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 ) ) <
class (mod. A ) , K
C
Y a compact
f o r a l l x E K. W e l e t LA(W) be t h e s e t o f a l l s u c h
subset, E,
functions.
Notice t h a t LA(V) i s a l w a y s a c l o s e d s u b s e t of C ( X ; E ) , c o n t a i n i m W. I t f o l l o w s 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 ) i f , and o n l y i f , LA(W) i s c o n t a i n e d i n t h e compact-open
closure
of
W
i n C(X;E).
Notice t o o t h a t LA(W) = L B ( W ) , i f B d e n o t e s t h e subalaebra o f C ( X ; M ) u e n e r a t e d by A and t h e c o n s t a n t Thus 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 )
functions.
i f , and o n l y i f ,
W
is
localizable under B i n C(X;E). When E = IK
,
e v e r y s u b a l s e b r a A C C ( X ; M ) i s a mod-
u l e over i t s e l f . The d e f i n i t i o n o f l o c a l i z a b i l i t y i s
motivated
by t h e c l a s s i c a l S t o n e - h r e i e r s t r a s s Theorem. I n d e e d , w e have t h e f o l l o w i n u r e s u l t which c o n n e c t s t h e n o t i o n o f
localizability
w i t h t h e u s u a l s t a t e m e n t o f t h e S t o n e - W e i e r s t r a s s Theorem. Theorem 1,
5
17, Nachbin [43] )
.
L e t A C C(X;M) be a M - s u b a t g e b r a , and f E C ( X ; M ) . T h e n f E LA(A) if, a n d o n l y if, t h e f o l l o w i n g
PROPOSITION 1 . 2
(See
let two
4
COMPACT
- OPEN
TOPOLOGY
conditions are s a t i s f i e d : (1) f o r e v e r y x E X s u c h t h a t f ( x ) # 0 , t h e r e e x i s t s q E A s u c h t h a t q ( x ) # 0; ( 2 ) f o r every x,y E X such t h a t f ( x ) # f ( y ) , e x i s t s q E A such t h a t g ( x ) # u ( y ) .
there
( a ) Suppose f E L A ( A ) . L e t x E X b e s u c h t h a t f ( x ) # 0 . A s s u m e t h a t q ( x ) = 0 f o r a l l 9 E A. L e t Y C X be t h e equivalence PROOF
class (mod. A ) t h a t c o n t a i n s x , and l e t K = { X I . By h y p o t h e s i s , t h e r e i s q E A s u c h t h a t I f ( x ) - q ( x ) I < E = \ f ( x )1 . Since q ( x ) = 0 , t h i s i s a c o n t r a d i c t i o n . T h e r e f o r e (1) i s s a t i s f i e d . The p r o o f t h a t ( 2 ) i s s a t i s f i e d i s a n a l o a o u s , so w e o m i t the details. Suppose now c o n d i t i o n s (1) and ( 2 ) are satisfied. L e t Y C X b e an e q u i v a l e n c e class (mod. A ) . By ( 2 ) , f i s conthen s t a n t on Y. L e t u E IK b e i t s c o n s t a n t v a l u e . I f u = 0 , CJ = 0 E A c o i n c i d e s w i t h f on Y. A s s u m e now t h a t u # 0 . By (11, t h e r e i s g E A s u c h t h a t q ( x ) # 0 , where x E Y i s an a r b i t r a r y p o i n t f i x e d i n Y . Then g ( y ) = u ( x ) f o r a l l y E Y. Theref ore h = ( u / q ( x ) ) q b e l o n g s t o A and h ( y ) = u = f ( y ) f o r a l l y E Y. Hence f E L A ( A ) .
5
3
PRELIMINARY LEMMAS
I n t h i s s e c t i o n w e s h a l l o b t a i n two lemmas t h a t w i l l b e u s e f u l i n t h e "approximate p a r t i t i o n o f u n i t y " needed i n t h e p r o o f o f t h e main theorem o f t h i s c h a p t e r . The second o f t h o s e lemmas i s due t o J e w e t t [32], who employed it i n h i s proof of a v a r i a t i o n o f t h e S t o n e - l d e i e r s t r a s s theorem. I t i s a c o r o l l a r y o f t h e c l a s s i c a l Weierstrass polynomial a p p r o x i m a t i o n theorem, b u t w e p r e f e r t o p r e s e n t J e w e t t ' s d i r e c t p r o o f , t o make our v e r s i o n o f t h e S t o n e - W e i e r s t r a s s theorem i n d e p e n d e n t o f Weierstrass theorem. LEMMA 1 . 3
Let A
c Cb(X;IR) b e a s u b a z g e b r a c o n t a i n i n g t h e con-
s t a n t s , and l e t Y C X b e an e q u i v a l e n c e c l a s s (mod. A).For every E > 0 , and e v e r y o p e n s u b s e t U c X , c o n t a i n i n g Y , s u c h t h a t t h e c o m p l e m e n t of U i s c o m p a c t , we c a n f i n d CI E A s u c h t h a t 0 < u < ~
< 1, ~ ( y = ) 1 for a l l y
E Y,
and a ( t ) <
E
f o r t $ U.
-
COKPACT
PROOF Then
Choose x E Y. F o r e a c h f E A , l e t X f = { t E X ; f ( t ) = f ( x ) ) .
Y =
n{ x f ;
f E A, f is n o t constant} open
sets
f E A , f i s n o t c o n s t a n t ) . By c o m p a c t n e s s , w e c a n
find
i s c o v e r e d by t h e f a m i l y
The compact s e t X \ U {X\Xf;
a f i n i t e number of f u n c t i o n s f l , . . . , f n constant, such t h a t
c
(X\U)
1
c
(Xf
For each i = l , . . . , n ,
-
ai = ( f i
Then 0 <
-<
qi
< 1, and hi
= 1
-
n
1
i qo =
. Define (hl+ . . . +
Then qo E A , 0 < a
... n
X f
I Ifi
/2 qi
beins
-
n
1 c U. by
E A
1
fi(X)
]
2
.
belonas to t h e alaebra A, 0
1, and h i ( v ) = 1 f o r y E Xf
complement o f Xf
none o f t h e n
E A,
d e f i n e cri
2
fi(X))
of
. .. u ( X \ X f n ) .
1u
(X\Xf
Consenuentlv, Y
< hi
5
OPEN TOPOLOGY
i
,
<
< 1 for t i n the
and h i ( t )
hn)/n.
n
< 1, o o ( y ) = 1 €or y E X
0 -
f
f o r t i o r i f o r y E Y. I f t $ U , t h e n t f! X f
i
1
... n X f n '
f o r some i .
a Hence
n o ( t ) < 1. S i n c e t h e complement o f U i s compact,
m = sup C c r o ( t ) ;
t
E'
U } < 1.
For k E N s u f f i c i e n t l v l a r a e , mk < E . Then u = a 0 k helonas t o A; 0 < a < 1; o(y) = 1 f o r a l l v E Y ; and a ( t ) < E f o r a l l t $ U ,
as d e s i r e d . LEMMA 1 . 4
n o m ia l q:IR
-
Let 0 <
E
< 1
-
E.
Given 6 > 0 , t h e r e e x i s t s a p o l y -
IP s u c h t h a t
(a)
0 < cr(t) < 1, for a l l 0 < t < 1;
(b)
0 < q ( t ) < 6 , for a l l 0 < t <
(c)
1
-
6 < q(t)
5 1, for a l l 1 -
E;
E <
t < 1.
6
COMPACT
- OPEN
TOPOLOGY
PROOF ( J e w e t t [ 3 2 ] ) The p o l v n o m i a l q: IR IF will be o f t h e form q ( t ) = 1-(1 t m I n . Choose a n i n t e q e r r s u c h t h a t (3/4Ir<6. f
-
Then c h o o s e i n t e q e r s m a n d s s u c h t h a t
3 .-
1
< s < -
(l-EIm
1 (l-EIm
<
-r6 . _1m E
-
L e t n = rs a n d n o t e t h a t nEm < 6 , and < s(l
-
€1
m
-
<
< 1. Hence
L e t p ( t ) = (1
p(t) > 1
(3/4)
-
that t m l n . From ( i ) ,i t f o l l o w s 6, for all 0 < t < E . Hence q ( t ) = 1 p(t) < 6, for
-
all 0 < t
5 E . From ( i i ) ,it f o l l o w s t h a t p ( t ) < 6 for all 1- E < t 5 1. Hence q ( t ) > 1 - 6 , f o r a l l 1 - E 2 t 5 1, and t h e p r e s e n t lemma i s t r u e . W e now e s t a b l i s h t h e main a p p r o x i m a t i o n t h e o r e m f o r v e c t o r - v a l u e d c o n t i n u o u s f u n c t i o n s , namelv a S t o n e - W e i e r s t r a s s t h e o r e m f o r modules. The p r o o f i s d u e t o ' s . Machado [ 3 8 1 . When , i t w a s p r o v e d by L. Nachbin (see Theorem 1, 5 1 9 , [40] I ,
E = K
u s i n g t h e classical Stone-Weierstrass theorem f o r alnebras,thich
i s i n t u r n a corollary o f t h e t h e o r e m f o r modules
established
by Nachbin. To see t h i s , it s u f f i c e s t o n o t i c e t h a t , by p r o p o s i t i o n 1.2,
t h e classical Stone-Weierstrass theorem states
every subalaebra A
c
C ( X ; R ) i s l o c a l i z a b l e under
itself
that in
H o w e v e r , Machado's p r o o f , r e l y i n a o n l y on Lemmas 1.3 and 1.4, p r o v i d e s i n p a r t i c u l a r a v e r v e l e q a n t and d i r e c t p r o o f o f t h e Stone-Weierstrass theorem. W e f i n a l l y remark that the f a c t t h a t w e are d e a l i n a w i t h vector-valued € u n c t i o n s , causes no a d d i t i o n a l d i f f i c u l t y . I n d e e d t h e p r o o f f o r E = Mwould j u s t s u b s t i t u t e estimates w i t h a b s o l u t e v a l u e f o r estimates with
C(X;IR).
seminorms on E.
COMPACT
5
4
-
7
OPEN TOPOLOGY
STONE-WEIERSTRASS THEOREM FOR MODULES
THEOREM 1.5 A C C ( X ; I R ) be a subalgebra. W C C ( X ; E ) is localizable u n d e r A in C ( X ; E ) .
