

+

! I f 1 I",$

= sup {v(x)p(f(x));

w h e r e v E V and p E cs(E)

x E XI

.

When E = M , and no confusion may arise, we simply CVm(X) instead of CVm(X;M 1

.

write

L e t v : X * JR b e d e f i n e d b y v(x) = 1 f o r a l l EXAMPLE 5.2 x E X, and l e t V = {v}. T h e n CVaD(X;E) i s t h e v e c t o r s u b s p a c e of a l l f E C(X;E) t h a t v a n i s h a t i n f i n i t y . T h i s s p a c e i s usually uniform d e n o t e d b y Co(X;E). I t s t o p o l o g y u i s t h e t o p o l o g y of c o n v e r g e n c e o n X. The vector subspace of all f E C(X;E) such that the support of f is compact will be denoted by K(X;E). Obviously, K(X;E) c Co(X;E). If X is compact, K(X;E) = Co(X;E) = C(X;E).

W E I GH T E D A P P 0 X I M A T I 0 N

80

If p

E

cs(E) and K

supIp(f(x)); x

E

c X is a compact subset, then

KI < sup Ip(f(x)); x

E

XI

for all f E Co(X;E). This shows that the topology of convergence on X is stronger than the compact-open K induced by C(X;E) on Co(X;E).

uniform top0loqy

EXAMPLE 5.3 L e t X b e a l o c a l l y compact H a u s d o r f f s p a c e . Cons i d e r t h e d i r e c t e d f a m i l y V = I $ E Co(X;IR); $ 2 01. Then CVm(X;E) = Cb(X;E) a s v e c t o r s p a c e s and t h e t o p o l o g y d e f i n e d by t h e f a m i l y of seminorms f * sup ($(x)p(f(x)); x

E

!If!!

XI =

$?P

o n Cb(X;E) i s c a l l e d t h e s t r i c t t o p o l o g y and i t is d e n o t e d

by

[II]). The strict topology f3 is stronser than the compactopen topology induced on Cb (X;E) by C (X;E); on the other hand, f3 is weaker than the topology 0 of uniform Convergence on X. R.

(see

B U C ~

EXAMPLE 5 . 4 L e t V b e t h e s e t of a l l c h a r a c t e r i s t i c f u n c t i o n s o f compact s u b s e t s K C X. T h e n t h e Nachbin s p a c e CVm(X;E) is j u s t C(X;E) endowed w i t h t h e compact-open t o p o l o g y .

9

2

THE BERNSTEIN-NACHBIN APPROXIMATION PROBLEM

Let W C CVm(X;E) be a vector subspace which is an A-module, where A CC(X;lK) is a subalqebra. The B e r n s t e i n - N a c h b i n a p p r o x i m a t i o n p r o b l e m consists in asking for a description of the closure of W in CVm(X;E). Let P be a closed, pairwise disjoint coverinq of X. We say that W is P - l o c a l i z a b l e in CVm(X;E) if the closure of W in CVm(X;E) consists of those f E CVm(X;E) such that, siven any S E P, any v E V, any p E cs(E) , and any E > 0, there is some q E W such that v(x) p(f(x) - g(x)) < for all x

E

S.

E

W I E GH T E D A P P R O X I MAT I 0 N

81

The s t r i c t B e r n s t e i n - N a c h b i n a p p r o x i m a t i o n p r o b l e m consists in askinq for necessary and sufficient conditions for an A-module W to be P-localizable, when P is the set PA of all equivalence classes Y C X modulo XIA. In [ 4 6 ! , the sufficient conditions for localizability established by Nachbin (see e.q. Nachbin 14 31 ) were extended to the context of vector-fibrations, and a fortiori to vector-valued functions, in the case of modules over r e a l or s e l f - a d j o i n t c o m p l e x alqebras. In [401, the results of [46] were extended to the g e n e r a l c o m p l e x case in the same way that Bishop's Theorem generalizes the Stone-Weierstrass Theorem. Before statinq Definition 5.5, we recall that ?(lR? denotes the algebra of all IR -valued polynomials on 37". DEFINITION 5.5

L e t w b e a w e i g h t o n IR

n

.

The w e i g h t w i s s a i d

T(IR")

C Cw,(R?. n If w is a rapidly decreasinq weiqht on R , then w is called a f u n d a m e n t a l w e i g h t in the sense of Serqe Bernstein, n if ?(IRn) is dense in Cwm(IR ) . We shall denote by Rn the set of all fundamental weishts on IFn. We denote by 0; the subset of Rn consisting of those w E Rn which are s y m m e t r i c in the sense w(t) = w(\tl), for all

t o b e r a p i d l y d e c r e a s i n g a t i n f i n i t y when

t

E

,...,ltnl) if

IR", where It( = (Itl\

t = (tl,...,tn).

We denote by rl the subset of R1 consisting ofthose k y E R1 such that y E R1 for any real number k > 0. Let then rlS = rl n filS d and similarly We notice the inclusion Rn C $2: s d such that r dl c rl. Here Rn denotes the subset of all w E Rn

.

lul 5 d

rl

=

1 tj implies ~ ( u ->) r l n n1d .

~ ( t for ) all u,t

and then

E 37

DEFINITION 5.6 L e t P be a c l o s e d , p a i r w i s e d i s j o i n t c o v e r i n g of X. W e s a y t h a t W i s s h a r p l y P - l o c a l i z a b l e i n CV,(X;E) if, g i v e n f E CVm(X;E), v E V and p E cs(E), t h e r e i s some S E P such t h a t

inf(l If-gl\v,p;q

E

W} = inf(I(flS

-

u I s ~ I

V,P

:q

E

w].

82

WEIGHTED APPROXIMATION

DEFINITION 5.7 F o r e a c h v E V, p E cs(E), a n d 6 > 0 , we denot e by L(W;v,p,G) t h e s e t of a l l f E CVaD(X;E)s u c h t h a t , f o r e a c h e q u i v a l e n c e c l a s s Y C X (mod. A) t h e r e i s q E W s u c k t h a t

I 'fly -

qlyl Iv,p < 6 -

%

In our next definition, is the class of all ordinal numbers whose cardinal numbers are less or equal than 2 , where IX 1 is the cardinal number of X. For each u E , Pa is the closed, pairwise disjoint covering of X defined in@, Chapter 1.

Ix/

6

DEFINITION 5.8

We s a y t h a t t h e A-module W i s s h a r p l y l o c a l i z a and b l e u n d e r A i n CVaD(X;E) i f , g i v e n f E CVm(X;E), v E V,

p

E

cs(E), f o r e a c h a

E

6 there

e x i s t s an element S

U

8 Pa s u c k

that:

q

E

wl.

DEFINITION 5.9 We s a y t h a t a s u b s e t G(A) C A i s a s e t of gener a t o r s f o r A, i f t h e s u b a l g e b r a o v e r K g e n e r a t e d b y G(A) is d e n s e i n A f o r t h e c o m p a c t - o p e n t o p o l o g y of C(X;m); and we s a y t h a t a s e t of g e n e r a t o r s G(A) c A i s a s t r o n g s e t of g e n e r a t o r s i f , f o r any u E 6 and a n y S E Pa, t h e s e t AS n G(A) is a set of g e n e r a t o r s f o r t h e a l g e b r a AS ( R e c a l l t h a t AS = Ca E A; alS i s r e a l - v a l u e d ) ) . For e x a m p l e , t h e w h o l e a l g e b r a A i s a s t r o n g s e t of g e n e r a t o r s f o r A. A l s o , i f t h e a l g e b r a A h a s a s e t of g e n e r a t o r s G ( A ) c o n s i s t i n g o n l y of r e a l - v a l u e d f u n c t i o n s , then G(A) i s a s t r o n g s e t of g e n e r a t o r s f o r t h e a l g e b r a A. Similarly, a subset G(W) C W is a s e t of g e n e r a t o r s f o r W if the A-submodule of W qenerated by G(W) is dense in for the topology of CVm(X;E). Let us call G(W)* the r e a l linear span of G(W). L e t A C Cb(X;7R) b e a s u b a l g e b r a c o n t a i n i n g the c o n s t a n t s . For e a c h e q u i v a l e n c e c l a s s Y C X m o d u l o XIA, l e t Then, t h e r e b e g i v e n a c o m p a c t s e t K y C X, d i s j o i n t from Y.

LEMMA 5.10

WE I G H T E D A P P R O X I MAT I O N

83

.

t h e r e e x i s t e q u i v a l e n c e c l a s s e s Y1,. .,Yn C X modulo X I A s u c h A t h a t t o e a c h d > 0 , t h e r e c o r r e s p o n d f u n c t i o n s al,...,an i n s a t i s f y i n g the following properties: < 1, i = l,...,n; (a) 0 5 ai < ai(t) < 6 , f o r t (b) 0 -

(c) al

+...+

E

K i'

,

i = I,...,n;

an = 1 on X.

PROOF Let PA be the set of all equivalence classes Y C X modulo X I A . Select one element Y1 in PA, and let P be the collection of all elements Y E PA such that the intersection Y (3 K yi

is non-empty. Choose a real number 0 < E < 1 - €.For each Y E PA, U = there is by E A qiven by Lemma 1.3, 4 3 , Chapter 1, with X\Ky. Let By = { x E X; by(x) > 1 - E } . Clearly, Y C By, so that the collection {By; Y E PI is an open coverinq of the compact subset Ky c X. By compactness, there are equivalence classes

1 Y2,...,Yn in P such that Ky C B 2 u uBn, where 1 written Bi = By f o r Y = Yi, i = 2,...,n. For each

...

i = 2,...,n, there is a polynomial pi:IFt+IR (1)

Pi(l) = 1;

(2)

0 5 p p 1

(3)

0

L Pi(t)

(4)

1

-

5 1, t

E

[O,lJ;

< 6, t E

[O,E];

6 < Pi(t) 2 1, t E (1

we

have

index

such that

-

E,

1).

Indeed, apply Lemma 1.4, 4 3, Chapter 1, toset such polynomials. Consider qi = pi(bi) , where bi = by, €or Y = Yi' i = 2,...,n. Then qi E A , i = 2,...,n. Define a2 = g2 a3 = (1 - 92193

...............

W E I GH T E D A P P R 0 X I MAT I0 N

84

< 1; and For i = 2,...,n, it is easily seen that ai E A; 0 < ai Y = Yi. < qi(x) < 6 for all x E Kit where Ki = Ky with ai(x) -

Moreover, by induction, we see that a2 Let al = 1

+...+

-

an = 1

(a2 +...+

-

a,).

(1

-

-

q2) (1

Then al

E

q3)

... (1 - qn) .

A;

0 < al

and

5 1,

al +...+ an = 1 on X, which proves (a) and (c) of the statement. all To prove (b), it only remains to prove that a,(x) < 6 for x

E

Ky

1

j = 2,

. Now Ky 1 c

...,n, and

B2 u . . . u Bn, so that x

therefore 1

-

q . (x) < 6 ,

n

3

E

B 1' for some

and so

THEOREM 5.11 Suppose t h a t t h e r e e x i s t s e t s of g e n e r a t o r s G(A) and G(W), f o r A and W r e s p e c t i v e l y , s u c h t h a t : (1) G(A) c o n s i s t s o n l y of r e a l v a l u e d f u n c t i o n s ;

...,

(2) g i v e n any v E V , al, an E G(A), and p E cs(E), t h e r e a r e an+l,.. .,% with N > n, and w E

w(al(x)

%

E E

G(W) , G(A) ,

s u c h t h a t V(X)P(~(X))

,...,an(x) ,...,aN(x))

5

f o r a l l x E X.

Then W i s s h a r p l y l o c a l i z a b l e u n d e r A i n CVco(X;E).

We first remark that, since G(A) consists only of real valued functions, p = 2 and P2 = PA, where PA is the closed, pairwise disjoint partition of X into equivalence classes modulo X(A. Hence, all that we have to prove is that W is sharply PA-localizable in CVm(X;E). The proof will be partitioned into several lemmas, and to state them we need a preliminary definition. DEFINITION 5.12

L e t u s c a l l B t h e s u b a l g e b r a of Cb(X;R)of a l l

f u n c t i o n s of t h e form q(al,...,an), G(A),

and q E Cb(lR

n

where n > 1,

;lR ) a r e a r b i t r a r p .

al,

...,an

E

85

WE I GH T E D A P P R 0 X I MAT I 0 N

LEMMA 5.13

Assume t h a t G(A) c o n s i s t s o n t y of r e a l v a t u e d f u n c -

t i o n s . L e t f E L(W;v,p,X). T h e n , f o r e a c h E > 0 , t h e r e

bl,--.,bm

B, and gl,...,gm m

E

For each Y

PROOF

v(x) p(f (x)

-

E

exist

G(W) s u c h t h a t

E

PA, there exists wy

wy(x)) < h + ~ / 2 ,for all x

E E

G(W) * such

that

Y. Let us

define

-

Ky = {t E X; v(t)p(f(t) wy(t)) > A + ~/2). Then Ky is compact and disjoint from Y. Since the equivalence relations X'A andXIB

are the same, we may apply Lemma 5.10 for the algebra B. Hence, there exist equivalence classes Y1,...,Yn E PA such that toe& 6 > 0, there correspond hl,...,h n E B with 0 < hi 5 1 ; 0< hi(x)< for i = l,...,n. Moreover, 6 for x E K i I

hl

+...

where Ki = K Y i on X. Let us choose 6 > 0 such that nM 6 < ~ / 2 ,

= 1

hn where M = max +

{I

i = l,...,n),

If-wil

and wi = wy with Y=Yi

+...+

hnwn' We claim that v(x)p(f(x) - w(x)) < x + E , for all x E X. Indeed, n V(X)p(f(X) - w(x)) 5 X hi(X)V(X)P(f(X)- Wi(X))t i=l for all x E x. NOW, if k E Ki then hi(x) < 6, and therefore for i = l,...,n. Let w = hlwl

-

hi(x)v(x)p(f(X)

Wi(X))

< 6.)

If -

Wi!

IVIp

5 6 M;

on the other hand, if x $ Kit then the following estimate true :

-

hi(x)v(x)p(f(x)

is

wi(x)) < hi(x) ( A + ~/2).

Combining both estimates, we qet v(x)p(f(x)

-

w(x)) < nM 6

Since each wi

E

(A

+ €/2)(hl(x)+ ...+ hn(x))
0 , t h e m

i s w E W such t h a t IIw

-

bglIv,p < 6 .


86

PROOF and q w E

.

E

Suppose that b = q (al,.. ,an). Given v E V, p E G(W) there are an+l,...,aN E G ( A ) , where N > n,

CS (E)I

and

w(al (x), .. . ,an (x),.. . ,aN (x)1 for $ such that v(x)p ( q (x))
0 , we can find a real polynomial

.

q E ?(IRN) 11w - bql

IvIp

such that I Iq - r I l w < 6. From this it follows that < 6, where w = q(al, an' ...,aN)q E AW C W.

LEMMA 5.15

...,

S u p p o s e t h a t t h e h y p o t h e s i s of

Theorem 5.11 a r e sa-

t i s f i e d . T h e n , f o r e a c h f E CVm(X;E), v E V, and p E cs(E), we 4 E W} = SUp{inf{I ( f Y-glY/ !vlp;4EW};

have d = inf{ 1 If-ql

Y

IvIp;

E PAL

PROOF Clearly, c < d, where we have defined c = sup{inf{llfl~ - q l ~ l l ; u E w}; Y E PA VIP

.

To prove

reverse inequality, let E > 0 . For each Y E P A , there q y E w such that v(x)p(f (x) - qy (XI 1 < c + ~ / 3for all

the exists x E Y.

Therefore, f E L(W;v,p,c + ~ / 3 ) . By Lemma 5.13, applied X = c + ~ / 3and ~ / 3 ,there exist bl,...,bm E B and

.-

ql,.

G(W) such that m I If - 1 bigil I < (c + E / 3 ) i=l VIP

with

E

nr'

+

E/3.

there are By Lemma 5.14, applied with 6 = ~ / 3 m , wl,...,w m E W such that 1 Iwi - b.q.1 < ~/3m.From this it 1 1 < c + E , where q = w1 +...+ wm Since follows that 1 1 f - q 1 l v I p

IvIp

.

+

q E W, d < c

E.

Since

E

> 0 was arbitrary, d < c, as desired.