Every
A-modu l e
Since L A ( W ) = LB (W) , and W is an A-module if , and only if, it is a €3-module, where B denotes the subalsebra of C ( X ; I R ) generated by the A and the constants, we may assume without loss of generality that A contains the constants. Moreover, the case of a general X follows easily from the case of X compact, since for any compact set K C X , if Y c K is an equivalence class (mod. A I K ) , then there exists a unique equivalence class Z c X (mod. A ) such that Y = Z n K , and Y is compact subset of 2. Hence, we may also assume without loss of aenerality that X is compact. Let then f E L A ( W ) . We claim that f E k. Let O < E < 1 and p E cs(E) be siven. For everv equivalence class Y C X (mod. A ) , let wy E 147 be such that p ( f ( x ) - wy (x)) < ~ / 3for all x E Y. Then PROOF
uy
= ft E
x; p(f(t)- wy(t))<
E/31
is an open subset of X , containinq Y. By Lemma 1.3 gy E A , such that 0 < ay 5 1, g y ( y ) = 1 for all gy(t) < ~ / 3for all t 1 u y . Let
vy
= It E
x;
there y E Y,
is and
uy(t) > 1 - ( E / 3 ) 1 .
Then Vy is open and contains Y. Moreover V y c Uy. Indeed, if t $ U y r then ay(t) < ~ / 3 . If t E Vy were true, then (E/3) > > 1 - (~/3) , and this contradicts E < 1. This proves our claim. By compactness of X, there exist Y1,...,Y n equivalence classes u Vn, where for each i=1,. .,n,Vi (mod. A ) such that X = V1 u -1 denotes Vyr for Y = Yi. Let 0 < 6 < (3n (M+1)) E, where M is the constant max{llfllp, IIf - wlllp,...,!!f - wnll 1. To simP plify notation, we have written wi = wy, for Y = Vi. By Lemma 1.4, there is a polynomial q: P + IR such that
...
.
8
COMPACT
OPEN TOPOLOGY
(b)
0
5
T(t) < 6, for all 0
(c)
1
-
6 < cr(t)
L e t cri CI
= a
5
0
E A,
i 0
5
cri(t)
(2)
1
-
6 <
and U i
5
5t 5
5
ai
E/3;
1, f o r a l l 1 - ( E / 3 )
for Y = Y
cry,
0
(1)
f o r i=1,2,...,n,
i t e m 2.13,
-
(i=1,2
5 t 5
1.
,.. . , n ) .
Then
1 and
< 6 , i f t $ Ui
-_ < 1, i f x
rSi(X)
= Uyr for Y =
Y
i'
E
vi
F o l l o w i n a Rudin
[55],
l e t us define
(3)
hl
+
h2+
...+hn
= l-(l-crl)
(l-a2)
... (1-on).
< hi 5 1, - q i 5 1 a n d 0 - 1-0i -< 1, w e see t h a t 0 f o r a l l i = 1 , 2 , . . . , n . Given x E X , t h e r e i s some i n d e x i such t h a t x E V i . By ( 2 ) , q i ( x ) > 1 - 6. W e now u s e ( 3 ) and o b t a i n
I n view of 0 <
n
On t h e o t h e r h a n d , h i ( t )
< cri(t)
and f o r m u l a
(1)
lead t o
(5)
0
f o r a l l i=1,2,...,n. Let w =
5
hi(t)
< 6, i f t $ Ui
n C
hi wi
E
W. F o r e a c h x
E X,
we have
i=l
For t h e f i r s t term i n t h e rirrht-hand
t a i n , hy ( 4 )
,
t h e estimate:
side, we
oh-
CD!,'PACT
-
9
OPEN TOPOLOGY
< 61
fl Ip
< E/3.
TO evaluate the second term, let Ix
=
(1 < i < n; x
where C ' is the sum for i Therefore f E proof. COROLLARY
-1.6
E
e,
E
Ix and C " is the sum for i E Jx. as claimed, and that concludes the
( S t o n e - W e i e r s t r a s s t h e o r e m l . L e t A C C(X;IR
a s u b a l g e b r a and l e t f E C ( X ; E ) .
of A i n C ( X ; I R )
u 1 and Jx= {1 5 i 5 n;x&'Uij.
i f , and o n l g i f ,
he
Then f b e l o n g s t o t h e c l o s u r e t h e f o l l o w i n g two
conditions
are s a t i s f i e d :
(1) f o r e v e r y x CI
E A such
E X
s u c h t h a t f(x)
# 0,
there
i s
t h a t a(x) # 0 ;
(2)
f o r e u e r y x, y E X s u c h t h a t f ( x ) # f(v) , t h e r e
i s g E A s u c h t h a t a(x) # a ( y ) .
-
Since every alqebra is a module over itself, A = L A ( A ) , by Theorem 1.5. On the other hand, by Proposition 1.2, with M = IP, we have that f E L A ( A ) if, and onlv if, conditions (1) and (2) are satisfied. PROOF
10
5
COMPACT
5
-
O P E N TOPOLOGY
THE COMPLEX SELF-ADJOINT CASE
The following lemma is the key tu the reduction the complex self-adjoint case to the real case.
of
LEMMA 1.7 L e t A C C(X;C) b e a s e l f - a d , j o i n t s u b a l g e b r a , and l e t E b e a l o c a l l y c o n v e x v e c t o r s p a c e o v e r C . L e t B C C(X;R) he t h e s e t {Re f; f E A ) . Then (1) B C A;
(2) B is a s u b a l g e b r a o f C(X;IF). F o r e v e r y v c c t o r s u b s p a c e W C CtX;E) we h a v e : ( 3 ) IJ is a n A-module
if, and o n l y i f , F' is a
B-mo-
dw l e ; ( 4 ) LA(V) = LB(W).
-
PROOF For everv f E A , Re f = (f+?)/2. Hence f E P implies Re f E A , which proves (1). Clearlv, R is a vector subspace o f C(X;a 1. NOW, if f , 9 E A , then by (1), Re f and Re g belong to A.Hence (Re f) (Re q ) E A. Since (Re f)(Re a) is real-valued, it follows that (Re f) (Re u ) E B. Thus (2) holds. Let now W be a vector subspace of C(X;E).Since BCF, W is an FA-module implies that V is a B-module. To prove the converse, it is sufficient to prove that A c B + i B. Let f E A, f = u + i v. By definition, u E B. On the other hand v = Re a, where CT = -i-f. Hence v E B, and the Droof of ( 3 ) is complete. Finally, ( 4 ) follows from the fact that x E v (mod. A ) if and only if x v (mod. B). To prove this last fact, notice that x f y (mod. A ) implies the existence of f E A such that f (x) # f ( y ) If Re f (x) # Re f (y), then x v (mod. B). If on the other hand, Im f(x) # Im f ( v ) , then CT = -i-f is such
*
.
.
v (mod. B) in this that CT E A and Re f (x) # Re a(y) Hence x case too. The converse, x E y (mod. A ) * x f v (mod. B), follows from B C A . This completes the proof of the present lemma. THEOREM 1.8 L e t A C C(X;(c) b e a s e l f - a d j o i n t s u b a l g e b r a , and l e t E b e a l o c a l l y c o n v e x s p a c e o v e r C . Then e v e r y A-submodule W C C(X;E)is l o c a l i z a b l e u n d e r A i n C(X;E).
COMPACT
PROOF
-
11
OPEN TOPOLOGY
L e t B = {Re f ; f E A).
BV Lemma 1 . 7 ,
is a
P.I
By theorem 1 . 5 , W is l o c a l i z a h l e under B i n C(X;E).
However, by p a r t ( 4 ) o f Lemma 1.7, LB(F!)
= LA(T”).
-
B-module.
w.
Hence $(W) = Therefore,
LA(W) = W , a s d e s i r e d .
L e t A C C ( X ; C ) be a s e l f - a d j o i n t
COROLLARY 1 . 9
subaZgebra,and
l e t f E C ( X ; C ) . The f o l l o w i n g s t a t e m e n t s a r e e q u i v a l e n t .
that
(1)
f b e l o n g s t o t h e c l o s u r e of A;
(2)
g i v e n x , y E X and
Ig(x)- f ( x ) I < (3)
E r
E
Iq(y)- f ( y ) 1 <
> 0, t h e r e i s cy E A E;
(a) f o r every x, y E X such t h a t f ( x ) # f ( y )
t h e r e i s a E A such t h a t
U(X)
there
f belongs t o LA(A).
(1) * (2). Obvious. (2) * ( 3 ) . L e t x , y
PROOF
,
# q ( y ) ; and
(b) f o r e v e r y x E X s u c h t h a t f ( x ) # 0 , i s (I E A s u c h t h a t q ( x ) # 0 ; (4)
such
E X b e such t h a t f ( x )
# f ( y ) . Defi-
ne E = I f ( x ) - f ( y ) l . Then E > 0 , and by (2), t h e r e i s such t h a t I q ( x ) - f ( x ) I < ~ / 2and l u ( y ) - f ( y ) I < ~ / 2 .
9 E A
I f q ( x ) = s ( y ) , then E =
-< a c o n t r a d i c t i o n . This proves p a r t (a) of ( 3 ) . A s i m i l a r men t p r o v e s p a r t ( b ) . ( 3 ) * ( 4 ) , by P r o p o s i t i o n 1 . 2 . ( 4 ) * (1), by Theorem 1.8, s i n c e e v e r y a l g e b r a
arqu-
is
a
module o v e r i t s e l f . COROLLARY 1.10
be a closed s e l f - a d j o i n t
subgiven
x, y
E
REMARK.
Let A
c
a l g e b r a , a n d l e t f E C(X;E). T h e n f E A i f , and o n l y if,
X there i s q
E A
C(X;C)
such t h a t q ( x ) = f ( x ) and q(y) = f ( y ) .
F o r f u r t h e r r e s u l t s see Arens
[I].
12
-
COMPACT
OPEN TOPOLOGY
We w i l l f i n i s h t h i s p a r a g r a p h by p r e s e n t i n g t w o a p p l i c a t i o n s o f Theorem 1 . 8 . EXAMPLE I .
Let
be t h e c l o s e d i n t e r v a l
X
L = C f E C([O,lJ
-
W1
;
a);
IIO,l]
f(0) = 01
C
IR
.
Let
.
b e t h e v e c t o r s u b s p a c e o f C ( [0,1] ; a ) g e n e r a t e d by t h e functions t t ' n + l f o r n = 0 ,1 ,2 ,3 , . . . . c l e a r l y , W l i s n o t a
Let
s u b a l g e b r a o f C ( LO,
1-1
;a)
.
- module,
However, W1 i s a n A
i s t h e v e c t o r s p a c e g e n e r a t e d by t h e f u n c t i o n s n=0,1,2,3,
... .
where
A
t + t2n f o r
I n f a c t , A i s a separating s u b a l g e b r a o f ~ ( L 0 , 1 J ; a ) ,
which i s s e l f - a d j o i n t a n d c o n t a i n s t h e c 0 n s t a n t s . B ~Theorem1.8, f E C(
each
[O,lj ;a) x E [O,l]
belongs t o
. Now -
5
Wl(x)
if
,
and o n ly i f
0 if
=
,
f ( x ) E CJ1 (x) f o r
W1(x)
x = O , and
=
a
if
Wl = L i n C ( l O , l ] ; a ) . W 2 b e t h e v e c t o r s u b s p a c e o f C ( [O,ll ; a ) gene r a t e c ! by t h e f u n c t i o n s t I+ t 2 n lf o r n = 1 , 2 , 3 , . . . . W 2 is an a l g e b r a , b u t w e p r e f e r t o look a t t h e f a c t t h a t W2 i s a l s o an A - module. Again W2(x) = 0 i f x = 0 , a n d W2(x) = a i f O < x 5 1. By Theorem 1 . 8 , w e have 0 < x 5 1. Hence
L e t now
EXAMPLE 11. and l e t
Let
X = IRn
E
.
b e a complex Banach s p a c e w i t h norm
Let
E(IRn
:a)
and
E(mn ; E ) b e the vector spaces
of i n d e f i n i t e l y d i f f e r e n t i a b l e f u n c t i o n s on and E , r e s p e c t i v e l y . Then
E(IRn;
11,
/I
a) c
IR"
C(IRn
with values i n
;a) i s a s e p a r a t -
i n g s u b a l g e b r a which i s s e l f - a d j o i n t and c o n t a i n s t h e c o n s t a n t s . The same a s s e r t i o n s are t r u e w i t h r e s p e c t t o t h e algebra D ( I R n
;a)
c o n s i s t i n g of a l l
the
f E E(R";C)
w i t h compact s u p p o r t . Both
s p a c e s C ( I R n ; E ) and D ( B n ; E ) are v e c t Q r subspaces of C ( I R n ; E ) which a r e A - modules f o r A = E (IRn; a ) o r A = D ( I R n ; a ) . Here
D (IR"; E ) i s t h e v e c t o r s p a c e o f a l l
f E E (IRn ;E ) which have com-
pact support. Now f o r e a c h
x
E
R n , t h e r e i s some
GX
E z)(IRn
;a)
s u c h t h a t @,(x) = 1. Hence \7(x) = E , f o r W = E (IR" ; E ) or for FI = U(IRn;.E). By Theorem 1 . 8 , b o t h E(IR11;E) a n d D ( m n ; E ) are dense i n t h e c o m p a c t - open t o p o l o g y o f C ( I R n ; E ) .
COMPACT
5
6
-
13
O P E N TOPOLOGY
SUBMODULES OF C ( X ; E )
L e t u s begin w i t h t h e followinq c o r o l l a r y t o
theo-
r e m 1.8. THEOREM 1.11 L e t A c C(X) be a s e p a r a t i n g s e l f - a d j o i n t a l g e b r a and l e t DJC 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 A function f
A-module.
E C(X;E)
suban
b e l o n g s t o t h e c l o s u r e of
W r
if
and o n l g i f , f ( x ) b e l o n g s t o t h e c l o s u r e of w ( x ) = { a ( x ) ; a E k! i n E, for e a c h x E X. S i n c e A i s s e p a r a t i n a , each e q u i v a l e n c e c l a s s
PROOF
(mod. A ) i s a s e t reduced t o a p o i n t , Y = (a(x);
E
CI
{XI,
1
Y C X
and W I Y = W(x) =
W] c E. Usina t h e above theorem w e can p r o v e a r e s u l t
on
i d e a l s i n f u n c t i o n a l u e b r a s , due t o I . Kaplansky. I f E i s a loc a l l y convex s p a c e endowed w i t h a j o i n t l y c o n t i n u o u s m u l t i p l i c a t i o n , then C(X;E)
becomes an a l a e b r a ( o v e r t h e same f i e l d of E )
when w e d e f i n e o p e r a t i o n s p o i n t w i s e . Now t h e problem arises
of
characterizina the closed r i a h t (resp. l e f t ) ideals I C C(X;E). Suppose t h a t f o r e v e r y x E X a c l o s e d r i a h t ( r e s p . l e f t ) Ix C E i s criven, and l e t I = { f E C ( X ; E ) ;
ideal
f ( x ) E Ix f o r a l l x w .
Clearly, I is a closed r i q h t (resp. l e f t ) ideal i n C(X;E). s h a l l p r o v e t h a t an9 c l o s e d r i g h t ( r e s p . l e f t ) i d e a l i n
W e
C(X;E)
h a s t h e above form, i f E h a s a u n i t . THEOREM 1.12
(Kaplansky)
Let E be a l o c a l l y convex space
en-
dowed w i t h a j o i n t l y c o n t i n u o u s m u l t i p l i c a t i o n . Assume t h a t E has a u n i t , L e t I C C(X;E) be a c l o s e d r i g h t f r e s p . 1 e f t ) i d e a Z . F o r e a c h x E X, l e t 1, b e t h e c l o s u r e of I ( x ) i n E. T h e n 1, i s a c l o s e d r i g h t f r e s p . l e f t ) i d e a l i n E, and I = ( f E X ( X ; E ) ; f ( x ) E I x for a l l x E X I . PROOF
For e v e r y x E X , I ( x ) i s c l e a r l y a r i q h t ( r e s p .
i d e a l i n E.
left)
S i n c e t h e m u l t i p l i c a t i o n i n E i s j o i n t l y COntinuOuS,
t h e c l o s u r e o f any r i g h t ( r e s p . l e f t ) i d e a l i n E is a
right
( r e s p . l e f t ) i d e a l . Hence Ix i s a r i g h t ( r e s p . l e f t ) i d e a l i n E .
t h a t I is a C(X)-module. I n d e e d , i f f E I and g E C ( X ) r C ( X ; E ) b e t h e f u n c t i o n x w g ( x ) e , where e i s t h e u n i t
W e claim
let h
E
14
COMPACT
o f E.
-
OPEN TOPOLOGY
I f I is a r i a h t ideal, then q ( x ) f ( x ) = a ( x ) [ f (x)e!= f ( x ) [ s ( x ) e ] = f ( x ) h ( x )
f o r a l l x E X , and t h e r e f o r e ideal is treated similarly,
f = f h E I . The case of a
left
hv w r i t i n a
s ( x ) f ( x ) = q ( x ) [e f ( x ) ] = [ q ( x ) e ] f ( x ) = h ( x ) f ( x ) . I t r e m a i n s t o a p p l y Theorem 1.11 t o t h e s e p a r a t i n s ( s e l f - a d j o i n t
i n t h e complex case) a l g e b r a A = C ( X )
,
and t h e c l o s e d
A-module
I.
U n d e r t h e h y p o t h e s i s of T h e o r e m 1 . 1 : t h a t t h e a l g e b r a E i s s i m p l e . Then any c l o s e d two-sided
assume ideal
COROLLARY 1 . 1 3
c o n s i s t s of a l l f u n c t i o n s v a n i s h i n g o n a c l o s e d s u b s e t o f X. W e f i r s t r e c a l l t h a t E i s s a i d t o be s i m p l e i f i t
PROOF
n o two-sided
i d e a l s c t h e r t h a n 0 and E. L e t N
s u b s e t o f X.
Clearly, Z(N) = If E C(X:E);
x
E
N} i s a c l o s e d t w o - s i d e d i d e a l o f
C
X be a
f(x) = 0
all
for
C(X;E).
C o n v e r s e l y , i f I i s a c l o s e d two-sided C(X;E),
has closed
ideal
in
l e t u s d e f i n e N ={x E X; f ( x ) = 0 f o r a l l f E I } . C l e a r -
l y , N i s c l o s e d i n X and I
t o prove Z ( N )
c
Z(N).
C I, let f E Z(N),
t h a t f j? I . By Theorem 1 . 1 2 ,
i.e.
To prove t h e converse,
and assume
hv
t h e r e i s an x E X
contradiction such
that
f ( x ) J?! Ix. S i n c e Ix i s a t w o - s i d e d i d e a l , 1, = IO};
the
Ix = E b e i n g i m p o s s i b l e b e c a u s e f ( x ) E E.
# 0. Since
f E Z ( N ) , . x j? N .
Hence f ( x )
case so
However, Ix = (0) i m p l i e s I ( x ) = 0 , a n d
x E N, a contradiction. COROLLARY 1 . 1 4
L e t A and W b e a s i n Theorem 1 . 1 1 .
Then
d e n s e i n C ( X ; E ) i f , and o n l y i f , W(x) i s d e n s e i n E, for x E
W
i s each
x. W e s h a l l d e n o t e by C ( X ) Q E , t h e vector s u b s p a c e o f
C(X;E)
x
-+
c o n s i s t i n r r o f a l l f i n i t e sums o f f u n c t i o n s o f t h e
f ( x ) v , where f E C ( X ) a n d v E E. C l e a r l y , C ( X ) 49 E
form
is
a
C ( X I -module.
THEOREM 1.15
Let X be a Hausdorff space such t h a t C(X)
r a t e s t h e p o i n t s of X.
T h e n C ( X ) eP E is d e n s e i n C ( X ; E ) .
sepa-
COMPACT
PROOF
-
15
OPEN TOPOLOGY
L e t W = C ( X ) Q E. By h y p o t h e s i s , W i s module
s e l f - a d j o i n t s e p a r a t i n g s u b a l q e b r a , namely C ( X )
W(x) = E , f o r e a c h x 1 . 1 4 above.
E X.
a
over
.
Moreover,
The r e s u l t now f o l l o w s from C o r o l l a r y
Theorem 1.15 above c a n b e used t o d e r i v e vector-valued v e r s i o n s o f theorems on d e n s i t y o f s p a c e s of s c a l a r - v a l u e d f u n c t i o n s . T h i s i s done t h r o u g h t h e f o l l o w i n g .
L e t X b e a Hausdorff space such t h a t C(X) s e p a r a t e s
COROLLARY
t h e p o i n t s of X.
L e t W C C(X;E) be a s u b s e t such t h a t A
Q E cW,
w h e r e A i s t h e s e t {I#I o f; E E l , f E W). If A i s d e n s e C(X), t h e n W i s dense i n C(X;E)
in
.