PROOF OF THEOREM 5.11 Let f E CVm(X;E), v E V, and p E cs(E) be qiven. Let Z be the quotient space of X by the equivalence relation X l A , and let n : X + Z be the quotient map. By Lemma 1 of [46j the map -1 -1 2

E

z

-+

' lfln

(2)

-

qln

( 2 )I 1

VIP

Id E I G H T E D A P P R O X I MAT I 0 N

is upper semicontinuous and null at infinity on Z, W. Hence the map defined by -1 -1 h(z) = inf {Ijfln ( z ) - q l n ( z ) I IVrp;

87

for

each

g E

CJ

E W}

for all z E 2 , is upper semicontinuous and null at infinity on Z too. Therefore h attains its supremum on Z at some point z . -1 Consider the equivalence class Y = IT ( z ) modulo XIA. On the other hand, the supremum of the map h is by Lemma 5.15 equal to d. Thus, we have found an equivalence class Y C X modulo XIA such that inf {IIf-gl !v,p; q

E

W} = inf { I \fly - g ~ Y ~ ~ v ,q pE ;w}.

By the remark made before Definition 5.12 the module W is sharp-

ly localizable under A in CVm(X;E). THEOREM 5.16 S u p p o s e t h a t t h e r e e x i s t s e t s of g e n e r a t o r s G(A) and G(W), f o r A and W r e s p e c t i v e l y , s u c h t h a t : G(A) is a s t r o n g s e t of g e n e r a t o r s f o r A; (2) g i v e n a n y v E V, p E cs(E), al,. ,an E G(A) a n d q E G(W), t h e r e e x i s t s w E RS such that n v(x)p(q(x)) 5 w(lal(x) Ir...,lan(x)I ) f o r alZ x E x. T h e n W i s s h a r p l y l o c a Z i z a b l e u n d e r A i n CVm(X;E). (1)

..

PROOF

u

E

Let f E CVm(X;E), v E V, and p E cs(E) be qiven. Let Assume that for each T > u we have found an element P l such that

6.

Sl E

(a) S T C (b)

S

u

for all

infII If-gl

IVrp;

p < T;

g E W} = infh If's - g ! S 1 1 : T T V,P

9 E WI.

FIRST CASE. u = T +thesis there is S E 1 be the subalqebra of Theorem 5.11 applied there is a set S

G.

1 for some T E By the induction hypoP T such that (a) and (b) are true. Let A T all a E A such that a \ S T is real-va1ued.B~ to the alqebra AllSl and the module WlS, such that - Pl+l


88

inf I If g EW

IS -

On the other hand, SECOND CASE

IVrp

sume that inf {

=

G has

Define no predecessor. P, and S,C S , for all T < 0. As-

E

(7

< u } . Then S u E

I If IS, -

fined d = inf { I If

-

q

~

q IS

IVrp:

inf 1 If I S - 9 ' S I V*P qEW

by construction.

S a c S,,

. The ordinal

fl {ST; 1

S , =

q IS I

IVrp;

I

q E W } < dr where we have de-

q E W

There exists q E W such that

I

If

1 . (The case d

IS,

-

g

Isn I

= 0 is trivial).

IVrp
0 (one says then that V > 0 on X).

'

EXAMPLE 5.23 For any subset S c X, the characteristic function of S will be denoted by xs. Let xf(X) = {AxF; X > 0 and F c X,F finite}. Then V = xf(X) is directed set of weiqhts on X and FVb(X,E) = FVm(X,E) = F(X,E). The topoloqy wv in this case is the topology w of pointwise convergence. Let xc(X) = { A x y ; X > 0 and K c X I K compact). EXAMPLE 5.24 Then V = xc (X) is a directed set of weiqhts on X and FVb ( X , E ) = FVm(X,E). The topology wv in this case is the topology K of compact convergence, determined by the family of seminorms 1lfll = KIP s u p {p(f(x)); x E K), when K ranqes over all compact subsets of X, and p ranges over all continuous seminorms on E.

92

WEIGHTED

EXAMPLE 5 . 2 5

L e t K+(X)

APPROXIMATION

be t h e set of a l l c o n s t a n t

func-

0

-I-

i s a d i r e c t e d s e t of w e i q h t s and FVb(X,E) = B ( X , E ) , FVm(XrE) = B o ( X , E ) . The t o p o l o q y t i o n s o n X . Then V = K ( X )

t h i s case i s t h e t o p o l o q y u o f u n i f o r m c o n v e r q e n c e , by t h e f a m i l y o f seminorms 1 l f l

IP

on

X

wv

in

determined

= sup { p ( f ( x ) ) ; x E X I ,

when

p r a n q e s o v e r a l l c o n t i n u o u s seminorms o n E . I f V i s a d i r e c t e d s e t of w e i q h t s on X , c l e a r l y a n d C V m ( X , E ) a r e t h e i n t e r s e c t i o n s FVb(X,E) n C ( X , E ) a n d

CV ( X , E )

b FV,(X,E)

CI

C ( X , E ) r e s p e c t i v e l y . T h o s e spaces are e q u i p p e d

When V = x f ( X ) ,

t h e t o p o l o g y i n d u c e d by wv.

C ( X , E ) e q u i p p e d w i t h t h e t o p o l o s y w of p o i n t w i s e

When V = x c ( X ) , topology Cb(X,E)

CVb(X,E)

= CVm(X,E) = C ( X , E )

of c o m p a c t c o n v e r q e n c e . When V =

K

and C V m ( X r E ) = C o ( X , E )

with

=CVm(XrE)=

CVb(X,E)

converqence. the

equipped w i t h

+ K (X),

=

CVb(X,E)

both equipped w i t h t h e

topolocy

u of u n i f o r m c o n v e r g e n c e . DEFINITION 5 . 2 6

(SUmnerS)

o n X and f o r e v e r y u we w r i t e U

5

V.

E

If

U an V

are directed s e t s

U t h e r e is a v E V s u c h t h a t u

I n case U

5

V and V

5

U,

weights

of

5

then

v,

v e w r i t e U = V.

L e t U be a d i r e c t e d s e t o f w e i g h t s on X and @:X

a mapping. T h e n , f o r e v e r y V o n X s u c h t h a t U < V o

@

the

-P

X

map-

p i n g f * f o @ i s a c o n t i n u o u s l i n e a r mapping f r o m F V b ( X , E ) i n t o v E V F U b ( X , E ) . I n d e e d , g i v e n f E FVb(X,E) a n d u E U , c h o o s e such t h a t u

5

v o @,

T h e n , f o r a n y c o n t i n u o u s seminorm p

on E ,

w e have

Hence t o g e t a c o n t i n u o u s l i n e a r mappinq f r o m

the

s p a c e FVm(XrE) i n t o FUm(XrE) i t i s s u f f i c i e n t t o assume t h a t f o r e v e r y compact s u b s e t K

c

X t h e i n t e r s e c t i o n of @

-1

(K) w i t h

s u p p o r t o f a n y u E U i s c o m p a c t . I n d e e d , i f f E FVm(X,E) u E U , choose v E V such t h a t u < v o @ . W e know t h a t f E F U b ( X , E ) . NOW, q i v e n a n y c o n t i n u o u s seminorm p o n E

the and and

0 , t h e r e e x i s t s a compact s u b s e t K c X such t h a t v ( x ) p ( f ( x ) ) -1 f o r a l l x 6 K . L e t K ' be t h e i n t e r s e c t i o n of @ (K) with -1 t h e s u p p o r t o f u . L e t x 6 K ' . Then e i t h e r x k @ (K) or x is -1 Then (K). n o t i n t h e s u p p o r t of u. Suppose f i r s t t h a t x 6 b E

7

0 be such that llgl1 < M for all q E S,and K C X P be a compact subset of X such that v(x) < E / ~ ( M + I If I for all x I$ K. Let g E S be such that 1 If-9'1 < E/2 IvI', where KrP I IvI I = sup {v(x); x E XI. Then

IP)

1

If-gll

: ' . Y i , U ( Q i )

=

>

(2)

i s c o n t i n u o u s i n t h e t o p o l o g y i n d u c e d by C ( X , E ) , i n w h i c h case 0

it can be uniquely continuously extended t o C o ( X , E ) ,

K(X)

@ E i s dense i n C

0

(X,E).

I n order t o characterize the dual

o f C ( X , E ) as a s e t o f E l - v a l u e d bounded 0

since

W

Radon m e a s u r e s

w e h a v e f o f i n d n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s measures t o be i n t e g r a l . I f L E C o ( X , E ) '

for

such

to

l e t us r e t u r n

the

( 1 ) . The transpose

E l - v a l u e d b o u n d e d Radon m e a s u r e u d e f i n e d y by W

o f u i s t h e n a l i n e a r map f r o m E i n t o M b ( X ) ,

u'

on X ,

t h e s p a c e o f all y F E t h e r e cor-

b o u n d e d Radon m e a s u r e s o n X . H e n c e , . f o r e v e r y responds a unique r e g u l a r Borel measure < u ' ( y ), x B > ,

II such t h a t 1) ( B ) = Y Y f o r a l l B o r e l s u b s e t s B of X . S i n c e L i s c o n t i n u o u s

t h e r e e x i s t s a c o n t i n u o u s seminorm p on E and a c o n s t a n t such t h a t

IL(f)

1 5

I IP

kl If

/ < y , u ( g )>

I

=

for all f

IL($ 8 y)

E

1 1. k

Hence

Co(X,E).

p ( y ) 11(1)I

I

T h e r e f o r e , t h e b o u n d e d Radon m e a s u r e u' ( y ) h a s norm k p ( y ) , and t h e c o r r e s p o n d i n g B o r e l measure

I Ily(B) I 5 I

ILlyI

I 5k

P(Y)

-t

B

-t

Y

(B) belongs t o E ' .

ri(B)

,

Y

II

< -

is such t h a t

c

the

X,

C a l l t h i s map k j ( B ) . The s e t

map function

of a l l B o r e l subsets of X

d e f i n e d o n t h e O-ring

w i t h values on E '

LJ

Iu' ( y )

-

T h i s shows t h a t , f o r a f i x e d B o r e l s u b s e t B y

k > 0

and

i s t h e n c o u n t a b l y a d d i t i v e . I n d e e d , i f {BnI i s

a c o u n t a b l e f a m i l y o f d i s j o i n t B o r e l subsets o f X a n d B d e n o t e s i t s union, t h e n f o r an a r b i t r a r y y

-

E E

w e have

m

-

CY,

11

(Bn)

m

T h i s shows t h a t I I ( B ) =InZl l l ( B n ) i n t h e s e n s e o f E ' .

For

W

any

o f d i s j o i n t Borel s u b s e t s o f X I whose f i n i t e families {B } , i 1 ~ 1 1 f o r each u n i o n i s X , a n d { y i l i E I o f e l e m e n t s o f E w i t h p ( y1. ) :
0 i s s a i d t o h a v e

finite

p-.T n m I: 7)ar.l:a t i o n . On t h e o t h e r h a n d , f o l l o w i n g J . Dieudonng, t h g o r s m e d e Lebesgue-Nikodym.

17,

Canad. J . Math.

SUr

le

3 (1951), 129-

1 3 9 , an E l - v a l u e d bounded Radon m e a s u r e o n X i s s a i d t o b e p-dcmW

inated

u

i f t h e r e i s a p o s i t i v e bounded Radon m e a s u r e

on X

such

that I < y , u ( @ ) >2 l u(l@l)p(y) f o r a l l y E E and @

(4)

€ K(X).

The a r g u m e n t s c o n t a i n e d i n I. Sing=,

sm

lesappfications

l i n s a i r e s i n t 6 g r a l e s d e s e s p a c e s de f o n c t i o n s c o n t i n u e s . I , R e v . Math. P u r e s A p p l . 4

( 1 9 5 9 ) , 391-401,

and N .

P. C k , Lineartrans-

f o r m a t i o n s o n some f u n c t i o n a l s p a c e s , P r o c . London Math. Soc.(3) 1 6 ( 1 9 6 6 ) , 705-736,

f o r Banach s p a c e s E c a n b e e x t e n d e d t o p r o v e

t h e following. THEOREM 5 . 3 8

L e t u b e a n E;-vaZued

b o u n d e d R a d o n mcsasurc o n X.

Thcn t h c f o l l o w i n g a r e e q u i v a l e n t : (a)

u is i n t e g r a l ,

(b)

u is p - d o m i n a t e d ,

(c)

P on E l u has f i n i t e p-semivariation,

for some c o n t i n u o u s

seminorm

f o r some c o n t i n u -

ous seminorm p on E . W e d e n o t e by b$,(X,E')

t h e s e t of a l l Z ' - v a l u e d boundW

e d Radon m e a s u r e s o n X w h i c h s a t i s f y ( a ) o r (b) o r ( c ) . COROLLARY 5 . 3 9

(1) a n d

T!ie

THEORErl 5 . 4 3 and

The c o r r e s p o n d e n c e L

f--t

u s(7t

a r g u m e n t s i n Wells [ G 7 ] T h i , corrc:;pon&,ic-c-

)

'

forniuIn::

and P k ( X , E ' ) .

show t h a t

L + + u s P t up b!

( 2 ) i:; a v e c t o r isomorphism bctwpen Cb(X,E) , B )

%(X,E'

hi{

up

( 2 ) i s u v e c t o r i s o m o r p h i s m b e t w e e n Co(X,C)

formu%u::

'

(1)

trnd

. W e a p p l y t h e above r e s u l t s t o c h a r a c t e r i z e

of t h e N a c h b i n s p a c e C V m ( X I E ) f o r V i n a

t h e dual

certain interval

of

d i r e c t e d s e t s o f w e i g h t s , f o l l o w i n g t h e s a m e p a t h a s W.H.Summer~ A r e p r e s e n t a t i o n theorem f o r b i e q u i continuous completed

tensor

WE I GH T E D A P P R O X I N A T I 0 N

99

p r o d u c t s o f w e i g h t e d s p a c e s , T r a n s . M a t h . SOC. 146 ( 1 9 6 9 ) , 1 2 1 131. THEOREM 5 . 4 1

+

Co(X)

5

1,i~f V l>cj

V C B(X).

11

( ~ , L I > ~ ~ C ~ , ' :C: P. ! L

T / i i > t ~L i i c

(1) iiizd ( 2 ) i s

I N U ~ ~ : :

~

~

~

o f ~ < ' i g / / t :O:

~

,

L

X

with

U p ~/Iy

~f O i l - ~

~ I

U ~ S L ' L~

+t P

w o i o r ~~ ~ ~ o r n o r p h i sh ri 2rt~w c 1 ' n CVm ( X , E )

ti

'

arid

%(X,E').

PROOF

L e t L t CVm(X,E)

S i n c e V C B(X) , i t f o l l o w s f r o m P r o -

I .

c

p o s i t i o n 5.28 t h a t Co(X,E)

C V m ( X , E ) and t h e t o p o l o g y

by w v i s w e a k e r t h a n t h e u n i f o r m t o p o l o g y

0.

Hence t h e r e s t r i c -

t i o n of L t o C o ( X , E ) , s a y M , b e l o n g s t o C ( X , E ) ' . 0

y ) = L(@ 8 y )

< y , u ( @ ) >= M ( d , 8 F

E,

to

According

i f w e d e f i n e u ( @ ) f o r e a c h I$ E K(X) by

Corollary 5.39,

for all y

induced

(1)

t h e n u E %(X,E').

Conversely, l e t u F % ( X , E ' ) .

By Theorem 5 . 4 0 ,

if we

d e f i n e L o v e r K(X) 8 E b y L(C Qi

8 yi)

=

c

(2)

t h e n L c a n be e x t e n d e d u n i q u e l y t o a B - c o n t i n u o u s t i o n a l over C b ( X , E ) .

+

Since Co(X)

s i t i o n 5.28 t h a t CVm(XfE)

c

d u c e d by D i s w e a k e r t h a n

i*i

C

V'

b

5

V,

linear

func-

i t follows f r o m

Propo-

(X,E) a n d t h a t Eie t o p o l o g y Hence L c a n b e e x t e n d e d

in-

uniquely

t o a n w V- c o n t i n u o u s l i n e a r f u n c t i o n a l o v e r C V m ( X f E ) . The r e s p o n d e n c e s e t up by (1) a n d ( 2 ) i s o b v i o u s l y o n e - t o - o n e

corand

l i n e a r . This ends t h e proof. W e t u r n now t o t h e g e n e r a l c a s e of a r b i t r a r y N a c h b i n spaces. Consider E'-valued W

Radon m e a s u r e s u o n X ,

o u s l i n e a r m a p p i n g s u f r o m K(X) i n t o E ' inductive l i m i t topology. x

0

For every x

W' E

i .e.,

continu-

when K(X) h a s its usual E = (E')' W

the

mapping

u i s a n u m e r i c a l Radon m e a s u r e d e f i n e d by =

<X,U(@)>

A complex o r e x t e n d e d r e a l - v a l u e d

function

f

i s s a i d t o be i n t e g r a b l e f o r u i f f o r e v e r y x E E i t i s i n t e g r a b l e f o r x o u , i n w h i c h case u ( f ) i s t h a t e l e n e n t o f E*= ((E;)')* f o r which

~

,


100

<x,u(f)>=

f d(x

u)

(1

I X

for all x

E.

E

Similarly, we say t h a t a function f is locally integrable f o r u i f f o r every x x

0

F: E

it is locally integrable

for

u.

V a n d u F M (X,E'). S i n c e v i s l o c a l l y i n t e g r a b l e

Let v

PROOF

b

f o r u, a n d t h e r e f o r e v u ( a ) = u ( v

a)

f

T* f o r a l l @

F

K(X),

let

us d e f i n e a l i n e a r f u n c t i o n a l o n K ( X ) B E by Lo: @ .

1

B x . ) = i:<x. , v u ( @ i ) > . 1

1

T h e r e e x i s t s a c o n t i n u o u s seminorm p on E a n d a posi-

t i v e bounded Radon measure p on X s u c h / < x , u ( @ ) >5 I for all x

I J ( I @ O. P ( X )

. Let

E and @ E K ( X )

E

that

L1(X,ji)

(4) be t h e space

of

all

f u n c t i o n s , w i t h i t s u s u a l L - s e m i n a r m and L1(X,u,E 1 P t h e s p a c e o f a l l E - v a l u e d f u n c t i o n s which a r e u-integrable w i t h P t h e seminorm 11-integrable

I If1 Il where E

P

f ),

11(p

d e n o t e s E endowed w i t h t h e s e m i n o r m p o n l y . By ( 4 )

I

p i n g t from L 1 ( X , u ) on L 1 ( X , v )

=

Q E

w e c a n e x t e n d u t o a c o n t i n u o u s l i n e a r mapi n t o E ' C E ' . Define a l i n e a r functional T P

by

ui

T(C

Q x.) = 1

1 < xi , t ( u i ) >

For t h e s t e p functions f = C

IJJ i

Q

x . where t h e I I ~ ' s 1

are c h a r a c t e r i s t i c f u n c t i o n s o f p a i r w i s e d i s j o i n t Bore1 s u b s e t s B . o f X I w e have 1

I T ( f ) I = IC < xi , t ( $ i ) > l

5

C[<xi,t(lQi)>l

for all x

p

0

t>

O b v i o u s l y u ( @ ) E E*. S i n c e

E.

E

<w 4, x

w $, x

'1,

t>l =

11

I IQl

=

(I

E

=

i.e.,

+

+

v u = v'u'

v"u"

+

<x,v' 'u'

F: Vt$,(X,E')

'

( $ 8 jo

($)>

a n d O ( L ' + L")

= @(L') +

The p r o o f t h a t O ( X L ) = X O ( L ) i s t r i v i a l .

O(L").

T h i s e n d s t h e p r o o f of T h e o r e m 5 . 4 2 . Let

each

v

F: V

V

b e a d i r e c t e d s e t oi' w e i g h t s o n

+

X

Such

i s c o n t i n u o u s , i . e . , V C C (X) = {f E C ( X ; R): f Let

that

2

01.

u E n b ( X ; E). T h e r e i s a c o n t i n u o u s s e m i n o r m

p

W E I GH T E D A P P R O X I M A T I 0 N

103

on E , and a positive and bounded Radon measure li

on

X

such

that

for all y

F

E and q5

defined on K(X) and then

F

K(X) . Since

u

E

E4b (X , E ' )

, then

E can be extended to an element of

Q

C o ( X , E)'

+

K (X) can be extended to be a positive and bounded + Radon measure on X , which will be the least !-I E Mb(X) satisfying (1). By definition,

defined on

v F V is continuous, the operator

Since by our hypothesis any Tv(f)

=

!Ip

vf naps CV_(X,E) into Co(X,E). Let D

1;

g

-

-1 Tv ( DP ) . Let

-

$

th

T* denote the transpose map of V

v c c+ (X).

PROOF:

= {g E Co(X, El;

. Then Tv .

1 1 , if p is a continuous seminorm on E

PROPOSITION 5.43: i'li

P

Let T//t

The operator

V h c a d i r t 7 c t p d s c t of wi.ight::

Dv

on

=

IP

X

Y/

Tv

is a continuous

linear

map from

CVm(V,E) into Co(X,E), and therefore continuous in the weak topologies. Hence TC is continuous in the corresponding weak *topologies. N o w Do is weal:

P

therefore T;(Do)

P

*-

is also weak

compact by Alaoglu's Theorem,and

*-

compact. Since

solutely convex, it follows from the

Bi

- polar

T;(Do)

P

is ab-

Theorem that

IJ E I GH T E D A P P R O X I M A T I 0N

104

? Tv(D. * 01,1 0 0 . On the other hand,

Tc(Do)

-1

0 70

r

00

Tv (D ) = L TC(D P P and DOo = D (the last inequality follows again bythe Bipolar P P Theorem). Hence = 1-

P

P

COROLLARY 5.45:

PROOF: IIu

!Ip

If p is a continuous seminorm on E,then D

THEOREM 5.46:

#

0

{u t: Mb(X,E');

=

P 2 1 1 follows easily from Corollary 5.39. BY

5.43 above,

L

Tf V C C+(X), t h c n

0

Proposition

D Z f p = Tc(Di); while it is clear that

I,i,i>t W

of CV,(X,E)

b e n v e c t o r .slnhspacc'

b u a n ~ s t r ~ pfo ni n~t o f W

1

0

()

Dv,p

.

if f o r

nnd l e t

g E C(X) the

r f l < : t r 7 ' ( , / i o nof 9 t o the. . s u p p o r t of L i : bciundcd a n d real-valued,

L(gw) = 0 . : u p p ( i r t of L.

wh?'ii

PROOF:

Let

for Qvcry

u

lip

E \J,

thcn

L # 0 be an extreme point of

ollary 5.45 above, L

11

w

g

i s ron.:tant

W 1 (1 Do

VrP

.

on the

By Cor-

vu, where u E Mb(X,E') is such that 2 1. Since L is extreme, it follows that llu i l p = 1.We =

may assume without l o s s of generality that u and vu have the c 1 on the support of u. Let e = gu. same support and that 0 5 g = 5 l@(x),for all x f X and @ E I d C

c o r n p a r t s u p p o r t a n d n n f i d ~ n t i c a 7 7 yz c r o . a71

n E N

PROOF:

.

Thcn

Since

I#I

yM

f u n c t i o n on R w i t h

P u t Mn =

I/

@(n) i I f o r 2

is n o t fundamental.

C(M), the class

F

is not fun-

yM

C(M) is not quasi-analytic

and the conclusion follows from Theorem 2. REMARK 7: The above corollary piovides a simple counterexample to localizability (see 5 3 1 of Nachbin

1 . Notice also that, in this case, is a continuous and positive function by ReyM mark 6 . COROLLAIZY 2: and

L e t w hi: a w i i i g h t o n

p(IR) i:; n o t d e n s c .

thc ciass

PROOF:

t

E

IR s u c h t h a t ?(El)

I/

Let 14n = wtnii, f o r a l l n C(M) is n o t q u a s i - a n a l y t i c .

Since

w

5

y

and Remark 1. LEMPI1A 3 :

[43]

For

M

IR. T h c n w

P4 '

C Cwmm) E

IN. T h c n

the conclusion follows from Theorem

Sized, L e t

u ( t ) = (1 + ; t i ) y M ( t )

is f u n d a m e n t a l

i f , a n d o n Z y if,

yM

for>

2

nil

is fundcinii7n-

tul.

,

in view of Kemark 1 it is enough to prove

PROOF:

Since

that

is fundamental when y

w

be defined by

'M

: 4 P

< w =

Mn+l

M

is fundamental. In fact,let M '

for all

n t: N . for

t # 0, we have


116 for ‘1)

It; I_ 1. so, there exists a positive constant C such that

‘ C yp4, -

.

Since y

is fundamental, it follows from Theorem2

M

that C ( M ) is quasi - analytic. Then

C ( P 4 ’ ) is a quasi - analytic

class, whence y M , is fundamental by Theorem 2. So,from Remark

1, we conclude that

w

is fundamental.

Put Tp4(t) = sup {

PROOF:

Let

01

E N l

for all

be as in Lemma 3. From this and Theorem

have that the class

t f IR.

2 , we

C ( I 1 ) is quasi- analytic if, and Only

T,,

is fundamental. Since rem 3 follows from Theorem 1. ili

Nil, n

=

{P E

if,

(PI 5 TrIl, Theo-

117


REFERENCES FOR CHAPTER 5.

[ll]

BUCK

GLICKSBERG

[2 6 1

KLEINSTUCK

[ 351

MACHADO and PROLLA

,

,

,

[40]

NACHBIN

[42]

NACHBIN,

:LACHADO and PROLLA

PROLLA

[SO]

[43J

1391

, [51]

PROLLA and MACHADO SUMMERS TODD

[64]

1651

WELLS

ZAPATA

r 6 q

[68]

[52:

,

[451 1461

[41]

C H A P T E R

THE SPACE C o ( X ; E )

Let

v: X

IR

-+

s p a c e CVJX;E) space

E

X

6

W I T H THE UNIFORM TOPOLOGY

be a Hausdorff space. I f

V = { vl

,

i s t h e c o n s t a n t f u n c t i o n 1, t h e n the is t h e space Co(X;E) , f o r each l o c a l l y

(see Example

5.2,

where Nachbin convex

is

the

suba;’GcDra, a c a

let

Chapter 5.) Its topology

t o p o l o g y o f u n i f o r m c o n v e r g e n c e on X.

under

be a n

a

be

Cb(X;M)

W i s sharply localizabk

Then

A-submodule.

i n Co(X;E).

A

PROOF.

c

Let A

THEOREM 6 . 1 . W C Co(X;E)

Since

A C Cb(X;X),

we can apply C o r o l l a r y 5.21,

5

3,

Chapter 5.

Let

COROLLARY 6 . 2 .

that

A

i s self-adjoint.

A

and

b e a s i n t h e o r e m 6.1..

W

Then

Assume

i s l o c a l i z a b l e under

W

in

A

Co(X;E).

PROOF.

Since

is self-adjoint

A

a s t r o n g set of g e n e r a t o r s f o r

5

of theorem 5.20,

,

sume t h a t

if,and

Let

PROOF.

Assume

T h e n K ( X ; IK) 8 E Since

self-adjoint =

and

X

that

Then

X

f o r a l l x E X.

i s dense i n

W

E

,

’

PA

.

Co(

.

As-

X;E

)

f o r e a c h x E X.

is d e n s e i n C o ( X ; E ) .

i s l o c a l l y compact

E is a

=

i s a S o c a Z l y compact I k u r d o r f f

subalgebra of C b ( X ; I K ) .

K(X;M) Q

P2

b e a s i n Corollary 6 . 2

W

o n l y i f , W(x) i s d e n s e i n

COROLLARY 6 . 4 .

space.

A

i s separatin?.

A

Tm A i s

hypothesis

3 , C h a p t e r 5 . On t h e o t h e r h a n d , s i n c e G ( A )

c o n s i s t s o n l y of r e a l v a l u e d f u n c t i o n s , p = 2 and Thus W i s l o c a l i z a b l e u n d e r A i n C o ( X ; E ) . COROLLARY 6 . 3 .

u

then G(A) = R e A

satisfying the

A

K(X;M)

-

K ( X ; E )is a s e p a r a t i n g The v e c t o r

subspace

module s u c h t h a t W ( X )

I t r e m a i n s co a p p l y C o r o l l a r y 6 . 3 a b o v e .

W =

= E

r

! J I T H T H E UNIFORri TOPOLOGY

C,(X;E)

DEFINITION 6 . 5 . Let W C Co(X;E) he a The S t o n e - W e i e r s t r a s s h u l l o f W i n C o ( X ; E )

119

vector

subspace

, d e n o t e d by

Ao(W),

A ( W ) n Co(X;E).

i s the s e t

(For t h e d e f i n i t i o n of

A(W)

, see

Definition 4.12.

5

The . a r g u m e n t s used i n t h e proof of Lemma 4 . 1 6 ,

2,

C h a p t e r 4 , show t h a t A

0

= L (A Q E ) = LA(W) A

(W)

when E i s a l o c a l l y convex Hausdorff space; and W C C o ( X ; E ) a v e c t o r space i n v a r i a n t u n d e r c o m p o s i t i o n w i t h e l e m e n t s

is of

E' B E.

L e t W C C o ( X ; E ) b e a v e c t o r s u b s p a c e . We say i s a S t o n e W e i e s t r a s s s u b s p a c e i f A o ( W ) C where the that W b a r d e n o t e s t h e u n i f o r m closure of W i n C o ( X ; E ) . DEFINITION 6 . 6 .

w,

(Stone Weierstrass)

THEOREM 6 . 7 .

Suppose

v e x Hausdorff space. Every se l f - a d j o i n t

i s a l o c a l l y con-

polynomial

s u b s p a c e , i.e.

i s a Stone-Weierstrass

W C Co(X;E)

E

algebra

f o r every f E

f b e l o n g s t o t h e u n i f o r m CZosure of W i n Co(X;E)

E Co(X;E),

if,

and o n l y i f :

(1) for g E W

such t h a t

any x E X,

such t h a t f ( x ) # 0

PROOF. A-module

is

with f ( x ) # f ( y ) ,

there i s g E W

g(y).

B y t h e previous r e m a r k s , A ( W )

o 9; a E E l , g E W}.

where A =

there

g(x) # 0;

( 2 1 for any x , y E X, If(x) #

such t h a t

,

0

= LA(W) = LA(A 0 E )

-

By C o r o l l a r y 6 . 2 applied t o t k

A 0 E , w e have L A ( A 63 E ) = A Q E . S i n c e

m i a l algebra, A Q E C

r.Hence -

T h e converse

,

Ao(W)

W C Ao(W)

C

W

i s polyno-

7.

is true,

whenever

E

is

Hausdorff.

Suppose E i s H a u s d o r f f . L e t W C CO(X;E) be a s e l f - a d j o i n t p o l y n o m i a l a l g e b r a . Then W i s d e n s e i f and o n l y i f , W i s s e p a r a t i n g and e v e r y - w h e r e d i f f e r e n t f r o m z e r o .

COROLLARY 6 . 8 .

THEOREM 6 . 9 .

Suppose

E

i s a l o c a l l y convex Hausdorff space.

120

Co(X;E)

c

Let W

W I T H T H E UNIFORI.1 TOPOLOGY

= { a o f;

under

be a v e c t o r subspace which i s i n v a r i a n t

Co(X;E)

c o m p o s i t i o n w i t h e l e m a n t s of E ' g E a E E',

f E W).

, and

let

A =

The f o l l o w i n g c o n d i t i o n s a r e

equiv-

alent: (1) W i s l o c a l i z a b l e under A i n Co(X;E). W i s a S t o n e - Weierstrass s u b s p a c e .

(2)

( 3 ) A i s a S t o n e - Weierstrass

subspace.

.

B y p r e v i o u s r e m a r k , A ( W ) = LA(W) Hence (1) and 0 are e q u i v a l e n t . A s s u m e ( 2 ) , and let f E Co(X) be an e l e m e n t

PROOF.

(2)

of

L e t E > O be g i v e n . C h o o s e a E E ' w i t h a f 0 , and V E E , a ( v ) = 1. L e t g = f 8 v. O b v i o u s l y g E Ao(W). B y hypothesis, g € L e t p E c s ( E ) be w i c h t h a t Ia(t) I < p(t) ,

Ao(A). with

r.

-

f o r a l l t E E . L e t h E W be chosen so t h a t p ( g ( x ) h ( x ) 1 < E f o r a l l x E X. Hence I f ( x ) - ( a o h) ( X I \ < E f o r a l l x E X. S i n c e a o h E A , f E i,and therefore A i s a Stone-Weierstrass subspqce. F i n a l l y , assume ( 3 ) . S i n c e = Ao(A) , A is a closed

s

s e l f - a d j o i n t subalgebra of C o ( X ) . Indeed Ao(A) = A(A) n C o ( x ) , and by P r o p o s i t i o n 4.15, 5 2 , C h a p t e r 4 , h ( A ) i s a self-adjoint subalgebra of C ( X ) L e t B = .By C o r o l l a r y 6 . 2 a p p l i e d to B Q E , w e have L B ( B 8 E ) = B Q E . Hence LA(W) = t h e B-module - - because = LA(A Q E) C LA(B 8 E ) = LB(B Q E) = B Q E C A Q E ,

.