PROOF
The s e t A 8 E c o n s i s t s of a l l f i n i t e sums of
of t h e form x
+
f ( x ) v , where f E A and v E E . Suppose t h a t
i s dense i n C ( X ) , i . e .
-
w; by
Ti
= C ( X ) . Then C ( X ) Q E =
o t h e r hand, A Q E C -.By C
functions On
A
the
Hence C ( X ) Q E is
h y p o t h e s i s A Q E C W.
Theorem 1.15 W i s d e n s e i n C ( X ; E ) ,
8 E.
and t h e proof
done. I f Z C X i s a c l o s e d s u b s e t , and M C E i s v e ct or subspace, then W(Z;M)
x
E
Z) i s a c l o s e d C(X)-submodule. I f Z1
W(Z1;M).
In particular, W(Z;M)c
o t h e r hand, i f M1
c
W({x);M)c n{W({x);H); M
C Z2,
f{W({x);M); l
t h e n W(Z;M1)
M2,
a closed
= ( f E C(X;E);f(x) E M
x
for
then
W(Z2;M)C
E 2).
On
In
C W(Z;MZ).
c H , codim H = 1). T h i s
the
particular, suggests t h a t
W((x1; H ) are maximal p r o p e r c l o s e d C(X)-modules, and t h a t , f a c t , each p r o p e r c l o s e d C(X)-module
all
is the intersektion
in of
a l l m a x i m a l p r o p e r c l o s e d C(X)-modules t h a t c o n t a i n it. THEOREM 1.16
Every p r o p e r c l o s e d C(X)-module W C C(X;E)
c o n t a i n e d i n some p r o p e r c l o s e d C ( X ) - m o d u l e V of one (hence maximal) i n C(X;E). Moreover, W i s t k e
is
codimension intersection
of a l l maximal p r o p e r c l o s e d C ( X ) - m o d u l e s t h a t c o n t a i n i t . PROOF
L e t W C C(X;E)
b e a p r o p e r c l o s e d C(X)-module.
f E C ( X ; E ) b e a f u n c t i o n which d o e s n o t b e l o n g t o W.
Let
Since W i s
c l o s e d , by Theorem 1.11, t h e r e i s x E X such t h a t f ( x ) does n o t b e l o n g t o t h e c l o s u r e o f W(x) i n E . By t h e Hahn-Banach theorem, t h e r e i s I#I E E ' s u c h t h a t I # I ( f ( x ) )# 0 , w h i l e
I#I(q(x))= 0
for
COMPACT - OPEN TOPOLOGY
16 all 9
E
W. L e t H b e t h e k e r n e l o f
$
in
E,
and
define
V = { g E C ( X ; E ) ; g(x) E H). C l e a r l y W c V, and f & V. S i n c e t h e map T: h + h ( x ) is a c o n t i n u o u s l i n e a r map from C ( X ; E ) i n t o E , and V is t h e k e r n e l o f $ o T , V i s a c l o s e d v e c t o r s u b s p a c e of codimension one i n C ( X ; E ) . I t remains t o n o t i c e t h a t V is a C ( X I -module. COROLLARY 1.17 C(X;E)
and $
A l l maximal p r o p e r c t o s e d
a r e o f t h e form { g
E C(X;E);
C(X)-modules of $(g(x)) = 0 ) f o r some x E X
E El.
W e can g e n e r a l i z e t h e above r e s u l t s t o A-submodules,
where A is any s e l f - a d j o i n t s u b a l g e b r a . Indeed w e have t h e f o l lowing r e s u l t .
L e t A C C(X) be s e t f - a d j o i n t s u b a l g e b r a . Every p r o p e r c l o s e d A-submodute W C C ( X ; E ) i s c o n t a i n e d i n some ?rape r c l o s e d A-submodule 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 ) in C ( 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 o f a22 m a x i m a t proper c l o s e d A-submodutes V i n C(X;E) t h a t c o n t a i n i t . THEOREM 1.18
PROOF
L e t f E C(X;E)
b e a f u n c t i o n o u t s i d e o f W. S i n c e W =
by Theorem 1.8 t h e r e is s o m e x E X such t h a t f l [ x ] d o e s belofig t o t h e closure o f Wl[x] i n C ( [ x ] ; E ) . e q u i v a l e n c e class (mod. A ) t h a t c o n t a i n s x) Theorem, t h e r e is $ E C ( [x]; E )
'
(Here [x]
. By
w,
not
is
the
t h e Hahn-Banach
such t h a t $ ( f 1 [x])
# 0,
while
Y.
$(ql [x]) = 0 f o r a l l g E L e t V C C ( X ; E ) be t h e set {u E C ( X ; E ) ; $(ql [x]) = 0). I t is clear t h a t V is an A-module, c o n t a i n i n g W, and t h a t f
V. S i n c e t h e map T: u
c o n t i n u o u s l i n e a r map from C ( X ; E )
-+
qI[x]
is a
i n t o C ( [x] ; E l , when eachspaoe
c a r r i e s i t s own compact-open t o p o l o g y , V = k e r ( $ o T )
is
a
c l o s e d vector subspace o f codimension one i n C ( X ; E ) . COROLLARY 1 . 1 9 L e t A C C(X) be a s e l f - a d j o i n t subatgebra. A l l maximal' p r o p e r c l o s e d A - s u b m o d u t e s V C C ( X ; E ) a r e o f t h e f o r m
f o r some x E X and $
E C([x];E)'.
COMPACT
5
7
-
17
OPEN TOPOLOGY
AN EXAMPLE: A THEOREM OF R U D I N
L e t E b e a r e a l l o c a l l y convex Hausdorff s p a c e , and
l e t F b e a ( c o m p l e t e ) l o c a l l y convex Hausdorff s p a c e o v e r C . L e t
Ff(E;IR)
b e t h e s u b a l g e b r a of C ( E ; I R )
of E and by t h e c o n s t a n t mappings. L e t
g e n e r a t e d by t h e d u a l E ' F ( C ; F ) be t h e
vector
s u b s p a c e of a l l f u n c t i o n s o f t h e form n
p(z) = where n E IN
, ai
E F,
i=O,l
be t h e v e c t o r s u b s p a c e (
c ai zi i=o
,...,n.
Ff(E;R )
If X
c
@ F(C;F)
E x C, let
) I x,
i.e.
?c
C(X;F?
9 consists
of t h e r e s t r i c t i o n s t o X of f i n i t e sums o f p r o d u c t s of t h e form (t,Z) A
c
r--)
q ( t ) p ( Z ) , where q E
q(E';IR)
Let
and p E F ( C ; F ) .
C ( X ; I R 1 b e t h e s u b a l g e b r a c o n s i s t i n g of a l l f u n c t i o n s of the
form ( t , Z )
E X
-+
9
q ( t ) , where q E F f ( E ; I R ) . Obviously,
is
an A-module. F o r each t E E , l e t K t = (2 E C ; ( t , Z )
E X).For each
t E E , such t h a t K t # 0 , l e t Xt = { ( t , Z ) E X ; Z E Kt). s e p a r a t e s t h e p o i n t s o f E , t h e e q u i v a l e n c e classes Y
Since
c
X
E'
(mod.
are p r e c i s e l y t h e sets X t d e f i n e d above. I n t h e n e x t Theorem w e s h a l l suppose t h a t X i s comp a c t and f o r e a c h t E E,Kt i s a c a n p a c t s u b s e t o f CC which h a s a c o n n e c t e d complement. I n t h i s case, w e d e n o t e by CA(X;F) the s e t of a l l g E C(X;F) s u c h t h a t , f o r each t E E , t h e mapping, A)
gt:
z
-+
g t ( Z ) = g ( t , Z ) i s holomosphic i n t h e i n t e r i o r o f K t .
Assume X i s a c o m p a c t H a u s d o r f f s p a c e . CA(X;F) n i s c o n t a i n e d i n t h e c l o s u r e of .f i n C(X;F), i . e . f o r each g E CA(X;F), g i o e n E > 0 and r E c s ( F ) , t h e r e e x i s t p o l y n o m i a t s ql, qm E Tf(E;IR) and p l , . . . , p m E T ( C ; F ) such t h a t m r ( q ( t , Z ) - C q i ( t ) pi(Z)) < E i=l f o r aZZ ( t , Z ) E X .
THEOREM 1 . 2 0
...,
PROOF
Let q E
CA(X;F). For each t E E , w i t h Kt
holomorphic i n t h e i n t e r i o r of Kt.
# 0 , qt
By t h e v e c t o r - v a l u e d
is
version
o f M e r g e l y a n ' s Theorem (see B i e r s t e d t [5], and B r i e m , L a u e r s e n , and P e d e r s e n [ 9 ] )
, the
f u n c t i o n qt belongs t o t h e c l o s u r e
of
18
COMPACT
T ( C ; F ) / K t i n C(Kt;F).
-
OPEN TOPOLOGY
Therefore, qiven
E
t h e r e e x i s t s p E P ( C ; F ) such t h a t r ( n t ( Z ) L e t f = (1 C3 p ) I X .
Z E Kt.
for all (t,Z)
E Xt.
€ 9 and
Then f
> 0 and
-
p ( Z ) ) < F: f o r a l l r(T(t,Z)- f(t,Z))< E
I t follows t h a t q E LA(?).
g belongs t o t h e c l o s u r e of
r E cs(P),
9 , RED.
By Theorem
1.5
When E = IR" a n d F = C , t h e above r e s u l t w a s p r o v e d by Rudin ([54], T h . 4 ) . H i s p r o o f i s a p a r t i t i o n of u n i t y a r a u m e n t combined w i t h M e r a e l y a n ' s Theorem. I n h i s book [56] Rudin d e r i v e s i t from B i s h o p ' s t h e o r e m (see t h e p r o o f o f Theorem 5 . 8 , [56] ) When E = IR I , where I i s a n a r b i t r a r y set of i n d i c e s ,and F = C , Theorem 1 w a s p r o v e d by Chalice [13]. H i s v e r y short p r o o f i s b a s e d on de Branges lemma t h a t s a y s t h a t f o r a n y ext r e m e p o i n t p of t h e u n i t b a l l o f A', t h e s u p p o r t o f p i s a s e t o f a n t i s y m m e t r y f o r A , i f A i s any f u n c t i o n a l q e b r a . T h i s lemma i s a n e s s e n t i a l s t e p i n G l i c k s b e r ' q ' s p r o o f [27] of B i s h o p ' s Theorem. Our n e x t r e s u l t i s a c o r o l l a r v o f Theorem 1.20 abcve A s a i n , when I = ( 1 , n } , K = [0,1] c IR, it i s d u e t o Rudin ([54], Theorem 3 ) ; and when I i s a n a r b i t r a r y i n d e x s e t , and K c IR i s a n a r b i t r a r y compact s u b s e t o f t h e l i n e , i t i s d u e t o C h a l i c e ( [13] , Theorem 2 ) C h a l i c e p r e s e n t s a d i r e c t p r o o f , v e r y
.
.
...,
.
s h o r t , b a s e d a a a i n on d e B r a n a e s lemma a n d Merclelyan's Theorem.
L e t K be a compact s u b s e t of IR. If f E C ( K ; C ) and ui E C ( K ; I R L i E I, a r e such t h a t f and t h e ui s e p a r a t e t h e f p o i n t s of K, t h e n t h e f u n c t i o n a l g e b r a A on K g e n e r a t e d b y and t h e ui i s C ( K ; C ) COROLLARY 1 . 2 1
.
PROOF
I Consider t h e r e a l l o c a l l y convex Hausdorff s p a c e E= lR.
(XI 1 f (x) is a homeomorphism o n t o a compact s u b s e t X c E x C . F o r e a c h t E E , l e t Kt = { Z E C; ( t , Z ) E X I . Each s u c h compact s u b s e t h a s a con-
The mapping Ip d e f i n e d on K b y x
+
( (ui
n e c t e d complement i n C , and t h e i n t e r i o r of Kt fore,
CA(X;C)
E ?(C;C)
E
There-
= C(X;C).
L e t h E C(K;C).
For each
i s empty.
Then q = h o 4
> .O t h e r e e x i s t q l , .
..q,,,
E
Tf
-1
belongs t o C(X;C). I (IR ;IR) and p l , . ,pm
..
such t h a t
m
I c
j=l
q j ( t ) Pj(Z)
-
s(t,Z)l
<
E
COMPACT
for a l l ( t , Z )
E X.
-
19
OPEN TOPOLOGY
Now e a c h ( t , Z )
E X i s of t h e form
@(XI
for
each some x E K . T h e r e f o r e u ( t , Z ) = h ( x ) . On t h e o t h e r h a n ? I ; W ) i s a f i n i t e sum o f homogeneous polynomials of
qj E
Ff(lR
,...,n ,...I ) ,
(k EC0,1,2
t h e form k$k
with @ E E ' = @
IR.
For
i E I
t h e r e e x i s t s a f i n i t e set F C I such t h a t
each such @,
@((ti)iE = I) 1 iEF where ai E IR
aiti
f o r a l l i E F. Hence
f o r a l l x E K.
C o n s e q u e n t l y , t h e r e e x i s t s a f i n i t e sum o f
geneous p o l y n o m i a l s
.f E T~ ( m ' ~ J ; m FC ),
1 finite,
homo-
I F I = car-
d i n a l o f F , such t h a t
f o r a l l t = ( u ~ ( x ) ) where ~ ~ ~ F,= ( a l ,
...,a IF1 I *
NOW t h e f u n c t i o n
m
e
c
j QF(ual,.."u
1) pj(f) alFl KI b e l o n g s t o A , and w ( x ) = 1 q j ( t ) p . ( Z ) for a l l x E K, if 3 j=1 ( t , Z ) = $ ( x ) . Therefore I w ( x ) - h ( x ) I < E , f o r a l l x E K. Hence w =
j=1
(
finite
P. = C ( K ; C 1 ,
t h e a l q e b r a A i s d e n s e and c l o s e d i n C(K;@), i . e . ,
as d e s i r e d . REMARK.
W e s h a l l present a
proof
of
the
vector-valued
Mergelyan's Theorem i n C h a p t e r 8 . A s i m p l e p r o o f , which i s avail-
able when t h e space E 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 , i s
pres-
e n t e d i n C h a p t e r 4 . A l o c a l l y convex s p a c e E h a s t h e approximat i o n p r o p e r t y , i f q i v e n K C E compact p E c s ( E ) ,
E
> 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
i.e.
f o r a l l x E K.
BISHOP'S THEOREM I n t h e theorems of t h i s p a r a a r a p h X d e n o t e s a
pact Hausdorff s p a c e and E a seminormed s p a c e , w i t h
corn-
seminorm
COMPACT
20
t
I It1 1 .
+
-
OPEN TOPOLOGY
L e t A C C(X;Ip.) be a suba lo e b ra and W
[XI
F o r e a c h x E X , w e d e n o t e by
A-module.
c
an
C(X;E)
be e q u i v a l e n c e c l a s s
(mod. A ) t h a t c o n t a i n s x , a n d Wx d e n o t e s t h e v e c t o r s p a c e W [x! c o n t a i n e d i n C ( [ x ] ; E ) . Moreover, w e w r i t e I / f j / =s u p I I / f ( t ) I ( ; t ~ X ) and = sup { l l f ( t ) l l ; t E Usina t h i s notation,
[XI}.
Ilfl;xll\
t h e f o l l o w i n q s t r o n q e r form o f Theorem 1 . 5 c a n b e p r o v e d Buck [ 1 2 ] , Theorem 2 , pq. 8 7 ; C l i c k s b e r q [ 2 G ] ,
For e v e r y f
THEOREM 1 . 2 2
f-sl
I
inf
XEX
qEw
{ I If-q(
Since
IIf1
cx;
E
W } and
[XI
[XI
c
-
--
x
let
E X,
XI.
= s u p { ~ ( x ) x; E
X(x) < d for all x
X,
-
Tr- 1 1
1;s E W}; and f o r e a c h x
Il;cl
h(x) = inf {llfl[X]-ul
pcr.419).
E C(X;E),
= sup
Let d = inf
PROOF
(See
Hence X < d.
E X.
To p r o v e t h e r e v e r s e i n e q u a l i t y , l e t 0 <
c
1.WiLi-
o u t l o s s o f a e n e r a l i t y w e may assume t h a t A c o n t a i n s t h e
cons-
t h e r e e x i s t s wx E W s u c h t h a t
t a n t s . For each x E X ,
I If(t)Let UxC
x
wx(t)l I <
+ ~ / 3f o r a l l t
X b e t h e open s u b s e t
Then [x] C U x .
E
It
E
[xl.
E X;llf(t)-wx(t)ll<
Proceedina e x a c t l y a s i n t h e proof of
+ ~ / 3 1 . Theorem
1 . 5 ( r e c a l l t h a t w e h a v e assumed A C C ( X ; I R ) ) w e f i n d q E W s u c h
I If(x)-q(x)l(
that the
X +
<
+
X
E
f o r a l l x E X, i.e.
1
\f-ql( < X
> 0 was arbitrary, d < A. p r o o f o f t h e Theorem.
Thus d <
REMARK
E.
E
c
if A
Lemma 1 . 7 ,
55,
C
C(X;IR).
C(X;C)
I n p r o v i n s t h e reverse
is a self-ad,ioint
+
E.
T h i s completes
I n t h e proof o f t h e i n e n u a l i t y h < d, we did not
the fact that A d < A,
Since
use
inecrualitv
s u b a l o e b r a , we may
t o s u b s t i t u t e B = R e F = { R e f; f E A } f o r A
use in
t h e p r o o f . Hence t h e f o l l o w i n a r e s u l t i s t r u e . THEOREM 1 . 2 3
W
C
Let A
C
C ( X ; E ) a n A-moduZe.
C(X;C)
be a s e l f - a d j o i n t
For e a c h f E C ( X ; E ) ,
s u b a l g e b r a , and
COMPACT
COROLLARY 1.24
(Buck [12])
-
21
OPEN TOPOLOGY
Let W
c
C(X;E)
be a
C(X)-module.
F o r e a c h f E C ( X ; E ) we haoe
I I f-ql I
inf 9EW
= sup XEX
An even s t r o n u e r
I J f ( x ) - q ( x )1 1 .
inf qEW
form of Theorems 1.22 and 1.23
is
a v a i l a b l e . To p r o v e t h i s w e n o t i c e t h e f o l l o w i n g
[39], pq. 1261 L e t X and Y be compact H a u s d o r f f s p a c e s and n a c o n t i n u o u s mapping f r o m X o n t o Y. F o r e a c h u p p e r s e m i c o n t i n u o u s f u n c t i o n 9: X -+ IR define h:Y -> IR b y LEMMA 1 . 2 5 (Machado an.d P r o l l a
h ( y ) = s u p { ? ( X I; x E n f o r a l l y E Y.
-1
(y)1
Then h i s u p p e r s e m i c o n t i n u o u s o n Y.
PROOF For e a c h y E Y , t h e s e t {y} i s c l o s e d i n Y , therefore -1 -1 TI ( y ) i s compact i n X. Hence t h e r e is a n a E TI ( y ) such t h a t -1 h ( y ) = q ( a ) = s u p { q ( x ); x E n ( y ) 1, b e c a u s e q i s upper s e m i -
.
.
c o n t i n u o u s . So h is w e l l d e f i n e d from Y t o IR L e t r E IR s e t { x E X ; q(x) > r } i s c l o s e d , whence compact i n X . C a l l
The it
S i n c e n i s c o n t i n u o u s , n ( X r ) i s compact i n Y . W e claim t h a t n ( X r ) = {y E Y ; h ( y ) 1. r } , which p r o v e s t h a t h i s upper semi-
Xr.
continuous. Indeed, i f y E n ( X r ) ,
t h e n y = n ( x ) f o r some x E X r ,
-1
and t h e n h ( y ) > q ( x ) 2 r. C o n v e r s e l y , i f y j! s ( X r ) and t E n &I, t h e n q ( t ) < r ; it f o l l o w s t h a t h ( y ) < r , b e c a u s e t h e r e is some p o i n t t o E n - l ( y ) s u c h t h a t h ( y ) = g ( t o ) . T h a t completes
the
proof. THEOREM 1.26
be an A-module. that
Let A
c
C(X;IR)
Let f E C(X;E).
b e a s u b a l g e b r a and l e t WCC(X;E) There e x i s t s a p o i n t x E X such
inf 9EW
PROOF
L e t Y b e t h e compact Hausdorff space t h a t is t h e
t i e n t o f X by t h e e q u i v a l e n c e r e l a t i o n d e f i n e d by A ,
and
quo-
let
22
IT
COFIPACT
: X
+
-
OPEN TOPOLOGY
Y b e t h e q u o t i e n t map. Then, by Lemma 1 . 2 5 t b e
func-
tion
i s u p p e r s e m i c o n t i n u o u s on Y , f o r e a c h q E T.7.