A 0 E C

m.B y

therefore LA(W) Let

C

Y

Lemma 4.1,

v,

5

1, C h a p t e r 4 , A Q E C W ,

.

and

w h i c h proves (1)

be a & a b e d

c o m p a c t H a u s d o r f f space X. T h e n

s u b s e t of a

locally

is also a locally

Compact

non-empty Y

then the restriction f l Y belongs t o C o ( Y ; E ) L e t us c a l l Ty the r e s t r i c t i o n o f T y : C ( X ; E ) + + C(Y;E) t o t h e subspace C o ( X ; E ) . T h e n Ty: C o ( X ; E ) + Co(Y;E)

Hausdorff space, and i f f E Co(X;E)

i s a c o n t i n u o u s l i n e a r map. LEMMA 6.10.

map

Ty:

PROOF.

For any c l o s e d non-empty s u b s e t Y C X , t h e l i n e a r + C (Y;E) i s a t o p o l o g i c a l homomorphism. 0

Co(X;E)

The s a m e proof of Lemma 3 . 2 .

a p p l i e s . Indeed, t h e s e t

C,(X;E)

WITH THE

121

UNIFORM TOPOLOGY

F = { x E X; p ( g ( x ) ) 1. E } i s t h e n compact and d i s j o i n t f r o m Y. THEOREM 6.11. L e t Y be a c l o s e d non-empty s u b s e t of a l o c a l l y compact Hausdorff s p a c e X, and l e t E b e a n o n - z e r o F r i c h e t s p a c e . Then Co(X;E) l y = Co(Y;E).

I

PROOF. L e t w = c ~ ( x ; E ) y. S i n c e x i s locally compact, Co(X) is s e p a r a t i n g and everywhere d i f f e r e n t from z e r o . Hence t h e s a m e i s t r u e o f Co(X) (P E C Co(X;E). Taking r e s t r i c t i o n s t o Y and a p p l y i n g C o r o l l a r y 6.3 (or C o r o l l a r y 6.4, s i n c e K(X) C W Co(X)), w e see t h a t W i s d e n s e i n Co(Y;E). W e claim t h a t i s c l o s e d i n Co(Y;E). L e t M be the K e r n e l of the map Ty i n Co(X;E) S i n c e Ty i s ' c o n t i n u o u s , M i s c l o s e d . The s p a c e Co(X;E) i s a FrGchet space, b e c a u s e E i s F r g c h e t . The q u o t i e n t of a F r g c h e t space by a c l o s e d s u b s p a c e i s a F r g c h e t s p a c e . T h e r e f o r e Co(X;E)/M is complete. By Lemma 6.10, Co(X;E)/M and Ty(Co(X;E)) = W are l i n e a r l y t o p o l o g i c a l l y i s o m o r p h i c . Hence W i s complete too, and t h e r e f o r e c l o s e d i n Co(X;E).

.

FQ3MARK 6.12.

can choose

When E is a Banach s p a c e , and f E CO(Y;E)I g E Co(X;E), glY = f , such t h a t I I f 1 l y = 1141

Ix-

we

REMARK 6.13. L e t u s now c o n s i d e r a p a r t i c u l a r case of v e c t o r f i b r a t i o n s . Namely, w e w i l l c o n s i d e r vector spaces L of c r o s s sections s a t i s f y i n g the following conditions: (1) X is a l o c a l l y compact Hausdorff s p a c e ; (2) e a c h E x i s a normect s p a c e , whose norm w e

by t

+

II

t

denote

II

f € L, t h e f u n c t i o n x -c is upper semi-continuous and v a n i s h e s a t i n f i n i t y on X.

( 3 ) f o r e a c h cross-section -t

I

If (x)

II

I n t h e language o f [39],

w e s a y i n t h i s case t h a t L is a Nachbin s p a c e o f cross-sections, and endow it w i t h t h e topology of t h e norm

IlflI !

If

(x)

=

SUP

{I

The above s u p is f i n i t e , is campact and t h e map

I I 211

; x E XI.

the f (XI

I

set { x i s upper

E

X; semi-

122

Co( X;E)

W I T H THE

UNIFORM TOPOLOGY

continuous. W e have t h e r t h e f o l l o w i n g s t r o n g f o r m o f t h e StoneWeierstrass Theorem. Let

THEOREM 6 . 1 4 .

be a Nachbin s p a c e of c r o s s - s e c t i o n s

L

8a-

t i s f y i n g c o n d i t i o n s ( 1 ) - ( 3 ) o f Remark 6 . 1 3 and assume t h a t L and -module. Then, for e v e r y CblX)-submodule W C L i s a Cb(X) e v e r y f E L , tle h a v e . d = inf

IIf

-

g / I = sup

inf

XEX

(JEW

gEW

-

IIf(x)

g(x) 1

1

= c.

PROOF.

Clearly, c < d . To prove t h e r e v e r s e i n e q u a l i t y , let E > 0 . Oor each x E X , t h e r e e x i s t s wX E W such t h a t I I f (X) - wx(x) 1 1 < c + E / 2 . L e t ux = { t E x; I I f ( t ) w x ( t l \ l < C+E/21 Then Ux i s an open s u b s e t of X , c o n t a i n i n g t h e p o i n t x , and i t s complement i n X i s compact. By Lemma 5 . 1 0 , 5 2 , Chapter 5, applied t o t h e algebra A = Cb(X;IR), t h e r e e x i s t funcxl, xn E X,such t h a t t o e a c h 6 > 0 , t h e r e correspond for t i o n s a , . . . , a n E cb(x;IR) w i t h 0 5 ai 5 L , 0 5 a i ( x ) < 6 x E X Let

f (x) = v

map-

v

E Ex c h o o s e

f E L

such

JI E A x , ~ ~ ( $ 1 clearly a linear functional. 0 . L e t v # 0 be g i v e n i n Ei Choose f E L w i t h f ( x ) = v. By c o n d i t i o n ( 3 ) , t h e r e i s a n e i g h b o r h o o d U o f x i n X , whose complement i s compact,and s u c h t h a t , f o r a l l t E U , I I f ( t ) I I < (1 + c ) - I I f ( x ) I I . S i n c e X i s l o c a l l y compact, t h e r e is g E K(X) s u c h t h a t 0 5 g _< 1 , g ( x ) = 1 and g(t) = 0 f o r a l l t 6! U. S i n c e L i s a K(X) -module, g f E L a n d 119 f I I < (1 + E ) I If ( x ) I I .Now(g f ) ( x ) = v and so that

= Jl(f). Since

124

C o ( X;E)

(v) I

W I T H T HE

= 1$(9 f )

E >

0 be giv-

e n . Then U = { t E X ; I } f ( t ) 1 1 < E } i s open, c o n t a i n s x , and c g < 1, g ( x ) = 1 and x\u s compact. Choose g E K ( X ) w i t h 0 f o r t E U. Choose v E Ex such t h a t 1 Ivll < l , $ ( v ) i s g(t) = 0 r e a l and Q ( v ) > 1 E . Choose h E L w i t h 1 I h l I l l a n d h ( x ) = v. L e t m = g h S i n c e L i s K(x)-module, m E L. On t h e o t h e r hand I l m l -< 1 and $ ( m ( x ) ) = $ ( v ) > 1 E . F o r t e'u, m ( t ) = 0; for t E u, ( I f ( t )/ I < E . Hence I I f + m l I 5 1 + E . E x a c t l y as in Lemma 1.35, 9 1 0 , C h a p t e r 1, one shows t h a t

-

.

-

l$,(m) and

1$,(f

whence

IJll(f)

This proves $,I

$,

E Ax

t h e proof.

-

+ m)

-

-

I

4

K

$,(f

+ m)

I

$,(m)

-

Q2(f)

I

< 8

< 4 K ,

6

$, JI, E Ax. T h e r e f o r e $, + $, = 2 $ E Ax i m p l i e s n B', which c o n t r a d i c t s JI E E ( A x . n B') T h i s ends

.

Co( X;E)

126

W I T H T H E UNIFORM TOPOLOGY

REFERENCES FOR CHAPTER 6 . BROSOWSKI and DEUTSCH MACHADO a n d PROLLA STROBELE

[63]

[lo]

[39]

C H A P T E R

7

THE SPACE Cb(X;E) WITH THE STRICT TOPOLOGY

We start with the Stone-Weierstrass Theorem for algebras and modules. The first such Theorem was obtained by Buck himself (see Buck Ell] ) :!?,-densityof subalqebras of Cb (XI and ts-density of Cb(X)-modules in Cb(X;E) when E is finite-dito mensional. The latter result was qeneralized by Wells E6671 the case of any locally convex space E, and subspaces W C Cb (X;E) that are A-modules, where A = If 'E Cb(X); f (x) C [0,1]1. Further results were obtained by C. Todd (see his Theorem 3, [65]). Our first theorem subsumes all those earlier results. L e t X b e a l o c a l l y compact H a u s d o r f f s p a c e , and THEOREM 7.1 vector l e t A c Cb(X;M) b e a s u b a l g e b r a . L e t W C Cb(X;E) b e a s u b s p a c e w h i c h i s an A-module, w h e r e E i s a l o c a l l y convex s p a c e . T h e n W i s s h a r p l y l o c a l i z a b l e u n d e r A i n Cb(X;E) e q u i p p e d w i t h t h e s t r i c t t o p o l o g y B.

PROOF

Apply Corollary 5.29 since A c Cb(X;M 1 .

THEOREM 7.2

L e t A and W b e a s i n Theorem 7.1. Assume t h a t

A

i s s e l f - a d j o i n t . T h e n W i s l o c a l i z a b l e u n d e r A i n Cb(X;E) w i t h t h e s t r i c t t o p o l o g y !?,.

PROOF Since A is self-adjoint, then G(A) = Re A U I m A is a strong set of generators for A satisfyinq the hypothests of Theconorem 5.20, § 3 , Chapter 5. On the other hand, since G(A) sists'only of real valued functions, p = 2 and P2 = PA. Thus W is localizable under A in Cb(X;E) with the strict topology !?,. COROLLARY 7.3

L e t A and W b e a s i n Theorem 7.2. Assume

A i s s e p a r a t i n g . T h e n W i s R-dense W(X)

i n Cb(X;E) i f , and o n l y

that

if,

i s d e n s e i n E, f o r a l l x E X .

PROOF

For each x

E

XI there is some I$

E

Co(X) with

@(XI

> 0.

128

Cb(X;E)

!IITH

T H E S T R I C T TOPOLOGY

Hence the condition is necessary. The sufficiency follows Theorem 7 . 2 . COROLLARY 7 . 4 PROOF

K(X)

The s p a c e

Q

K(X) and W =

K(X)

E.

Q

Hau s do r f f in t h e space

L e t X and Y be two l o c a l l y compact

s p a c e s . Then (Cb(X) Q Cb(Y)) Q E x

E is R-dense i n Cb(X;E).

Apply Corollary 7 . 3 , with A =

THEOREM 7 . 5 Cb(X

from

is B-dense

Y; E).

PROOF A = Cb(X) Q Cb(Y) is a self-adjoint separating subalgebra of Cb(X x Y) and W = A Q E is such that W(x) = E, for each x E X. It remains to apply Corollary 7 . 3 . In fact the following stronger version of Theorem 7.5 is true. L e t X and Y be two l o c a l l y compact

THEOREM 7 . 6

s p a c e s . Then (K(X) Q K(Y)) t3 E is R-dense i n t h e

Cb(X

x

PROOF

Hausdorff space

Y ; E).

Similar to that of Theorem 7 . 5 .

DEFINITION 7 . 7

L e t W C Cb(X;E) b e a o e c t o r s u b s p a c e .

S t o n e - W e i e r s t r a s s R-hull t h e s e t A(W)

of W i n Cb(X;E), d e n o t e d by

The

Ap(W)

is

fl Cb(X;E).

(For the definition of A(W) , see Definition

4.12,

5 2 , Chapter 4 ) .

An obvious modification of the proof of Lemma

5

4.16,

2 , Chapter 4 shows that

AB(W) = LA(A 8 El = LA(W) , when E is a locally convex Hausdorff space, and W C Cb(X;E) a vector subspace invariant under composition with elements E’ Q E. DEFINITION 7 . 8 L e t W C Cb(X;E) be a v e c t o r s u b s p a c e . We where t h a t W is a S t o n e - W e i e r s t r a s s s u b s p a c e i f AR(W) C b a r d e n o t e s t h e R - c l o s u r e of W i n Cb(X;E).

w,

THEOREM 7 . 9

(Stone-Weierstrass)

is of

say the

S u p p o s e E i s a Z o c a t t y convex

Cb( X;E)

W I T H T H E S T R I C T TOPOLOGY

129

H a u s d o r f f s p a c e . E v e r y s e l f - a d j o i n t p o l y n o m i a l a l g e b r a WCCb(X;E) i s a S t o n e - W e i e r s t r a s s s u b s p a c e . In p a r t i c u l a r , f o r e v e r y f E Cb(X;E), f b e l o n g s t o t h e R - c l o s u r e of W i n C b ( X ; E ) i f , and only if:

(1) for a n y x E X , s u c h t h a t f(x) # 0 , t h e r e i s g E W s u c h t h a t g(x) f 0 ; (2) for any x,y E X, w i t h f (x) # f ( y ) , there is q E W s u c h t h a t g(x) # g(y). PROOF By a previous remark, Ag(W) = LA(W) = LA(A @ E l , where AA = {I$ o 9 ; 4 E E ' , q E W). By Theorem 7.2 applied to the module A @ E, we have LA(A @ E) = A @ E . Since W is a polynomial alqebra, A Q E c i . Hence A (W) C i . The converse,ii c A6(W), is 6 true whenever E is Hausdorff. COROLLARY 7.10 S u p p o s e E is H a u s d o r f f . L e t W C C b ( X ; E ) be a s e l f - a d j o i n t p o l y n o m i a l a l g e b r a . The n W i s B-dense i f , and o n l y i f , W i s s e p a r a t i n g and everywhere d i f f e r e n t f r o m z e r o . REMARK

For further results and counter-examples see

Haydon

THEOREM 7.11 Suppose E i s a l o c a l l y convex Hausdorff space. L e t W C Cb(X;E)be a v e c t o r subspace which i s i n v a r i a n t under c o m p o s i t i o n w i t h e l e m e n t s of E ' Q E and l e t A={4of; $ € E n , fEW). The foZZming c o n d i t i o n s a r e e q u i v a l e n t : (1) W i s l o c a l i z a b l e u n d e r A i n C b ( X ; E ) . (2) W i s a S t o n e - W e i e r s t r a s s s u b s p a c e . (3)

A i s a Stone-Weierstrass subspace.

PROOF By previous remark, Ae(W) = LA(W). Hence (1) and (2) are equivalent. be Assume ( 2 ) , i.e. AB(W) C i . Let f E C b ( X ) an be given. element of Ag(A). Let v E C , ( X ) , v 1. 0 , and E > 0 Choose I$ E El and u E E with @(u) = 1. Let g = f 8 u.Obviously, that g E Ag(W). Let p be a continuous seminorm on E such I I $ ( t ) 1. < p(t) for all t E E. By hypothesis, there is h E W such that v(x)p(q(x) h(x) ) < E , for all x E X . Hence v(x) If(x) ( 4 o h)(x)l < E , for all x E X . Since 4 o h E A,

-

-

130

f

E

Cb(X;E)

WITH T H E S T R I C T TOPOLOGY

X I the B-closure of A in Cb(X), i.e. Ae(A) C A.

Finally, assume (3). Since = AR(A), is a B-closed self-adjoint subalgebra of Cb(X). Indeed, A e ( A ) = A ( A ) n Cb(X) , and by Proposition 4.15, 5 2, Chapter 4, A ( A ) is a self-adjoint By Theorem 7.2, applied to the subalgebra of C(X). Let B = polynomial algebra B Q E, we have LB(B Q E) = B Q E. - -Hence L A ( W ) = LA(A Q E) C LA(B 4 El = LB(B Q E) = B Q E C A 0 E, be-

x.

cause ?i 0 E c A Q E. By Lemma 4.1, § 1, Chapter 4 , and therefore LA(W) C which proves (1).

w,

A @

E C W,

REMARK From Proposition 4.15, 2, Chapter 4, and the following facts: (1) A e ( W ) = A ( W ) fl Cb(X;E) : and (2) R is stronger than the compact-open topology: it follows that A ( W ) is the smallest p-closed B self-adjoint polynomial algebra contained in Cb(X;E) which contains W . THEOREM 7.12 E v e r y p r o p e r f3-closed Cb (X)-module W C Cb (X;E) is c o n t a i n e d i n some p r o p e r p - c l o s e d Cb(X)-module V o f c o d i m e n s i o n o n e ( h e n c e m a x i m a l ) i n Cb(X;E). M o r e o v e r , W i s t h e i n t e r s e c t i o n of a l l m a x i m a l p r o p e r p - c l o s e d Cb(X)-modules t h a t contain it. Let W C %(X;E) be a proper 8-closed Cb(X)-module. Let f E Cb(X;E) be a function such that f $! W . Since W is B-closed, and Cb(X) is separating, by Corollary 7.3, there is x E X such that f ( x ) @ W ( x ) in E. By the Hahn-Banach theorem, there is 4 E E' such that $(f(x)) # 0, while @ ( g ( x ) ) = 0 for all g E W . Then V = Ig E Cb(X;E): I$( g ( x ) ) = 0) is a p-closed Cb(X)-module of codimension one in Cb(X;E), containing W , and f $! V.

PROOF

COROLLARY 7.13 A l l m a x i m a l p r o p e r R-Closed Cb(X)-modules of C (X;E) a r e o f t h e f o r m {g E Cb(X;E); $ ( g ( x ) ) = 0) f o r Some b x E X and 4 E E'. Before proceeding we need the following elementary properties of (Cb(X;E), B ) which were proved by Buck [ l ] .

Cb(X;E)

131


PROPOSITION 7.14 L e t X be a l o c a l l y compact Hausdorff l e t E b e a l o c a l l y c o n v e x H a u s d o r f f s p a c e , a n d l e t f3 b e s t r i c t t o p o l o g y o n Cb(X;E). T h e n

(1)

i f < i s t h e compact-open

(2)

t h e u n i f o r m t o p o l o g y , t h e n K 5 R 5 u; t h e t o p o l o g i e s R a n d u h a v e t h e same

space, the

topo2og.y and i f

u

is

bounded

sets; o n a n y a-bounded s e t i n Cb(X;E), t h e

(3)

topology

f3 a g r e e s w i t h n2 for a suitable p E cs(E) .Suppose first that {xn) has a convergent subsequence, say {x I . Let "k x

+

"k

X,

x

+

E X. Choose t# E Co(X) such that @(x

"k

) =

$(x) = 1

.

for all k E N Since S is strictly bounded, there exists a constant M > 0 such that 1 If II < M for all k E N Hence "k @ I p-

.

%

(X 1 ) 5 M for all k E N , a contradiction to "k chop(f (x ) ) > nk. Therefore, {xn) is discrete and we may "k "k ose a sequence of compact sets Kn with xn E Kn, but the KA s p(f

"k

are pairwise disjoint. Take $n E Ci(X) with range in [O,ll ,with

Cb(X;E) W I T H T H E S T R I C T TOPOLOGY

132

support contained in Kn, and $,(xn) = 1. Let $(x) = C cn $,(XI , + -1/2 for Then 6 E Co(X) , and $(xn) = c where cn = p(fn (x,) ) n' 1/2 > all n E N On the other hand I Ifn[ I > p(fn(xn)) I 4rP contradicting the strict boundedness of S.

.

.

(3) Let S c Cb(X;E) be a a-bounded subset. By (11, we have K / S< B l S . Conversely, assume T c S is (BIS)-closed.Let g E S be in the ( 0 and < M $ E Ci(X) be given.Let M > 0 be such that 1 If 1 I for all P f E S. Let K c X be a compact subset such that $(x) < E / ~ M for all x 1 K. Choose f E T such that IIf-glIKIp < E / ( ~ I $ I I+~ 1).

Let x

E X.

If x

$(X)P(f(X)

-

E

K, then we have

g(x)) < 1141

Ix . €/(I

161

Ix

+

1)
0 be given. Since {fa} is B-Cauchy, {+fa) is a-Cauchy, and thus converges to a function g E Cb(X;E) in the topology 6. Since fa + f in the topology K , then fa(x) + f(x) for every x E X. Therefore, g(x) = $(x)f(x) for all x E X I i.e. g = $ f. Notice that each $fa E Co(X;E) , which is a-closed in Cb(X;E). + is Therefore $f E Co(X;E) for all $ E Co(X). The proof of ( 4 ) then complete if we establish the following

.

LEMMA 7.15 L e t f E C(X;E), and s u p p o s e t h a t $f + e v e r y 6 E Co(X). T h e n f E Cb(X;E). PROOF

(Buck [llJ).

E

Co(X;E)

If f (X) were not bounded in E, then,

for

for

Cb( X ; E )

WITH THE STRICT TOPOLOGY

133

some p E cs (E), there would exist a sequence { x 2 in X that p(f (x,)) > n2 for all n E N. Since f is continuous, is discrete and we may choose a sequence of compact sets with xn E Kn and the KA s are pairwise disjoint. Take $ n E with range in [0,1] , supp 4 n C Kn and 4,(xn) = 1. Let $

Then all

E

n

E

(XI

= C

+

Co(X), $ (x,)

=

cn $n(x),

where

cn=p(f(xn))

such Kn C:(X)

-1

cn and therefore p($ (xn)f (xn))

{ Xn)

=

. lfOr

N. Thus $ f @ Co(X;E), a contradiction.

REMARK 7.16. The proof that (Cb(X),B ; has the approximation property was first established by Collins and Dorroh [14]; their argument being a thorough recasting of de Lamadrid's proof for compact X and the uniform topology (171, pg. 164). When X is completely regular and Cb(X) is equipped with the generalized strict topology Tt , Fremlin, Garling and Haydon( b5] ) , Theorem 10) showed that (Cb(X),Tt) has the approximations property. Their proof is different from and simpler than the proof of Collins and Dorroh. The result of [25] was generalized to Cb(X;E) by Fontenot [24] , who considered the case in which E is a normed space with the m e t r i c a p p r o x i m a t i o n p r o p e r t y , and Cb(X;E)

is equipped with the

cally convex topology on

0,

topology, i.e. the finest lo-

Cb(X;F) which agrees with the compact

-open topology on norm bounded sets. It X is locally then B = 0, . THEOREM 7.17. let

Let

compact,

X b e a l o c a Z l y c o m p a c t n a u s d o r j f s p a c e , and

E b e a normed s p a c e w i t h t h e m e t r i c a p p r o x i m a t i o n p r o p e r -

t y , Then

PROOF:

(Cb(X;E), B

)

hus t h e approximation property.

See Fontenot [24].

134

Cb(X;E)

W I T H THE S T R I C T TOPOLOGY

Let E be a locally convex Hausdorff space, and let X and Y be two locally compact Hausdorff spaces. Let u:Y -* X be from a continuous mapping. We denote by TU the linear mapping Cb(X;E) into Cb(Y;E) defined by composition with u, i.e. T f = u. f o u for all f E Cb(X;E). Let us assume that, for every

+

+

Co(Y), there exists J, E Co(X) such that @ < IJJ o u. Then TU is is (B,B)-continuous. Whenever TU is continuous and u(Y)

@ E

closed in X we say that u is B - a d m i s s i b l e . For example, if Y c X is a c l o s e d subset, and u : Y + X is the inclusion mappinq, it + + follows from Theorem 6.11 that Co(X) IY = Co(Y) and therefore u is B-admissible. THEOREM 7.18

Let u : Y

+

X b e a R - a d m i s s i b l e c o n t i n u o u s proper

mapping. Then TU i s an open mapping f o r t h e s t r i c t t o p o l o g i e s .

PROOF Let us consider the 0-neighborhood base consisting of all subsets of the form U = ig where

@ E

C:(X)

E

,p

W = Ih

E

Cb(X;E); @(x)p(q(x)) E

cs(E) and

E

E,

x E XI

E,

y E Y)

0. Let

Cb(Y;E); $(y)p(h(y))

where 0 = $I o u. We claim that J,

in

E

6 1 , then Kg is compact, and if y E V g then u(y) E K6 n u(Y). Since u(Y) is closed in X, K = K6 n u(Y) is compact and therefore V6 is com-

pact, because it is closed and contained in the compact set -1 u (K). (Recall that u is a proper mapping). Therefore W is an open R-neighborhood of 0 in Cb(Y;E). Clearly, TU(U) c W n TU(Cb(X;E)). Conversely, let h

E

W n TU(Cb(X;E)).

Let g E Cb(X;E) be such that h = g o u, i.e. h(y) = q(u(y)) > E ) . Then F C X is all y E Y. Let F = (t E X; @(t)p(g(t)) pact and disjoint from u(Y), because h E W. If F = j8, E K(X) g E U, and therefore h E TU(U). If F # 8, choose 0 < < 1, n(x) = 1 for all x E u(Y), and n(t) = 0 for -

for comthen

I

all

Cb(X;E)

135


t E F. This is possible because X\u(Y) is an open neighborhood of the compact set F, and X is locally compact. Let f = n g E Cb(X;E) Then h(y) = g (u(y) = Q (U(y))g (U(y) = f (u(y) for all y E Y, i.e. h = TU(f). We claim that f E U. Let x E X. If x E F, then f(x) = 0, so $(x)p(f(x)) = 0 < E . If x j! F, then

-

$(X)P(fh)) = $(x)p(n(x)g(x)) Thus f

E

= n(x)$(x)p(q(x))

< $(x)p(g(x)F(E. and

U and h E TU(U). Hence TU(U) = W n TU(Cb(X;E),

TU(U) is relatively open in TU(Cb(X;E)) for all U is an open mapping, QED.

E

,

and

TU

REMARK For similar results on operators defined on Cb(X) by composition with a continuous mapping between completely regular spaces, when Cb(X) has the generalized strict topology Tt see Theorem 9 and its Corollary, Fremlin, Garling, and Haydon [25!. L e t u s now c o n s i d e r B i s h o p ' s Theorem f o r t o p o l o g y . When c27;.

In

[51]

E = Q

s u c h a Theorem was p r o v e d by

Glicksberg

w e p r c v e d a v e r s i o n o f B i s h o p ' s Theorem f o r Nachbin

spaces of v e c t o r - v a l u e d t h e case o f

strict

the

Cb(X;E)

f u n c t i o n s s u f f i c i e n t l y g e n e r a l t o cover

with t h e strict topology

6 - Here however

w e w i l l d e r i v e i t f r o m Theorem 5 . 2 0 o f C h a p t e r 5 . Let X be a locally compact Hausdorff space, let E be a locally conoex Hausdorff space, and let A C C b ( X ; Q ) be THEOREM 7.19.

a subalgebra. Let W C C b ( X ; E ) be a n A - module. Then f is in the 6 - c l o s u r e of W if, and only if flK is

- closure PROOF:

of

Take

c ~ ( K ; E ) for every A - untisymmetric set G ( A ) = A.

On t h e o t h e r hand

Then

i s a l s o s a t i s f i e d . Therefore Cb(X;E).

and

NOW

G(A)

A C Cb(X;E),

xA

= Pp

P7

E

$(X;E)

in

the

K

i s a s t r o n g s e t o f generators.

s o c o n d i t i o n ( 2 ) o f Theorem 5.20 i s s h a r p l y l o c a l i z a b l e under i n

+

a n d t h e r e f o r e g i v e n f E Cb ( X ; E ) , Q E Co(X)

p E c s ( E ) then

f l -~

glK

il+,p

136

Cb(X;E)

b e l o n g s t o t h e B - c l o s u r e of

In particular, f if

,

and o n l y i f

f o r any

K E


, f IK

xA .

W

belongs t o t h e B-closure of

in

Cb(X;E)

XIK i n

%(K:E)

N o t i c e t h a t i n f a c t w e have p r o v e d t h e " s t r o n g " version o f B i s h o p ' s Theorem.

Let

THEOREM 7.20.

Let

COROLLARY 7 . 2 1 . X

,A

+

W

and

@ E Co(X),

REMARK:

and

X , E , A

4

f E Cb(X;E),

+

E Co(X)

Assume

and

(El)\

W b e a s i n Theorem 7 . 2 2 ,

11)

i s a normed s p a c e o v e r Q : , a n d

a r e a s i n Theorem 7 . 2 3 . Then, g i v e n

f E Cb(X;E)and

we h a v e

I n t h e above s t a t e m e n t , i f

h E C b ( S ; E ) , where

i s any s u b s e t , (1 h \ I s = s u p { / I h ( t ) 11 ; t above C o r o l l a r y i m p l i e s B i s h o p ' s Theorem J u s t take

and

be given;then

p E cs(E)

E S

1

of

.

Clearly,

Glicksberg

S C X

the

[27

].

E = Q : . I n f a c t , t h e above f o r m u l a w a s e s t a b l i s h e d b y

G l i c k s b e r g i n t h e case o f compact

X

and t h e u n i f o r m t o p o l o g y i n

h i s p r o o f o f B i s h o p ' s Theorem. S e e G l i c k s b e r g

[26],

page 4 1 9 .

Cb( X;E)

N I T H T H E S T R I C T TOPOLOGY

&XEFERENCES FOR CHAPTER 7. BUCK

1113

C O L L I N S and DORROH

[17]

DE LAHADRID

[2 4 ]

FONTENOT FREMLIN,

GARLING and HAYDON

GLICKSBERG

[26]

HAYDON

[3O]

PROLLA

pl]

TODD WELLS

[14]

[65] [67]

, [27]

[25]

137

C H A P T E R

8

THE €-PRODUCT OF L. SCHWARTZ

5

1

GENERAL DEFINITIONS

Let E be a locally convex Hausdorff space, with topological dual E'. We denote by EA the space E' endowed withthe topology of uniform convergence on all absolutely convex comHausdorff pact subsets of E. The space EA is a locally convex space, whose topology is defined by the family of seminorms u

E

E'

+

sup {lu(x) 1 ;

x

E S]

where S c E is an absolutely convex compact subset of E. Since the absolutely convex compact subsets of E are, a fortiori , weakly compact, it follows from Mackey's theorem (Grothendieck [28] , Corollary 2 to Theorem 7, Chapter 11) that the dual (EA)' of EA is E (as a vector space). Let now E and F be two locally convex Hausdorffspaff We shall denote by &fe(E&;F) the vector space of all continuous linear mappings T : EA + F endowed with the topoloqy of uniform The space convergence on the equicontinuous subsets of E'. ae(EA;F) is then a locally convex Hausdorff space, whose topology is generated by the family of seminorms T * sup (p(T(u)); u

E

Vo}

where p ranges through a system r of seminorms defininq the topology of F, and V runs through a 0-neighborhood base in E, and we may assume V to be absolutely convex and closed. In fact, we < l} , where q runs through a may even assume V = {x E E; q (x) system A of seminorms defininq the topoloqy of E. Indeed, every Vo with equicontinuous subset S c E' is contained in some V = { x E E; q(x) < 1) and q E A. PROPOSITION 8.1

T h e ZocalZy c o n v e x s p a c e s

de(E;;F)

and

E

-

139

PRODUCT

&fe(Eh;F) a r e l i n e a r l y t o p o l o g i c a l l y i s o m o r p h i c . PROOF Let T E &,(E;;F). 1.ts transpose T I is a linear mapping Indeed, (EA)' = E, T' : F' + (E;)'. We claim that T' E%(F;;E). as a vector space. On the other hand, let V C E be an absolutely convex closed neighborhood of 0 in E. To prove continuity of T ' we must show that a neighborhood N of 0 in FA can be found such that T'(N) c V. Now the polar Vo = {u E E'; lu(x) 1 < 1, x E Vl is an equicontinuous subset of El, which is weakly compact.Since u(E',E) and the topology of E; induce the Same topology on the equicontinuous subsets, Vo is a compact subset of E;. Therefore K = T W O ) is an absolutely convex compact subset of F. Its polar KO is then a neighborhood of 0 in FA. Since T'(Ko) c V, the neighborhood N = KO is the thing we are looking for. The transposition mapping T + T' is therefore a linear isomorphism between 2 (EA;F) and 6f (FA;E). We claim that T + T ' is a homeomorphism. By symmetry it is enough to provecontinuity A 0-neighborhood base in the space de(FA;E) is ob-. S runs tained by taking all subsets U = {f: f (S) C W), when through all equicontinuous subsets of F' and W runs through a 0-neighborhood base in E. We may assume W to be absolutely conan vex and closed. If S C F' is equicontinuous, there exists such that absolutely convex closed neighborhood V of 0 in F S C Vo. On the other hand, Wo is an equicontinuous subset o f El. Therefore N = {T;T(W0 ) C V} is a neighborhood of 0 inde(E&;F). Since T(Wo) c V implies TI (vo) C Woo, and Woo = W , we see that T E N implies T' E U, i.e. T T' is continuous.

.

+

DEFINITION 8 . 2 ([59])

We d e f i n e t h e € - p r o d u c t of E and

F

by

setting

E

E

F =ge(F;;E).

By the above Proposition 1, we may identify E E F i.e. , E E F and F E E are linearly topoloqicalwith 5 .(E;;F), ly isomorphic. REMARK.