Hence
i s u p p e r s e m i c o n t i n u o u s on Y t o o , and t h e r e f o r e a t t a i n s
its
supremum. By Theorem 1 . 2 2 t h i s supremum i s d= i n f ( 1 ' f - q l ' ; c r E F ! ) . W e w i l l p r o v e now a r e m a r k a b l e f a c t d i s c o v e r e d S . Machado [37].
by
Namely t h a t Theorem 1 . 2 6 above i m p l i e s B i s h q ' s
Theorem. To do s o w e w i l l u s e B i s h o p ' s o r i q i n a l d e s c r i p t i o n t h e p a r t i t i o n o f X i n t o a n t i s y m m e t r i c sets ( S e e [8] 1. I f X i s a H a u s d o r f f s p a c e and A C C ( X ; I K ) salqebra,
is
b e t h e class of a l l o r d i n a l numbers
let
c a r d i n a l numbers a r e less o r e q u a l t o 2
"I,
a
whose
is
where ' X I
of
c a r d i n a l number o f X . F o r e a c h Q E , w e d e f i n e by f i n i t e induction a closed, pair-wise d i s j o i n t coverina
the
transof X,
d e n o t e d by P a . F o r u = 1, w e d e f i n e P1 = {XI. A s s u m e t h a t PT h a s b e e n d e f i n e d f o r a l l T
ordinals
< u. W e c o n s i d e r t w o cases.
I f a = ~ + €or l , some
(a)
T
E
, we
d e f i n e Pu as
f o l l o w s . L e t T C X b e a n e l e m e n t o f PT. L e t AT = I f E A ; f l T r e a l } . Then A T / T C C ( T ; I R ) a n d w e c o n s i d e r t h e p a r t i t i o n o f
is T
i n t o e q u i v a l e n c e classes (mod. + ) . The p a r t i t i o n P U i s
then
d e f i n e d as t h e c o l l e c t i o n of a l l s u c h e q u i v a l e n c e classes
when
T ranges o v e r PT.
I f u has no predecessor, i . e . u i s a l i m i t or-
(b) dinal, define x
5
y f o r x , y E X, i f , a n d o n l y i f , x and y
l o n q t o t h e same e a u i v a l e n c e class o f PT f o r a l l T < u . T h i s d e f i n e s Po f o r a l l 0 E , a n d Pu i s a
G
f i n e m e n t o f P T whenever a > 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)
,
S,.
(b) as-
W).
(The case d = 0 is trivial). It follows that there is some g such that
EW
I!flSo-91Sal!< d Let
U = {t
E
X;lIf(t)- g(t)l! < d}.
Then So c U, and U is open. By compactness of X\U there in < a such that finitely many indices i1 < i2 <...<
x\u
are
...
c (X\ST 1 u u (X\ST 1 . 1 n c S , it follows that S C U, which Since ST c S c 1 l2 n ' n ' the contradicts (b) for in EG , in < a. This proves (b) for ordinal a, and ends the proof of Theorem 1.28.
...
In his proof of Theorem 1.28 S.Machado 1371 uses Zorn's Lemma instead of the above transfinite argument. The idea of using Zorn's Lemma can be applied to uive a direct proof of the bounded case of the Rernstein-Nachbin approximation problem (see Machado and Prolla [41!). Notice that P 2 is the partition of X introduced by Shilov. Indeed P2 is the collection of all equivalence classes REMARK
COKPACT
-
25
OPEN TOPOLOGY
coarser of X modulo the algebra {f E A; f is real on X I . This of all properpartitions of X can be used sometimes instead of gA. It then follows from Theorem 1.28 that there exists some S E PZ(once f E C ( X ; E ) is given) such that dist(f,W) = dist(f'S ;WIS 1 . This formula in turn implies the followinu. COROLLARY 1 . 2 9
L e t A and W be as i n T h e o r e m 1 . 2 8 . L e t
fEC(X;E).
T h e n 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 f , and o n l y i f , flS 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 WIS i n C ( S ; E ) , f o r a l l w h i c h i s an e q u i v a l e n c e c l a s s m o d u l o t h e a l g e b r a f a E A ;
S C X CI
is
real on X I .
5
9
VECTOR FIBRATIONS
A v e c t o r f i b r a t i o n is a pair ( X ; ( E x ; x E X I 1 , where X is a Hausdorff topological space and (Ex; x E X ) is a family of vector-spaces over the same scalar field M. The product set n ( E x ; x E X ) is always provided with the structure of a product vector-space over M ; it is called the vector space of all c r o s s - s e c t i o n s of the vector fibrais then tion ( X ; ( E x ; x E X ) ) . A v e c t o r s p a c e of c r o s s - s e c t i o n s any one of the vector subspaces of n ( E x ; x E X I . A vector space W of cross-sections is said to be a m o d u l e o v e r a s u b a l g e b r a A c C ( X ; M ) , or an A - m o d u l e , if aw E W for any a E A and w E W, where aw is the cross-section (a(x)w(x); x E X) if (w(x); x E X ) = w. Any family v = (vX; x E X ) such that vx is a seminorm on Ex for each x E X is called a w e i g h t of the vector fibration ( X ; ( E x ; x E x)). We shall restrict our attention to vector fibratians and vector spaces L of cross-sections satisfynq the following conditions: (1) X is compact; ( 2 ) each Ex is a normed space, whose norm we denote by t - * I It1 1 ;
COMPACT
26
-
OPEN TOPOLOGY
( 3 ) if L is a vector space of cross sections,
each f on X.
L the function x
E
-f
I
If (x)
I
is upper
Eor semicontinuous
In this case we say that L is an u p p e r s e m i c o n t t n u o u s v e c t o r s p a c e o f c r o s s - s e c t i o n s , and endow it with the topo1 0 9 of the norm
THEOREM 1.30 ([is]
,
Lemma 4 )
vector-space o f cross-sections
L e t L be an u p p e r s e m i c o n t i n u o u s which i s n C(X)-module.For e v e r y
C(X)-submodule W c L, we h a v e
The proof is entirely similar to that of Theorem 8, only it is much simpler. Call p = inf I If-91). For
PROOF
5
1.22, each
9EW
x
E
X, let h(x) = inf
/f(x)-y(x)I
I
and put X
=
sup {X(x);xEXj.
gEW
(x) 5 p , for each x E X. Hence X < p. To prove the reverse inequality, let 0 < €.For each X, there exists wx E W such that I If (x)-wx(x) I I
Clearly x
E
f
-
wX
A
E
L, by condition ( 3 1 , U,=
I t E X ; IIf(t)- wx(t)II
is an open set containinq x. By compacteness of X, there exist xl,...,x n E X such that X = U1 u... u Un, where for each i = 1,2,...,n, Ui denotes Ux €or x = xi. Let ql,-..,qn E C(X) be a continuous partition of the unity subordinate to the covn erinq U 1 ,...,Un. Let y = C yi wi, where wi = w X for x = xi i=l We claim (i = 1,2,...n). Since w is a c X)-submodule, q E W. that, for each x E x, qi (XI I I f (x)- wi (XI for all i = 1,2
,... ,n.
Indeed,
COMPACT
both sides are zero. If x gi(X) 2 0 . Hence
E
-
27
OPEN TOPOLOGY
Ui, then
I If(x)-
wi(x)
II<
A
+ E,and
f (x n
n
c
i=1 A +
E,
for all x
E
X.
< A + E , and a fortiori, p 5 A + E. Since Therefore I If-gl 1 E > 0 was arbitrary, p 2 A . This concludes the proof of Theorem 1.30.
REMARK. Theorem 1.30 is a generalization of Corollary 1.24,§8. To see this, just consider the vector fibration Ex = E for all x E X, where E is a fixed normed space. And then take L=C(X;E). Obviously, if Wx = w(x) in E, then inf I UEWx
5 10
EXTREME FONCTIONALS
Let S be a subset of a vector space E. A point x E S is called an e x t r e m e p o i n t of S if x is notan intemal point of any segment of line in S , i.e., if a,b E S and 0 < t < 1, then ta +(l-t)b = x implies that a = b = x. The set of extreme points of the set S is denoted by E(S) Let us recall, for use in Lemma 1.33 below,that the c o n v e x h u l l of a set S C E is the smallest convex set in E that contains the set S , and it is denoted by co(S).When E is a topological vector space, the c l o s e d c o n v e x h u l l of S , denoted by co(S), is the closure of its convex hull. We shall be interested in this section in characlocally terizing E(S) , when S is a subset of the dual of some convex space L of functions, or more generally, of cross-sections. Let L = C(X), and let S be the unit ball of E = L'; the Arens-Kelley Theorem ([2], or Dunford and Schwartz 1201, pg.443
.
COMPACT
28
-
OPEN TOPOLOGY
s a y s t h a t E ( S ) are t h e e v a l u a t i o n s a t p o i n t s o f X Banach s p a c e , and S = u n i t b a l l of L ' , : E'
+
For e a c h x E X ,
b e d e f i n e d by
C(X;E)'
(f) = $ ( f ( X ) )
6,($)
f o r a l l f E C(X;E)
(a) (See S i n g e r Co(X;E)
a
then Singer characterized
E ( S ) a s f o l l o w s . L e t B i b e t h e u n i t b a l l of E ' .
l e t 6,
multiplied where E i s
by s c a l a r s o f a b s o l u t e v a l u e one. I f L = C ( X ; E ) ,
and $ E E ' .
Then
E(S) = U { b X ( E ( B i ) ) ;x E X I .
[fjo]).
This r e s u l t w a s generalized t o the
space
of a l l f E C ( X ; E ) which 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 Hausdorff s p a c e X , by Brosowski, Deutsch and M o r r i s . (See
[lo],
Lemma 3 . 3 )
and by S t r a b e l e [63].
I
The Arens-Kelley
theorem w a s e x t e n d e d by R.C.
Buck
and M C L i s
[12] i n t h e f o l l o w i n g d i r e c t i o n . I f L = C ( X ; E ) ,
a
C(X)-module, M # 101, l e t S b e t h e convex set M I n B ' , where B ' i s t h e u n i t b a l l o f L ' . For e a c h x E X , l e t Mx b e t h e c l o s u r e i n E o f M(x) = { f ( x ) ; f E M}.
(b)
E(M*
n
B')
Then = U { ~ ~ C E CnM B;)) :
;X
ex,
M~ # E } .