When E is quasi-complete (i.e., when the closedbounded

140

E

- PRODUCT

sets in E are complete), then E& has the topoloqy of uniform convergence on a l l compact sets of E . Indeed, in a quasi-complete space, the closed absolutely convex hull of a.compact.set is compact. PROPOSITION 8.3

If E and F a r e q u a s i - c o m p l e t e ( r e s p . c o m p l e t e ) ,

then the €-product E

E

F i s quasi-complete

(resp. complete).

See Schwartz L59I]. We now show that we may identify E QbE F with a subspace of E E F. To this end we first recalltk definition of E Q E F. If E and F are vector spaces over IK, then B ( E ; F ) denotes the vector space of all bilinear forms on E x F. The mapping f + f(x,y), for each pair (x,y) in E x F is then a linear form on B ( E , F ) , i.e. an element of the alqebraic dual B(E,F)* of B ( E ; F ) . This linear form is denoted by x 0 y. The mapping defined by #(x,y) = x Q y is then a bilinear mapping from E x F into B ( E , F ) * . The linear span of $ (E x F ) in B ( E , F ) * is called the t e n s o r p r o d u c t of E and F , and is denoted by E Q F. Each element u E E Q F is a finite sum of the form r u = I: x i 0 y i i=l PROOF

,...

,r. This.representation is not with xi E E, yi E F, i = 1,2 unique, but we can assume that {xi} and {yi} are linearly independent in E and F respectively. The number r is then uniqmly determined and it is called the r a n k of the element u E E Q F. There are several useful topologies on E 0 F , when E and F are locally convex Hausdorff spaces. We are interested here in the topology T~ of b i - e q u i c o n t i n u o u s c o n v e r g e n c e . We identify each element of E Q F with a l i n e a r form on E' 8 F ' by means of the formula (1)

(x

Q

y) (x' 0 y') = x'(x) y'(y)

extended by linearity. The topoloqy T~ is the topoloqy of uniform converqence on the sets of the form %(S x T), where S and T run through the equicontinuous subsets of E' and F ' respectively. Another way of characterizing T, is the followinq.

E

-

PRODUCT

141

Each element x Q y, by means of formula (1) defines a b i l i n e a r can f o r m on E' x Fi which is separately continuous, i-e., we U identify E Q F with a subspace of @(EA; FA), the vector space of all bilinear forms on EI, x FA which are separately continuous. The topology T~ is then the topology induced on E 8 F by x -topology, where and are the families of equithe continuous subsets of E' and F' respectively. r us define If u E E 8 F, say u = C xi 0 yi, let i=1

6: F'+Eby

r

for all y' E F'. The mapping 6 is obviously linear and does not depend on the particular representation of u. We claim that the map 6 belongs to 8 (Fi;E). Indeed, if the net yd, + 0 in F; , then yd,(yi) + 0 f o r all i = 1,2,...,r. Hence 6(y') + 0 in E.The mapping u + e is then a linear one-to-one mapping from E Q F onto a subspace of 8 (F;;E). We shall denote by E QE F the bEW of E Q F in ,(F;;E) = E E F, with the induced topology. Since the topology of E is the topology of uniform convergence on the equicontinuous sets of E', the topology induced by E E F on EQF is the topology T~ of bi-equicontinuous convergence. The completion of E Qc F will be denoted by E iE F, and it is called the i n j e c t i v e tensor product.

a

5

2

SPACES OF CONTINUOUS FUNCTIONS

In this section we establish a representation t h e m for the €-product of C(X) and E, when X isakm-space and E is a quasi-complete (resp. complete) Hausdorff space. Before proceeding, we recall the definition of a km-space. DEFINITION 8.4 A Hausdorff space X i s s a i d t o be a km -space if, f o r e v e r y f u n c t i o n f : X + IR s u c h t h a t flK is c o n t i n u o u s , f o r e a c h c o m p a c t s u b s e t K C X , t h e f u n c t i o n f i t s e l f is c o n t i n u ous.

142

E

-

PRODUCT

We mention that, when X is a km-space, and Y is a completely regular Hausdorff space, and f : X + Y is such that flK is continuous, for each compact subset K C X, then f E C(X;Y) The following result shows the equivalence between the completeness of C(X;m) endowed with the compact-open topology and the property of X being a km-space.

.

THEOREM 8.5

L e t X be a c o m p l e t e l y r e g u l a r H a u s d o r f f space.The

following conditions are equivalent.

(a) C(X;M) i s c o m p l e t e u n d e r t h e compact-open PO

to-

logy.

(b) C(X;M) i s q u a s i - c o m p l e t e u n d e r t h e compact-open top0 logy. (c) X i s a k m - s p a c e .

PROOF

See Warner [66]

THEOREM 8.6

, Theorem

1.

L e t X be a c o m p l e t e l y r e g u l a r H a u s d o r f f

space,

w h i c h i s a k I R - s p a c e , and l e t E be a q u a s i - c o m p l e t e l o c a l l y conv e x H a u s d o r f f s p a c e . Then C(X) E E and C ( X ; E ) a r e l i n e a r l y t o p o l o g i c a l l y i s o m o r p h i c . I f , moreover E i s complete, and C(X) E E a r e l i n e a r l y t o p o l o g i c a l l y . i s o m o r p h i c .

C(X)

€;

E

We first prove the following lemma. LEMMA 8.7

I f X i s a ( c o m p l e t e l y r e g u l a r ) Hausdorff space, w h i c h i s 'a k m - s p a c e , t h e mapping A : x -* 6, i s a c o n t i n u o u s ? p i n g f r o m X i n t o C(X;mC)A.

PROOF Since each one-point set { x ) is compact, the map 6, : f + f ( x ) belongs to C(X;M) I . Let us write F = C(X;M). By the definition of the weak*-topology u(F*,F), the map A : X FA and let is always continuous. Let K C X be a compact subset, pK be the seminorm f E F + sup { If ( x ) I ; x E K). Then < l } ' . Therefore, A maps compact sets into A(K) C {f E F; pK(f) topoequicontinuous sets. On these FA and FA induce the same logy. Hence AIK is continuous as a map from K into FA, for each compact set K C X. By the remark made after Definition 1, A is a continuous mapping from x into +

FA.

E

-

143

PRODUCT

PROOF OF THEOREM 8.6 A s in the proof of Lemma 8 . 7 , let us write F = C(X;M 1 . Define $ : de(FA;E) * G, by $(TI = T o A, where we have defined G = C(X;E). We claim that $ is injective. When E = IK , this follows from

&f (FA;K) = (FA)' = F = C(X;lK). imp1ies For the case of a general E, notice that + (TI = 0 u o(T o A) = 0 for all u E El. Hence (u o T) o A = 0, for all u E El. By the previous case, u o T = 0, for all u E El. Therefore T = 0. We now make the following CLAIM

The map

+

is onto C(X;E) = G.

PROOF OF CLAIM Let q E C(X;E). For each.u E El, consider so defined u o g E C(X;lK). The linear map T : EA + C ( X ; l K ) is continuous. Indeed, given K C X compact, let V = {f E C(X;M); If(x)l < 1, for all x E K}. Since q is continuous, q(K) is compact. Let K1 be the absolutely convex closed hull of g(K) in E. Since E is quasi-complete, K1 is compact. 0 C V. Therefore T is continuous. Since On the other hand, T(K1) (EA;C(X;IK ) ) Then, its transpose it is obviously linear, T E T' belongs tog(FA;E). To prove that $(TI) = q, notice that for every x E X and u E El, we have < (TI o A) (x), u > = < A(x), T(u)> =

8

= < 6,,

.

T(u) > = T(u) (x) = u(g(x)).

It then follows that +(T')(x) = g(x) for all x E X, i.e., proof of +(TI) = q. Thus $ is onto G, and this completes the the claim. To finish the proof of the Theorem, we must show that + is a homeomorphism. Indeed for any net (T,) in the space (FA;E) the following are equivalent statements:

ge

(1) The net T, + 0 in 2 .(FA;E). (2) The net Td, * 0 in ife(E;;F). ( 3 ) Td, u + 0 in F, uniformly in u E S, for eachequi(4)

continuous subset S C El. (T, o A) (x) + 0 in E, uniformly in x E K, each compact subset K C X.

for

144

E

(5)

-

PRODUCT

$(Ta) * 0 in G = C(X;E)

.

Since ae(F;;E) is by Proposition 8.1, 51, linearly topologically isomorphic to the space F E E = C(X) E E, thiscanpletes the proof of the first part of Theorem 2. Assume now that E is complete. Since X is a km-space C(X) is complete too, by Theorem 8.5. It then follows from Proposition 8.3 that C(X) E E is complete. Now when we identify C(X) E E and C(X;E) , the vector subspace C(X) Q E E c C(X) E E is identified with the set of functions f E C(X;E) such that f (XI is contained in some finite-dimensional subspace of E,i.e., with the space of all finite sums of functions of the form x + g(x)v, where g E C(X) and v E E. By Theorem 1.14, 56, Chapter 1, this space is dense in C(X;E). We have seen that C(X;E) is complete, therefore C(X)

iE E =

C(X)

E

E.

This completes the proof.

5 3 THE APPROXIMATION PROPERTY We recall that a locally convex space E has the app r o x i m a t i o n p r o p e r t y if the identity map e can be approximated, uniformly on every compact set in E, by continuous linear maps of finite rank. In [227, Enflo has shown that there is a Banach space which fails the approximation property. For an account of the approximation property on function spaces, in particular in Nachbin spaces, see the papers of Bierstedt [6] and Bierstedt and Meise [7]. The following result is due to L. Schwartz.The p m f of (3) * (2) given below follows Schaefer [55], Chapter 111,59, Proposition 9.2. THEOREM 8.7 L e t E be a q u a s i - c o m p l e t e l o c a l l y c o n v e x Hausdorff s p a c e . Then t h e f o l l o w i n g a r e e q u i v a l e n t . (1) E has t h e a p p r o x i m a t i o n p r o p e r t y . ( 2 ) E Q E F i s d e n s e i n E E F, f o r a l l l o c a l l y c o n v e x

E

- PRODUCT

145

s p a c e s F. ( 3 ) E QE F i s d e n s e i n E

E

F, f o r aZZ Banach

spaces

F.

.

(Schwartz pq) Let T Exe(FA;E) = E E F. Let Zc(E) denote the space of all continuous linear maps from E into E with the topoloqy PROOF

(1) 3 ( 2 )

of compact converqence. The mapping 0 : v + v o T from d C ( E ) into ge(~;:;~) is continuous, since T(S) is a relatively compact subset of E, for every equicontinuous subset S c F'. To see this, notice that the weak*-closure 3 of S is equicontinw the topoloqies of FA and too, S is weak*-compact, and on FA coincide. Hence 5 is compact in FA, and S is relatively compact in F;. Since T E~(F;;E), T(S) is relatively compact in E. NOW, if v E E' Q E, then v o T E F BE E, because (FA)' = F. On the other hand, the identity map e on E is such that O(e) = T.Hene, if e is in the closure of E' 0 E in the space zc(E), then T is in the closure of F QE E in =fe(F;:;E) = E E F.

s

(2)

--

( 3 ) . Obvious.

(3) (2). Let F be a locally convex space. Let B be a 0-neighborhood base of absolutely convex closed sets in F. For each V E B, let 6, : F + FV denote the canonical map set, (Schaefer [ S q , pg. 9 7 ) . Let S C E' be an equicontinuous and let V E B be given. Let us write $ = $v and G = FV. Let

-

W = (1/4) $(V) C G. Then $-'(W) C V. Since $(F) is dense in G, and by hypothesis E Q E G is dense in the space E E G, it follows that E Q E $(F) is dense in E E G. Hence, qiven T E E E F = 8e(EA;F), then $ o T EdP,(EA;G). Therefore, we can find

$(F) such that w(x) - ( $ o T) (x) E W for all xES. Suppase r r w = C xi Q $(yi). Then $ ( C x(xi)yi - T(x)) E W, for all i=l i=l r x E s, and then C x(xi)yi - T(x) E V, for all x E S Let i=l

w

E

v =

E

QE

C

i=l

xi

Q

yi. Then v

E

E

QE

F, and v(x)

-

T(x)

E

V, for

all

146

E

-

PRODUCT

x E S. Since S and V were arbitrary, E BE F is dense in space ~~P,(E;;F) = E E F.

the

(2) => (1). (Schwartz [59]). Take F = E;. By the Corollary to Proposition 5, Schwartz [59], (El is isomorphic to a subspace of E t EA Since E QE E ' C X c ( E ) c E E EA, and by hypothesis, E BE El is and dense in E E E;, it follows that E QE El is dense in$,(E), therefore E has the approximation property.

.

ic

COROLLARY 8.8 L e t X be a c o m p l e t e l y r e g u l a r H a u s d o r f f s p a c e , w h i c h i s a k m - s p a c e . Then C(X;IK) e q u i p p e d w i t h t h e compactopen t o p o l o g y has t h e a p p r o x i m a t i o n p r o p e r t y .

For every Banach space E, by Theorem 8.6, 5 2, the following are isomorphic spaces: C(X) GE E, C(X) E E, and C(X;E).

PROOF

COROLLARY 8.9 For e v e r y compact Hausdorff s p a c e X, t h e Banach s p a c e C(X) has t h e a p p r o s i m a t i o n p r o p e r t y .

5

4

MERGELYAN'S THEOREM

In this section we shall prove a vector-valued version of Mergelyan's Theorem. Let K C C be a compact subset such that C\K is connected. For every complete locally Hausdorf f space E over C, let A(K;E) denote the closed subspace of C(K;E) of all those f E C(X;E) which are holomorphic on the interiorof K. Mergelyan's Theorem states that A(K;C) is the closure in C(K;C) of all polynomials with complex coefficients. (Rudinpjn, Theorem 20.5). We shall prove a vector-valued version of this result, due independently to Bierstedt 151, and Briem, Laursen, and Pedersen [g]. We shall present Bierstedt's proof. We begin with the following result (proved by Bierstedt for Nachbin spaces) which is the key to the relation and between the approximation property for subspaces of C(X) subspaces of C (X;E)

.

E

-

147

PRODUCT

THEOREM 8.10 L e t X be a completely r e g u l a r Hausdorff space, w h i c h i s a k l R - s p a c e , l e t Y C C(X) b e a c l o s e d s u b s p a c e , and kt E be a c o m p l e t e l o c a l l y c o n v e x H a u s d o r f f s p a c e . Then Y E E i s l i n e a r l y t o p o l o g i c a l l y isomorphic w i t h t h e v e c t o r subspace of a l l f E C(X;E) s u c h t h a t u o f E Y, for a l l u E El. PROOF We first remark that, when E, F and G are three locally subconvex Hausdorff spaces, and F is a topological vector space of G, then F E E is identified with a subspace of G E E, that, i.e., F E E c G E E topologically. From this it follows Y E E is isomorphic with a subspace of C(X) E E = C(X;E) Let W = {f E C(X;E); u o f E Y, for all u E El]. Then u o f E (Y2'= = Y, for all f E Y c E, and u E El. Hence Y E E C W.Conversely, if f E W, the mapping u + u o f maps EA into Y, i.e. f E Y E E.

.

COROLLARY 8.11 L e t X and Y b e a s i n T h e o r e m 8.10. The lowing a r e e q u i v a l e n t .

fol-

(1) Y h a s t h e a p p r o x i m a t i o n p r o p e r t y . (2) F o r a l l c o m p l e t e l o c a l l y c o n v e x H a u s d o r f f spaces E, Y 8 E is d e n s e i n {f E C(X;E);u o f E Y, f o r a l l u E El}. ( 3 ) For a l l Banach s p a c e s E, Y Q E i s dense in {f E C(X;E); u o f E Y, f o r a l l u E El]. We can now prove the vector-valued version of Mergelyan's Theorem. THOEREM 8.12 If K C Q: i s a c o m p a c t s u b s e t w h i c h h a s a c o n n e c Hausdorff t e d c o m p l e m e n t , and E is a c o m p l e t e locally c o n v e x s p a c e o v e r C , t h e n A(K;E) i s t h e c l o s u r e i n C(K;E) of ?(C) 8 E. PROOF Let Y = A(K;Q). Since holomorphy and weak holomorphy coincide, A(K;E) = { f E C(K;E); u o f E Y, for all u E El}. The space A(K;C) has the approximation property (see [211). Hence, by the previous Corollary 8.11, A(K;(II) 8 E is dense in A(K;E). By Mergelyan's Theorem, the set Q ( C ) IK is dense in A(K;C). Therefore ( ? ( a ) 8 E) IK is dense in A(K;Q) 8 E and hence it is dense in A(K;E). Notice that the functions of $(a) Q E are of

148

the form

E

Z

*

" i C Z xi, n i=o

- PRODUCT

E N

, xi

E

E, i = 0,1,2,...,n.

Let us consider now the case of holomorphic functions on open subsets U C 6". If E is a complete locally convex Hausdorff space over C, then II(U;E) denotes the set of all holomorphic E-valued functions on U, endowed with the compact-topology. When E = C , we write simply II(U). n L e t U b e a n o p e n n o n - v o i d s u b s e t of C , and l e t THEOREM 8.13 E b e a c o m p l e t e l o c a l l y c o n v e x H a u s d o r f f s p a c e o v e r C. Then Y(U) GE E, H(U) E E and II(U;E) a r e l i n e a r l y t o p o l o g i c a l l y isomorphic. We first remark that H(U:E) = {f E C(U;E) u o f EII(U), topofor all u E El}. By Theorem 8.10, H(U) E E is linearly logically isomorphic with H(U;E). On the other hand, when we identify C(U) E E and C(X;E) , the vector subspace H(U) QE E C II(U) E E is identified with the set of functions f E FI(U;E) such that f(U) is contained in some finite-dimensional subspace of of E, i.e., with the space of all finite sums of functions the form x * g(x)v, where g E H(U) and v E E. This spaces is dense in II(U;E) (see Grothendieck 1251 1 . Since the latter space is complete, we have

PROOF

H(U)

GE E

=

.

€I(U;E)

This completes the proof.

.

n L e t U b e a n o p e n n o n - v o i d s u b s e t of C Then COROLLARY 8.14 II(U) h a s t h e a p p r o x i m a t i o n p r o p e r t y , when e q u i p p e d w i t h the compact-open t o p o l o g y . In Chapter 4 , 5 1, we defined the space H(E) of all holomorphic functions f : E + C defined on a complex Banach space E. The following result of Aron and SchottenloherT_3!shows the equivalence between the approximation property for E and for II(E) with the compact-open topology. L e t E b e a c o m p l e x Banach s p a c e . The THEOREM 8.15 are equivalent:

following

E

-

149

PRODUCT

(1) E has t h e a p p r o x i m a t i o n p r o p e r t y . (2) H (El endowed w i t h t h e compact-open

topology

has t h e approxima t i o n p r o p e r t y .

PROOF (1) 3 (2). Since E is a km-space, and H(E) is complete, hence closed in C(E), by Corollary 8.11, all that w e h e tQ prove is that H(E) 8 F is dense in the set W = {f E C(E;F); u o f E H(E), for all u E F') for all Banach spaces F. However, since E is a Banach space, W = H(E;F). Let then f E H(E;F), K c E compact and E > 0 be given. By uniform continuity of f on K, there exists a 6 > 0 such that x E K, y E E, 1 Ix-yl I < 6 imply 1 If (x) - f(y) 1 1 < ~ / 2 . By the approximation property, there exists u E E' Q E such that x E K implies I Ix-u(x) 1 1 < 6. We next remark that u(E) is finite-dimensional and f lu(E) belongs to the space H(u(E) ;F) Since H (u(E))10 F is dense in H (u(E);F), there exists g E H (u(E);F) f (t)I I < ~ / 2for all t E u(K). Let h = g o u such that 1 Ig(t) Then h E H ( E ) Q F, and for all x E K we have 1 If (x) h(x) (2) 3 (1). Since E has the approximation property if, and only if, E i has the approximation property, and since E; is a complemented subspace of H(E) , then (2) (1) follows from the fact that a complemented subspace of a space with the approximation property has the approximation property.

.

-

-

5 5 LOCALIZATION OF THE APPROXIMATION PROPERTY The results of this section are due to Bierstedt [6], who derived a "localization" of the approximation property for closed subspaces of certain Nachbin spaces. We will consider only the case of C(X) for X compact. THEOREM 8.16

L e t X be a compact Hausdorff s p a c e ,

l e t A C C(X)

be a s u b a l g e b r a , and l e t W C C(X) b e a c l o s e d A-module. If WIK C C(K) has t h e a p p r o x i m a t i o n p r o p e r t y , f o r e a c h maximal A-antisymmetric erty.

s e t K C X, t h e n W has t h e a p p r o x i m a t i o n

prop-

150

E

-

PRODUCT

PROOF By Corollary 8.11, 5 4 , we have to prove that,for each complete locally convex Hausdorff space E, the A-module W Q E is dense in If E C(X;E); u o f E W for all u E E'I = S. Let K C X be a maximal antisymmetric set for A, and let T = {g E C(K;E); u o g E WlK, for all u E El}. Since WIK is closed and has the approximation property, it follows from Coroland lary 8.11 that (WIK) Q E is dense in T. However, SIK c T (W 8 E) IK = (WIK) Q E. Hence (W Q E) IK is dense in SIK,for each maximal antisymmetric set K C X. By Theorem 1.27, 9 8, Chapter 1, W 8 E is dense in S. COROLLARY 8.17 L e t X be a compact Hausdorff s p a c e . Every c l o s e d i d e a l I C C(X) has t h e a p p r o x i m a t i o n p r o p e r t y . For the next example, let E be a locally compact > 1, be an open non-void subHausdorff space, and let U c 8 , n set. For R C U x E, open and non-void too, define for each xEE, the "slice" R, = {Z E U; ( 2 , ~ )E R l . Then, R, is an open subset of U. We define C i I ( R ) = {f E C ( Q ) ; Z -+ f(Z,x) belongs to H(Qx) for each x E E such that R, # $ 1 , equipped with the compact-cpen in fact, topology. Then C H(R) is a closed subspace of C ( R ) ; it is a closed A-module, where A is the algebra {f E C(fl); f is constant on R, x {XI, for each x E El. The maximal antisymmetric sets for A are the sets of the form R, x {XI, for each x E E idensuch that R, # 4. If Y = C I I ( R ) and K = Rx x {XI, we may Since I I ( Q x ) is nuclear,Y 1K tify Y I K with a subspace of I I ( R x ) is nuclear too. Hence Y(K has the approximation property. By Theorem 1 above Y has the approximation property. We have thus proved the following

.

THEOREM 8.18 C I l ( R ) has t h e a p p r o x i m a t i o n p r o p e r t y . Let K c (c x E be a non-empty closed subset such that Kx = {Z E (c; ( Z , x ) E K} is a compact subset of (c. Define f(Z,x) is analytic on the interior of CA(K) = {f E C(K); Z Kxt for each x E E such that the interior of K, is # $ 1 . We further assume that, for each x E E, the complement of Kx in C is connected. -+

E

THEOREM 8.19

- PRODUCT

CA(K), u n d e r t h e a b o v e h y p o t h e s i s , has t h e

151

UP-

proximation p r o p e r t y .

.

PROOF Let Y = CA(K) As a subspace of C(K) with the compactopen topology, Y is closed. Moreover, Y is an A-module, where A is the algebra {f E C(K); f is constant on K, x {XI, x E El. As before, the maximal antisymmetric sets for A are the sets K, x {XI, with K , # 4 . For each such x we may identify Y I X , w k m ana, x {XI, with a subspace of A(Kx) ={f E C(Kx) ; f is X = K lytic on the interior of Kx). By Mergelyan's Theorem, since a K x is connected, the polynomials are dense in A(Kx). Since Y contains the polynomials, i.e. the functions of the form (2,x) p(Z), where p is a polynomial, YIK is dense in A(Kx). Since A (Kx) has the approximation property (Eifler [21]),Y 1 K has the approximation property. -+

152

E

-

PRODUCT

REFERENCES FOR CHAPTER 8 . ARON and SCHOTTENLOHER

151

BIERSTEDT

, [6]

BIERSTEDT and MEISE BRIEM, EIFLER

[7]

LAURSEN a n d PEDERSEN

[21]

ENFLO

[22]

RUDIN

[55]

SCHAEFER

[5 7

SCHWARTZ

[ 591

WARNER

;3]

[66]

[93

C H A P T E R

9


5

1.

VALUED FIELDS

DEFINITION 9.1. L e t F b e a f i e l d . A ( r a n k o n e o r r e a l - v a l u e d l I 1 : F '+ ?!I s a t i s f y i n g t h e f o l v a l u a t i o n of F i s a m a p p i n g

-

lowing c o n d i t i o n s :

(1)

I

(2)

I

x x

I I

2

(3)

IXYI

(4)

Ix + yl 5 1x

i

x

for u z z

0,

E

F; x = 0;

= 0, i f a n d o n l y i f , =

I

x

I

-j

*

I

y 1, f o r a l l x , y E F; + (yi, f o r a l l x,y E F.

is a valuation of F , we say that (F,,- 1 a v a l u e d f i e l d or a f i e Z d w i t h v a l u a t i o n . If

)

is

Any field F can be provided with a valuation, namely the t r i v i a l valuation, defined as follows: i x 1 = 1 for all and

x E F , x # 0,

I x / =0

if

The field IK (IK = IR or C) with its value is another example of a valued field. DEFINITION 9 . 2 .

I

-

1

Let

(F

,I

-

1)

Ix + yi 5 max

(1

usual absolute

b e a v a l u e d f i e l d . Ve s a y

x ,y

i s n o n a r c h i m e d e a n i f , for a l l (5)

x = 0.

x

I

I

I

E

that

F, we h a v e :

yl).

The following example, known as the p - a d i c v a l u a t i o n , provides us with a nontrivially valued nonarchimedean field. EXAMPLE 9.3. Let F be the field Q of all rational and let p be any prime number . Every x E Q, x # 0, written in a unique way in the form x = $ . d

numbers can be

b

where a and b cannot be divided by p . We define the

p -adic

154

14ONAkCH IMEDEAII APPROXI MAT1ON THEORY

v a l u a t i o n of Q by setting

101

P

= 0.

Further examples of nonarchimedean valuations are provided by: (a) (b)

the trivial valuation on any field; any valuation on a field with characteristic p # 0. In particular, all valuations of a finite field are nonarchimedean.

DEFINITION 9.4.

Let

(F

,I

*

n o n a r c h i m e d e a n , we s a y t h a t

1)

b e a v a l u e d f i e l d . If

I

*

I

I *I

i s not

i s archimedean.

Regarding archimedean valuations we have the follmirlg result. (See ~ o n n a [ 7 3 ] 1 . F w i t h an a r c h i m e d e a n v a l u a t i o n is i s o m o r p h i c t o a s u b f i e l d o f t h e field C of a l l c o m p l e x n u m b e r s , a n d t h e v a l u a z i o n of F i s t h e n u p o w e r of THEOREM 9 . 5 .

(Ostrowoki's

Theorem) A f i e l d

t h e usual a b s o l u t e v a l u e .

Any valued field is a metric space. Indeed, for any x and y in a valued field (F , I I ) , define the distance between them by d(x,y) =

IX

-

YI .

One easily verifies that d is a metric on F . S u p p o s e m t h a t (F, 1 * 1 ) is nonarchimedean. Then

=

Thus, for all (6)

max(d(x

x ,y , z

E

, y ) , d(y , z ) 1 .

F we have

d(x,z) 5 max(d(x,y),d(y,z)).

155


The above i n e q u a l i t y i s c a l l e d t h e u l t r a m e t r i c i n e q u a l i t y .

As a f i r s t example of what ( 6 ) way i m p l y , c o n s i d e r on any v a l u e d f i e l d ( F ,

I

*

I)t h e

open b a l l ( r e s p . c l o s e d b a l l ) o f

< r}

< -

.

r)

< r l . ) nonarchimedean, yo E F

i n t h e c l o s u r e of

Br(x) such t h a t

-

lYo

xn

-+

yo

0

such t h a t

be

162


for all

t E X.

I

t = x, we obtain

Making

A

I

0 . T h e r e f o r e ( F , 1 * 1 ) i s n o n a r c h i m e d e a n too.

By c o n v e n t i o n 9 . 1 6 ,

x

ists

Let

€

witn

E,

F

, and

F

x # 0 , and

4.

.

If

Ix

w e set

I

110

11

E

= 1

= 0,

E

medean norm o n

9

I * 1 be the t r i v i a l i s a n y vector s p a c e over

F be a n y f i e l d , a n d l e t

v a l u a t i o n of

then

11

-

I/

c a l l e d t h e t r i v i a 2 n o r m o n E.

X

be a c o m p a c t H a u s d o r f f s p a c e a n d l e t

be a normed s p a c e over a v a l u e d f i e l d ( F

s p a c e over n o t e d by

,1

x E E with i s a nonarchi-

all

VECTOR- VALUED FUNCTIONS. Let

(F

for

(C(X ; E)

1) .

The

C(X;E)

11

[I 1

vector

F o f a l l c o n t i n u o u s E - v a l u e d f u n c t i o n s on

1):

for a l l

,I

(E,

X

i s a l s o a normed space over t h e v a l u e d

,

de-

field

j u s t define

f E C(X;E).

, 11 Let

/I)

When ( E

,/ I

-

11)

i s nonarchimendean,

i s nonarchimedean too.

A C C ( X ; F ) be a s u b a l g e b r a a n d l e t

be a v e c t o r s u b s p a c e w h i c h i s a n A - m o d u l e , i s t o describe t h e c l o s u r e of given a function

f

in

W

in

C(X ;E)

TJ

C C(X ;E)

c W.Our a i m more g e n e r a l l y

i . e . AW

, or

C ( X ; E ) t o f i n d t h e nonarchimdean distance

164

of f

NONARCHI FlEDEAN APPROXIMATION THEORY

from W , i.e. to find d(f

;W) =

inf

{I:

f - g

11

: g E W 1.

To solve this problem in the line of argument of Chapter 1 , we need a "partition of the unity" result. To this end, we shall adapt the proof of Rudin c 5 5 1 , section 2.13, to the nonarchimedean setting. Namely we shall prove the following. LEP4MA 9.18. and l e t

Y be a 0 - d i m e n s i o n a l

Let

c o m p a c t H a u s d o r f f space,

b e a f i n i t e o p e n c o v e r i n g of

V1,... ,vn

(~~i.1)

let

b e a nonarchimedean valued f i e l d . T h e r e e x i s t s f u n c t i o n s hi E C(Y;F) ,

i = l,...,n, s u c h t h a t

(a)

hi(y) = 0

(b)

11

(c)

hl +

hi

//

for all

5 1, i

...

+ hn

Vi

, i

=

l,...,n;

l,...,n:

=

=

y

1

on

Y.

PROOF. Each y E Y has a closed and open neighborhood W(y) c Vi for some i (depending on 11). By compactness of Y, there are points y 1 , where we such that Y = M 1 U .. . U W m , 1 Ym have set W . = W(y.) for each j = l,...,m. If 1 5 i 5 n, let

...

3

3

Hi be the union of those W

j

which lie in Vi

be the characteristic function of

.

Let

fi

E

C(Y;F)

H i , i = l,...,n. Define

hl = fl

h2 = (1 - fl) f2

. . . . . . . . . . . . . hn = (1 - fl) (1 - f2) Then

H i C Vi

hi(y)

=

0

implies that

for

y

!j

Vi

fi(y) = 0

too, i

=

...

(1 - fn-l) fn

for all

y

j?

Vi and

so

1,.. . ,n. This proves (a).Clearly

llhi I I 5 1, i = l,...,n, since hi takes only the values 0 and 1, which proves (b). On the other hand, y = H I U ... u H n and hl +

...

+ hn

=

1 - (1 - fl) (1 - f2)

... (1 -

fn).

165


y E Y , a t least one

Hence, g i v e n

fi(y) = 1

and t h e r e f o r e

h l ( y ) + , , . . . + h n ( y ) = 1. This proves

(c).

THEOREM 9 . 1 9 . A C C(X;F) be

Let

b e a n o n a r c h i m e d e a n normed s p l z c e .

E

a s u b a l g e b r a and l e t

s p a c e w h i c h is a n . & - m o d u l e . L e t

where

Let

W C C ( X ; E ) be a v e c t o r subf E C(X;E).

Then

PA d e n o t e s t h e s e t o f a l l e q u i v a l e n c e c l a s s e s S C X m o d u l o

X/A.

Before p r o v i n g Theorem 9.19 l e t u s p o i n t o u t t h a t

it

implies t h e following r e s u l t . THEOREM 9 . 2 0 .

Let

E

, A , \%? and

f

b e a s i n Theorem 9 . 1 9 .

Then

f b e l o n g s t o t h e u n i f o r m c l o s u r e of W i n C ( X ; E ) i f , and only i f , f l S i s i n t h e u n i f o r m c l o s u r e of lence class

modulo

S C X

WIS

in

C ( S ; E ) for e a c h equiua-

X/A.