P r e v i o u s l y , S t r o b e l e had proved (see [63]) t h e i n c l u s i o n I n d e p e n d e n t l y , Cunningham and Roy [lS] had ( b ) f o r C(X)-modules M
c
3 .
proved
L , where L i s a v e c t o r s p a c e of cross-
s e c t i o n s s a t i s f y i n g t h e h y p o t h e s i s o f Theorem 1 . 3 0 .
W e shall--
s e n t t h i s more g e n e r a l r e s u l t b e c a u s e w e s h a l l r e d u c e t h e case of a g e n e r a l A-module W c C ( X ; E ) , where A i s any s e l f - a d j o i n t s u b a l g e b r a , n o t n e c e s s a r i l y s e p a r a t i n g on X , t o t h e case of C(X)-modules of c r o s s - s e c t i o n s , t h r o u g h a q u o t i e n t c o n s t r u c t i o n , W e s t a r t w i t h S i n q e r ' s Theorem f o r c r o s s - s e c t i o n s . L e t B ' b e t h e u n i t b a l l of L ' , and f o r each x E X , l e t B i b e t h e u n i t b a l l of E i . The mapping 6,:
Ei
L'
defined
by
ax($)
(f)
= $(f(X))
f o r a l l @ E E i and f E L , i s t h e n a c o n t i n u o u s l i n e a r < 1, and t h e r e f o r e maps B i i n t o B ' . of norm N o t i c e a l s o t h a t 6,
i s one-to-one
if
mapping
29
COMPACT- OPEN TOPOLOGY
Lx = {f(x); f
E
L) = Ex. This leads to the following:
DEFINITION 1.31 A v e c t o r s p a c e L of c r o s s - s e c t i o n s is s a i d t o be e s s e n t i a l i f Lx = {f(x); f E L) = Ex, f o r a l l x E X . Clearly, the image of E; by 6, is contained in the ~ set { J I E L';f(x) = 0 s $(f) = 0, V f E Ll. Call it A ~ . Wclaim the following
.
LEMMA 1.32 L e t L be an e s s e n t i a l C(X)-module of u p p e r c o n t i n u o u s c r o s s - s e c t i o n s o v e r X . The mapping 6, i s a i s o m e t r y of Ei o n t o Ax f o r e a c h x E X . PROOF We saw already that 6, of norm < 1, from E; into Ax.
is a continuous linear
semi2 ineap
mapping
Let JI E A , . For each v E Ex, choose f E L such that well f(x) = v and put cx(JI)(v) = $(f). Since JI E Ax, cx( JI ) is claim defined on Ex, and it is clearly a linear functional. We Choose that ~ ~ ( $E 1E;. Let E > 0. Let v # 0 be given in Ex. f E L with f (x) = v. By the upper semicontinuity hypothesis, I If (t)I I < there i s a neighborhood U of x in X such that < g < 1, (1 + E ) I If(x) I I , for all t E U. Let g E C(X) satisfy 0 g(x) = 1 and g(t) = 0 for all t jZ U. Since L is a C(X)-module, gf E L, and moreover IIgf
I[<
( 1 + ~IIf(x) ) 11.
Now ( g f ) ( x ) = v and
Hence
I lcxl -< IbX1 -<
1. Since 6x and c X are inverses of each other, and
1, we see that 6, and cX are linear isometries.
LEMMA 1.33 L e t L be an e s s e n t i a l u p p e r s e m i c o n t i n u o u s v e c t o r x € XI. Then s p a c e of c r o s s - s e c t i o n s o v e r X . L e t Q = U(6,(B;); (a) Q is w e a k * - c l o s e d ; (b)
PROOF
(Q) = 8'.
(a) Suppose
$ E
L' is the weak*-limit of a net {$a} in
30
COMPACT
- OPEN
TOPOLOGY
= 6x ( $ a ), where x u € a
Q. Each $ a i s of t h e form $,
.
6,)
$a E B; S i n c e X is compact w e may assume t h a t a t o some p o i n t x E X . T h e n , f o r any f E L w e have l$(f)
I
= lim
< l i m E
converges
I$,(f) 1 = l i m I $ a ( f ( x a ) ) I
1) 5
s u ~ I l $ ~ .ll ll f ( x a )
Hence + ( f ) = 0 , i f f ( x ) = 0 , i . e .
i s some $
and
X
$ E Ax.
IIf(x)
11.
By Lemma 1.32
there
.
= $ I or e q u i v a l e n t l y , $ = Ex($)
EA such t h a t 6 , ( $ I
1
The above i n e q u a l i t y shows t h a t
l$l I=(
($11
-< 1,
i.e.
$ E B;.
T h e r e f o r e $ = 6x($) E Q , as d e s i r e d . (b) Since 6 x ( B i )
c
B',
Q
c
B'.
On t h e o t h e r hand B '
i s convex, and weak*-compact by A l a o g l u ' s Theorem: hence the G ( Qc) B ' o b t a i n s . (By G ( Qw)e mean t h e weak*-closed convex b a l a n c e d h u l l o f Q). L e t f E Qo = t h e p o l a r of Q i n L. F o r e a c h x E X , by t h e Hahn-Banach Theorem t h e r e is $ E B; such t h a t $ ( f ( x ) ) = ) ' \ f ( x )1 1 . Then 6x($) E Q. Hence I If ( x ) 1 1 = I $ ( f ( x ) ) 1 = 16x($) ( f ) I 5 1. T h i s shows t h a t I If 1 1 < 1, i.e. inclusion
00
Qo C B = u n i t b a l l o f L. Taking p o l a r s w e o b t a i n B ' c Q the
b i p o l a r Theorem, Qoo =
co(Q)
L e t L b e an e s s e n t i a Z
C(X)-moduZe of u p p e r s e m i c o n t i n u o u s c r o s s - s e c t i o n s T(B')
PROOF
= U{Gx(E(B;));
E X .
E
o v e r X . Then
X , Ex # {O)]
By Lemma 5 , Dunford and Schwartz [ 2 0 ] ,
T h e r e f o r e any J, E E ; B ' )
x
x
pg. 4 4 0 , E ( B ' ) C Q .
i s o f t h e form 6x($) f o r some $ E BAl , b e c a u s e $ = 6 x ( $ ) E E ( B ' ) and
But i n t h i s case $ E EiB;)
i s l i n e a r and one-to-one. C o n v e r s e l y , l e t $ E 6 x ( E ( B i ) ) f o r some x Ex # {O}. . L e t $ E E(BA) be such t h a t $ = 6,($). Notice 6
By
and t h a t e n d s t h e p r o o f .
(Cunningham and Roy [ l 5 ] ) .
THEOREM 1.34
.
X
$ E Ax fl B'
cause $ =
E X ,
and $ = ~ ~ ( $ 1But . i n t h i s case $ E E ( A x n B ' )
t ~ ~ ( $ )
E EiB;)
and c X is a n i s o m e t r y . T o complete
proof w e must e s t a b l i s h t h e f o l l o w i n g CLAIM 1.35
PROOF
J, E E ( B ' ) .
A s s u m e , by c o n t r a d i c t i o n , t h a t $ ft E ( B ' ) .
Then
that
, bethe
- OPEN
COMPACT
(JI1 + $,)/2 f(x) = 0, 1 If I $ =
I
for some Wl,$, 5 1, and 0 <
31
TOPOLOGY
B' and $l # $., Let f < 1 be given. Then E
E
E
L with
U = {t E X ; I If(t) 1 1 < € 1 is open and contains x. Let g E C ( X ) with 0 5 g 5 1, g(x) = 1 and g(t) = 0 for t P: U be chosen Since I / @ 1/ = 1, there is v E Ex such that 1 IvI I < 1, @ ( v ) is real Let and @(v) > 1 - E. Choose h E L with I lhl 1 5 1, h(x) = v. m = gh. Then I lghl 1 5 1, and @(m(x)) = @(h(x)) = @(v) > 1 - E On the other hand, m(t) = 0 for t )? U and 1 (f(t)I I < E f o r tEU, < imply I If(t) + m(t) I I < 1 + E for all t E X . Hence 1 If + ml I 1 + E . Now ]q1(m)1 < 1 and I$,(m) 1 < 1. Thus al(m) and $ ,(m)
.
are complex numbers in the unit disk whose mid-point 6,($)(m) = @ ( m ( x ) ) is real and > 1 - E . Hence
(1) I+l(m)- $,(m)
1
< 2 G 1 - 2 < 4
~ On the other hand, Ial(f+m) 1 5 1 + and
$(m) =
5
I$,(f+m) 1 < 1 +
Ilrl(f+m) and $*(f+m) are complex numbers in the disk of
Thus
E.
radius
mid-point $(f+m) = $ ( f ) + $ ( m ) = $(m) (because f (x) = 0 and JI E A x ) is real and > l - ~ .Hence 1 + ~ whose
(2)
-
Iql(f+m)
I
~ l ~ ( f + m )< 4 K
Combining (1) and ( 2 ) we qet
)f(,$I
- J12(f) I
whence Jll(f) - $,(f)
JI1 +
Jr
I
= 0. This proves that
= 2 $ E Ax too,
$,
< 8
a1
tradicts the fact that $
and $,
belong to
E (Ax fl B')
€
q1 Ax
-
$,
E Ax.
Since
n B', which con-
.
COROLLARY 1.36 ( S i n g e r ) L e t X be a compact Hausdorff spaceard dual l e t E b e a normed s p a c e . L e t B' b e t h e u n i t b a l l of t h e of C ( X ; E ) . Then
E(B') = w h e r e B;
x
E X,
u
{6x(E(B;));
x
E
XI ,
i s t h e u n i t b a l l of t h e d u a l E '
t h e mapping 6x: E' 6 x ( 4 J ) (f) =
-+
of E, and f o r
C ( X ; E ) ' i s d e f i n e d by
@(f(X))
each
32
COMPACT
COROLLARY 1.37
-
OPEN TOPOLOGY
[2]) L e t X be a c o m p a c t H a u s d o r f f s p a c e . L e t B' b e t h e u n i t b a l l of t h e d u a l of C(X;M). Then ( A r e n s and K e l l e y
x E X, X E K ,
E(B') = { A 6 x ; where h 6 ,
:
1x1
= 1)
,
C(X;X) * K i s d e f i n e d by X6,(f)
= X-f(x)
f o r a l l f E C(X;m).
We come now to the case of general C(X)-modules M C L. Since the case = L is covered by Theorem 1.34,we shall assume i # L. For any x E X, we define Mx = {f(x); f E M ) C Ex.
Mi
We can identify (Ex/Ex)' with , the unit ball being M i n Bi, the unit ball being MI il B' respectively (L/i) ' with M', We have then THEOREM 1.38 E!M*
.