The a b o v e Theorem 9 . 2 0 i s t h e n o n a r c h i m e d e a n a n a l o g u e of Nachbin's Stone

- Weierstrass

Theorem f o r m o d u l e s ( T h e o r e m 1.5)

a n d 9 . 1 9 i s t h e " s t r o n g " S t o n e - W e i e r s t r a s s Theorem €or m o d u l e s ( t e r n i n o l o g y of Buck PROOF OF THEOREM 9 . 1 9 .

[12] )

.

L e t us p u t

d = d(f;W) and

< d . To p r o v e t h e reverse i n e q u a l i t y , l e t Clearly, c o u t loss of g e n e r a l i t y w e may a s s u m e t h a t A t h e subalgebra A'

of

C(X;F) g e n e r a t e d b y

E

< O.With-

is unitary-Indeed, A

and t h e c o n s t a n t s

i s u n i t a r y , and t h e e q u i v a l e n c e r e l a t i o n s X / A and X / A ' are t h e same. Moreover, s i n c e W i s a vector space, W i s a n A-modul e i f , and o n l y i f , W i s a n A ' - m o d u l e . L e t Y be t h e q u o t i e n t s p a c e o f q u o t i e n t map

71

.

For a n y

S E PA

,since

X

modulo X / A ,

d ( f IS; P I I S ) < c +

with E

,there

166

NONARCHIMEDEAN APPROXIMATION 'iH EORY

exists

some f u n c t i o n

11

-

ws(t)

Then y E'

f(t)

,I

KS =

CX E

ws

+

< c

in the for a l l

E

-

X; ;/W,(X)

W

A-module t E S.

f(x)

I/

such t h a t

Let

c

+ €1.

i s compact a n d d i s j o i n t f r o m S. Hence, f o r e a c h y E Y , -1 S = IT (y). This implies t h a t (KS) , i f KS

IT

i s e m p t y . By t h e f i n i t e i n t e r s e c t i o n p r o p e r t y , t h e r e i s a f i n i t e

set Ki

{yl =

,

KS

, ... , ynl c for

S =

TI

such t h a t

Y

-1

TI

(K~n )

( y i ) , i = l,...,n.

s e t g i v e n by t h e complement o f

TI

,

(Ki)

... n

Let

= @ , where

(K,)

b e t h e open sub-

Vi

i = l,..

TI

., n .

Y

is a

0-

d i m e n s i o n a l c o m p a c t H a u s d o r f f s p a c e . Hence, by Lemma 9 . 1 8 t h e r e

exist functions

Put

E

hi(y) = 0

(b)

[/hi

(c)

hl TI,

// 5

1

,

gi

since

Moreover

and

. .., n ,

i = 1,

IIgi

(1

hi ( y ) = 0 f o r a l l y

1 , i = l,...,n,

g =

C

i=l

Ilg(x)

CJ. w i i

-

w e have

f(x)

where

I/

< c

+

and

,

wi

= ws

E,

for a l l

g1

with

+

space

, i=l,...,n.

E 'rr(Ki)

... +

g n = 1 on X .

-1

, i=l,...,n.

S =

x E X.

in the

gi(x) = 0 f o r a l l

Notice t h a t

n

E X

C(X;F), i = l , . . . , n ,

b e l o n g s t o t h e c l o s u r e of A

gi

,

x

E

i s c o n s t a n t on e v e r y e q u i v a l e n c e c l a s s o f X rnodulo X / A .

gi

i = 1,2,...,n.

Then

i = l,...,n;

hn = 1.

C ( X ; F ) , for; e a c h

Let

,

i = 1,.. . , n ;

s o t h a t w e have

By Theoren! 5 . 1 2 , x E Ki

such t h a t

y JZ Vi

for all

... +

+

,

C(Y;F), i = l , . . . , n

(a)

gi = h i o

each

hi

TI

(yi)

Indeed,

for

any

167


NOW, for each 1 0

Let

be g i v e n . Then

f E C ( X ; E ) be i n flK

is i n

C(K;E);

A (W)

.

Let

W

is

K C X

AIK C C(K;E)

is

N 0N A R CH I ME DEAll A P P RO X I MAT I 0N T H E0 R Y a separating unitary subalgebra of (A1 K ) - m o d u l e .

Since

local, then

fIK

Therefore a

g E W

for a l l

f E C(X;E)

PROOF:

X

-

,E ,A

and

-

{x

1c X

W

Thus

open c l o s u r e o f in

E ,

in

W

f o r each

and

E

E A(W).

COROLLARY 9 . 6 7 .

i s c o m p a c t , t h e c o n d i t i o n i s obvig E

> 0, t h e r e is

By Theorem 9 . 6 5 ,

Let

X

,E ,A

such t h a t

p a c t - open topology o f

x E

f E ?;j,

W

and

sume t h a t W c o n t a i n s t h e c o n s t a n t s .

PROOF:

C(X;E)

x E X.

Conversely, i f t h e condition is v e r i f i e d , then

e

f

i n C(K;E).

b e a s i n Theorem 9.65. Then

By c o n t i n u i t y t h i s i s s t i l l t r u e i n a n e i g h b o r h o o d X.

an

WIK is

.

g) < E

f ( x ) E w(x)

Since each x E X

given

PK(f

i s i n t h e compact

ously necessary.

is

W1 K

c a n be f o u n d s u c h t h a t

Let

and o n l y i f ,

i f ,

and

a n d b y Theorem 9 . 3 5 ,

b e l o n g s t o t h e u n i f o r m c l o s u r e of W i K

x E K, i.e.

COROLLARY 9 . 6 6 .

C(K;E) ;

,

f l K E A (Wl K )

197

of

U

x

as desired.

b e a s i n T4eorem 9 . 6 5 .

Then

W

in

As-

i s d e n s e i n t h e com-

C(X;E).

Apply C o r o l l a r y 9 . 6 6 ,

noticing that

for all

W(x) = E ,

x.

COROLLARY 9 . 6 8 . i s 0-dimensional

Let

X

and

and l e t

l e f t ) i d e a l , and f o r e a c h

E 1


c C(X;E)

x E X,

let

Assume

X

be a c l o s e d r i g h t ( r e s p . Ix b e t h e c l o s u r e i n

E

of t h e s e t I ( x ) = { f ( x ) ; f E I} then

Ix i s a c l o s e d r i g h t

.

(resp. l e f t ) ideal i n

I = {f E C(X;E);

E,

f ( x ) E Ix f o r a l l

and

x E X}

.

198

NONARCH I MEDEAN APPROXI MAT1 ON THEORY

PROOF:

The f a c t t h a t I = {f E

f ( x ) E Ix

C(X;E);

for all

x E X I

f o l l o w s f r o m C o r o l l a r y 9 . 6 6 a n d t h e h y p o t h e s i s t h a t I i s closed, i f w e c a n show t h a t

C(X;E)

is a unitary separating

subalgebra

i n t h e s e n s e o f D e f i n i t i o n 9 . 3 1 . An a n a l y s i s o f t h e p r o o f o f T h e orem 9 . 6 5 shows t h a t i n f a c t a l l w e n e e d t o p r o v e i s t h a t

IK

C (X;E)

Now

is 0-dimensional,

X

C (K;E), f o r a l l compact subsets K C X.

is separating i n

x # y

i n t h e sense t h a t given such t h a t

f(x) = 1

therefore

Cb(X;F)

in

X

f (K)

By K a p l a n s k y ' s Lemma, t h e r e i s a p o l y n o m i a l p ( l ) = 1, p ( 0 ) = 0

Ip(t)l 5 1

and

g = h C9 e

h = p o f . Then

define

/I 5

points

t h e r e i s some f E Cb(X;F)

f ( y ) = 0 . Now

and

separates

i s compact i n F.

p : F + F such t h a t

t E f(K). Let

for a l l

belongs t o

C(X;E),

g(x) =el

g(y) = 0

and

C(X;E)/K

i s s e p a r a t i n g i n t h e s e n s e of D e f i n i t i o n 9.31.

IIq(y)

The p r o o f t h a t f o r each

E ,

1

for all

I(x)

T h i s shows

that

is a r i g h t (resp. l e f t ) idealin

x E X I i s e a s y . Then

closed r i g h t (resp. l e f t ) i d e a l i n E

y E K.

us

Ix,being its closure, is E

a

f o l l o w s from t h e f a c t t h a t

is a topological algebra. Let

COROLLARY 9 . 6 9 .

that

E

X

and

E

be as i n Corollary 9 . 6 8 .

i s s i m p l e . Then any cZosed t w o - s i d e d

ideal i n

c o n s i s t s .,fa22 f u n c t i o n s v a n i s h i n g o n a c Z o s e d s u b s e t of o v e r , a n y maximal t w o - s i d e d c Z o s e d i d e a l i n

C(X;E)

for some p o i n t

x E X.

form

{f E C ( X ; E ) ;

PROOF:

f ( x ) = 0)

The p r o o f i s s i m i l a r t o t h e case o f

X

is

Assume C(X;E)

X.Moreof the

compact a n d

the

u n i f o r m t o p o l o g y , so w e o m i t t h e d e t a i l s .

9

10.

THE NONARCHIMEDEAN STRICT TOPOLOGY.

I n t h i s s e c t i o n X i s a ZocaZZy c o m p a c t and

E

Hausdorff space,

i s a n o n a r c h i m e d e a n normed s p a c e o v e r a l o c a l l y

valued f i e l d (F,

I

I).

ed continuous E - v a l u e d vex topology

On t h e v e c t o r s p a c e

Cb(X;E)

compact

of a l l bound-

f u n c t i o n s l e t u s d e f i n e a l o c a l l y F-con-

B , c a l l e d t h e s t r i c t t o p o Z o g y , by s e t t i n g

199


XI

P ( f ) = supt l / $ ( x ) f ( x ) l l ; x E

@

fo

all

,

f E Cb(X;E)

such t h a t , given

I$(x)

It

i

I$(t)I >

E X;

g e b r a , and l e t A-module.

x E X

for a l l

be

A C Cb(X;F)

W C Cb(X;E)

i s 6

Then W

o u t s i d e of

.

K

I$ E C(X;F)

c

such

X

I t follows that E

> 0.

separating unitary subal-

c!

be a v e c t o r subspace which

an

is

- local. l e t , u s d e f i n e w h a t wemean

B e f o r e p r o v i n g Theorem 9 . 7 0

by s a y i n r j t h a t W

is @ - l o c a l .

DEFINITION 9 . 7 1 .

If

any

which i s i n W

f E Cb(X;E)

K

i s compact and open f o r e v e r y

E }

Let

THEOREM 9 . 7 0 .

t h e r e i s a compact s u b s e t

> 0

E

< E

Co(X;F) de o t e s

C(X;F) c o n s i s t i n g o f a l l those

t h e v e c t o r subspace of that

@ E C o ( X ; F ) . Here

where

W

we s a y t h a t

C Cb(X:E),

t h e n i n t h e s t r i c t c l o s u r e of ?J

is @

W

- localif

l o c a l l y a t all p o i n t s of X in

is

Cb(X;E).

x E X I t h e r e i s a n o p e n and x i n X , t h e " - c h a r a c t e r i s t i c funct i o n @ K of K i s such t h a t $,(x) = 1 and Q K E C o ( X ; F ) . Hence, a l l f u n c t i o n s f i n % , t h e B - c l o s u r e o f W i s Cb(X;E) r. a r e i n !V l o c a l l y a t a l l p o i n t s o f X , i . e . Ab(W) 3 i. T h e r e 4 f o r e 141 i s B - l o c a l i f , a n d o n l y i f , Ab(W) = the B - closure Since,

f o r each p o i n t

compact neighborhood

of

W

in

Let

of

(Here

Cb(X;E).

LEPIYA 9 . 7 2 .

K

A

w,

z

Ab(W)

there e x i s t a f i n i t e set I

, ... , $ n

Q2

F o r e v e r y x E X,

x1

Kx

, x2 ,

c

X I not containing

... , xn

E X

i n t h e u n i f o r m c l o s u r e of A

for

t E Kx

for all

Cb(X;E)).


t h e r e be g i v e n a compact s u b s e t

$1

n

= z(W)

i = 1,2,

...,n ;

$l

let

x .Then

and f u n c t i o n s s u c h that $ i ( t ) = O

+ ... +

$n = 1

in

X

i

PROOF:

tion

I n t r o d u c e t h e nonarchimedean S t o n e BFX

3 X.

p a c t , t h e sets

f - Cech

T h i s i s d o n e as f o l l o w s . S i n c e

vr

= {a E F;

/a1 5 r }

F

compactifica

-

i s l o c a l l y com-

are compact, f o r

every

200

NONARCH IPIEDEAN A P P R O X I M A T 1 ON THEORY

r E IR

,

r > 0 . Now e a c h

e : X

r f > 0 . C o n s i d e r t h e may

f o r some

f

n

+

*

s i n c e t h e space

a

-

0

dimensional

t h i s mapping i s a t o o o l o q i c a l embedding, a n d of

e(x)

II

in

- valued

unique m n t i n u o u s F

B Cb (Y;F) + C(B,X;F)

and l e t

TI

(x) n X

77

(x)

fl

of

Y

: B,X

a compact 0 - C i m e n s i o n a l

f

+

.

BF X

to

a

The mppinq

C(R,X;F)

.

:.zt

B = BA.

modulo t h e e:Tuivalence re-

f3X

h e t h e q u o t i e n t ma?.

Y

has

Bf, i s t h e n a Sanach a l -

I-+

Cb(X;F) a n d

Consider t h e q u o t i e n t snace

77

istheclosure

BFX

f E Cb(X:F)

Rf

extension

d e f i n e d by

g e b r a isomornhism between B ,

space,

'f

A s i n t h e classical case, each

lation

Hausdorff

.

V

fEH

f

(f(t)IfEH ;

is

X

d e f i n e by

Vr

fEH x

f(X) c Vr

i s such t h a t

f E C (X;F) = H b

Hausdorff snace.

If

Then Y i s

x E XI

then

X modulo X / A . T h e r e f o r e i s d i s j o i n t f r o n Kx Thus

i s a n equivalence class i n

X = {x}

n ( x ) E' n ( K x ) .

and

I

(x) n X

IT

.

Hence

By t h e f i n i t e i n t e r s e c t i o n p r o r J e r t y , t h e r e i s a f i n i t e

set

{x1,x2

I

...,x n } c 1 n 1

By Lemma 9 . 1 8 ,

such t h a t

X

...

nn(Kx 1 = @ . n

there e x i s t functions

hi

E

C(Y;F), i = l 1 2 , . .

.

such t h a t (a)

hi(y) = 0

for all

y

E n(Kx,)

( i = 1, ...,n )

1

(b)

I / hi

(c)

hl

$i = hi o n

Put

I

+

5 1, f o r a l l h2

+

i = 1,2,

... +

hn = 1

..., n .

belongs t o t h e uniform c l o s u r e of that

$i = $i I X

of

in

A

,

i = 1,2,.

.., n

i = 1,2,...,n.

on Y .

By K a p l a n s k y ' s Theorem, B

in

C(B,X;F).

It is

b e l o n g to t h e u n i f o r m

C b ( X ; F ) a n d h a v e all t h e d e s i r e d p r o p e r t i e s .

'i clear

closure

201

NONARCHI MEDEAN APPROXIMATION THEORY

PROOF OF THEOREPI 9 . 7 0 . $ E Co(X;F)

of

> 0

E

For each x x i n X such t h a t

t E Ux.

for all

Let

Q1

and A

in

4

nb(w)

t h e r e is

E X,

= A(W,

gx E W

/ l Q ( (tf)( t ) - g x ( t ) )

x k? K x .

, $2 ,

f t

an d neighborhoodu,

In particular

= { t E X;

Kx

p a c t and

of

and

c

n c~(x;E). Let be g i v e n . W e may assume 11 Q I[ > 0.

Let

//

>

- .

E

1

.

By Lemma 9 . 7 2 t h e r e e x i s t

... , Q n

E CbiX;F)

Then

x l , x2

Kx

i s com-

,...,xn

belonging t o t h e uniform

closure

Cb(X;F) s u c h t h a t

(b)

i Q i ( t )5 1

(c)

Qi

+ $2 +

For each

(d)

t E X

for all

~

...

+ Q n = l

i = 1,2,...,n

l @ i ( t-) hi ( t )I

Approximation of Vector Valued Functions

Approximation of vector valued functions

Spaces of Vector-Valued Continuous Functions

Banach Spaces of Vector-Valued Functions

Advanced calculus: Of real-valued functions of a real variable and vector-valued functions of a vector variable

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Banach Spaces of Vector-Valued Functions (Lecture Notes in Mathematics)

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Volterra and integral equations of vector functions

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