(Cunningham and Roy [l5!l
n B')
I
=
n B;) 1 ; x
E
x,
-
M~
# E~I.
Before proceeding to the proof, we notice a corolof lary for L = C(X;E), M c L a C(X)-module, B' the unit ball L', for each x E X, Mx = { f ( x ) ; f E M) C E, and Bi unit ball of E'
.
COROLLARY 1.39 E(M
1
f l
(Buck 112!) I
B') = U(bxCECMx
fl Bi)); x E
X, Ex # E}.
PROOF OF THEOREM 1.38. We will identify isometrically L with another vector space L* of cross-section on X in such a way that the image M* = {O} Namely, for each f E L, let f* be the cross-section (f*( x ) ; x E X) defined by
.
f*(x) = f ( x ) + Ex i.e. f*(x)
E
,
Ex/Mx. Let L* be the image of L under the
linear
mapping f * f*; L* is a vector space of cross-section of t h e m tor fibration (X;(Ex/Ex; x E X)) We endow L* with the supremum norm :
.
-
COMPACT
I If*\I
33
OPEN TOPOLOGY
= sup
XEX
I If*(x) I I
= sup inf I If (x)
xEX qEM By Theorem 1-30, I If*[I < +-, and in fact
-
q(x)
1'.
(If*ll = inf { ~ ~ f - g q ~E ~ M};
Hence f* = 0 if and only if f E M, and since the quotient norm in L / i is precisely inf { I If-gl1; q E M}, the correspondence f* is an isometry between L / i and L*. f+fi It remains to prove that L* is an essential vector space of upper semicontinuous cross-sections, which is a C(X)module. Since
-
(qf)* (x) = q(x)f (x) + = q(x) f(x)
[
Mx
= 1
+ Fix, = q(x)f*(x)
Mx
if q(x) # 0, and (qf)*(x) = = 0. f*(x) if g(x) = 0, we see immediately that L* is a C(X)-module. To verify that L* is essential, let v be any element of the fiber Ex/gX. Then v=w+Kxr with for some w E Ex. Since L is essential, there is f E L f(x) = w. But then f*(x) = f(x) + Ex = w + gx = v. Finally, if f
1 If*(x) I I
E
L, the quotient norm
= inf qEM
I If(x) - g(x) 1
1
shows that x + I If*(x)l I is an infimum of upper semicontinuous functions, and therefore it is upper semicontinuous too. This completes the proo€ of Theorem 1.38. We come now to the case of A-modules W C L , where A is a self-adjoint subalgebra, not necessarily separating on X. However we restrain ourselves from the most general result and consider only the case L = C(X;E), where X is a compact HausdmT space and E is a normed space. Let Y be the quotient topologi(mod.A), cal space of X modulo the equivalence relation x E y and let n be the quotient map of X onto Y. It follows that Y is a compact Hausdorff space and €or each a E A, there is a unique b E C(Y) such that a = b o n. If we put B = {b E C(Y) ;a= b o n, a E A}, then B is a self-adjoint subalgebra, which is separating
COMPACT- OPEN TOPOLOGY
34
over Y, containing the constants, if A contains the cosntants. By the Stone-Weierstrass Theorem (Corollary 1.9, 55) , B is dense in C(Y), if A contains the constants. On the other hand the mapping a -+ b is a linear isometry of A onto B. Hence B is closed closed, if A is closed. Therefore B = C(Y), if A is a self-adjoint subalgebra containing the constants. For each -1 y E Y, let E = C(n (y); E). For each f E C(X;E), let f*(y) = Y f n”(y), for all y E Y. Then f* is a cross-section of the vecWe tor fibration (Y; (Ey; y E Y)). Let L = {f*; f E C(X;E) 1. claim that L is an upper semicontinuous space of m - s e c t i o n s , when endowed with the norm
1
1
If*( I = sup
{ I !f*(y)1 1 ;
y E Y).
This is a consequence of Lemma 1.25, by taking there
g(x) =
! If(x) I i . Since {n-l(y); y E Y) is a closed partition of X, the mapping f -c f* is a linear isometry of C(X;E) onto L.Notice -1 also that L is essential, i.e. L = {f*(y);f* E I.bC(n (y);E), Y when E is a Banach space. This follows from the Tietze Extension Theorem for Banach space-valued mappings on compact spaces (see Theorem 5.3 , [52] or Theorem 3.4 below) . THEOREM 1.40 L e t A c C(X) b e a s e l f - a d j o i n t c t o s e d s u b a t g e b r a , c o n t a i n i n g t h e c o n s t a n t s and L e t W C C(X;E) be an A-module where E i s a Banach s p a c e and X i s a compact H a u s d o r f f s p a c e . L e t M = {f*; f E W). Then
E ( w l n B I ) = U I S ( E ( M ’ ~~ 1 ) ) ;y E Y, Y Y Y
iiy #
E~I.
PROOF To apply Theorem 1.38 we must show that both M and L are C (Y)-modules. Indeed, let f* E L and b E C(Y). Since B = C(Y), there is a E A such that b o n = a. Hence
where x
E
n
-1
(y) is any point. Therefore bf*
E
L. In particular,
COMPACT
-
35
OPEN TOPOLOGY
M, then f E W, so af E W, and (af)* = bf* belonqs to M. The result now follows from Theorem 1.38, identifying C(X;E) and L by the isometry f * f*.
if f*
E
911. REPRESENTATION OF VECTOIi FIBRATIONS. In this paragraph we prove two propositions which together add up to an interesting representation result. The importance of this result lies in the illuminating role it plays as regards semicontinuity assumptions such as (3) of 9. Propositions 1.42 and 1.43 are due to L. Nachbin and appeared in [39] by his permission. DEFINITION 1.41.
Let
(X;(Ex;x
E X))
be a vector f i b r a t i o n sat-
isfying conditions
(1)
5 9 , i.e.,
and ( 2 ) o f
(1) X
i s compact;
each
(2)
i s a normed s p a c e , whose norm we
Ex
d e n o t e by Let
11
11 .
L C v ( E x ; x E X) be a v e c t o r space o f
tions. A representation of
r
:L
+
L
cross-see-
i s a l i n e a r map
C(F;M)
F i s a c o m p a c t H a u s d o r f f s p a c e p r o v i d e d w i t h a continuous TI : F + X s u c h t h a t , f o r a22 f E L and x E X , the o n t o map equa li ty
where
11
Ilf(x)
= sup
11
r(f) ( y ) I ; y
E
n-'({x~)
I
results.
PROPOSITION 1.42.
* Under t h e c o n d i t i o n s o f D e f i n i t i o n 1 . 4 1 ,
L admits a representation
i s a norm o n L , and
i f
r, then
r i s a l i n e a r i s o m e t r y of
(L,((
*
11)
into
36
COf.1PACT
(C(F; JK)I each
f
/I
1),
E L,
I
x
the function
i.e.,L
s a t i s f i e s condition
PROOF:
For each
f
E L,
-
11
where
- OPEN TOPOLOGY i s t h e sup
I,, il I-+
I/f(x)1 1
f
Moreover,for
i s u p p e r semicontinuous,
5 9.
( 3 ) of
write
- norm.
=
r(f). By the definition
of
representation, the equality 1 1 f (x) 1; = sup{ I f ( y ) l ; y E n-'({x})) is true f o r each x E X. Since is continuous and onto, -1 (TI ({x}); x E X ) is a partition of F into non-empty, compact subsets; and since f E C(F;K) for-each f E L , it follows that
It is now clear that r is a linear isometry of
(L,
!IF). Let f be a fixed element of L .
(C(F; l K ) , I / I
f = r(f) E C(F;K) of equations
is related to
/ / f /I :
X =
TI
11
*
11)
into
Since
(F) + IR by the Set
the upper semicontinuity of I/f1 1 follows from Lemma 1.25. The proof is now complete. The following converse of Proposition 1.42 is true. PROPOSLTION 1 . 4 3 . sume t h a t
L
( X ; (Ex;x E X ) )
Let
i s e s s e n t i a Z and t h a t
c o n t i n u o u s , for each
f
E L.
and
x v iIf(x)
L be a s above.As-
I(
i s upper semi-
Then t h e r e e x i s t s a r e p r e s e n t a t i o n
of L .
PROOF: The proof consists of the cofistruction of a Hausdorff space F jointly with a continuous onto map and a linear map r : L -+ C(F; IK) such that
for all
f
E
L
and all
x
E
X.
compact TI : F X -+
-
COMPACT
37
OPEN TOPOLOGY
F i r s t , observe t h a t t h e function
f E
B(Ei)
.
L
11
-+
f
I/
=
= s u p { l l f ( x ) l i ; x E X 1 i s a norm o n L Let , x E X , and B ( L ' ) denote t h e closed u n i t b a l l s of t h e continuous duals
E i ,
x E X ,
and
as t o p o l o g i c a l subspaces of weak- star
,respectively,
L'
t h e s e continuous d u a l s provided with t h e r e s p e c t i v e
orem. F ,
F
, Ex) ,
-
, x E X, and B ( L ' ) a r e c o m p a c t H a u s d o r f f s p a c e s by t h e A l a o g l u - B o u r b a k i Theo (Ei
topologies
x E X,
o (L' , L ) . B ( E i )
and
p u r e l y as a s e t , i s d e f i n e d t h r o u g h t h e e q u a t i o n
B ( E i ) , x E X.
This
:F
that
i s a d i s j o i n t u n i o n , o r sum, o f t h e s e t s
s e t i s p r o v i d e d w i t h t h e n a t u r a l o n t o map f o r each
(x
,$ 1
E F
,
TI
(x
,9)
= x.
IT
.+
X
such
The b u l k o f t h e p r o o f c o n
-
sists o f p r o v i d i n g F w i t h a topology. Define w : F + B ( L ' ) a s follows. For each ( x , 8 ) E F, t h a t i s , f o r each x E X and $ E
B(E4) , t h e v a l u e of
such t h a t
w
XI
9
w
at
( f ) = @ ( f( x ) )
(x
, 9)
is the functions w
for all
XI9
* IK
:L
f E L. This d e f i n i t i o n o f
w i s j u s t i f i e d by t h e f o l l o w i n g a s s e r t i o n :
(1) w x I 9
E B ( L ' ) f o r each
Indeed, w (2)
f o r each
XI9
f E
x
E X
and each
0
EE(C').
i s obviously l i n e a r and, L r
Iw,,,(~)I
5 /If(x)lj 5 IlflI.
=I@(f(X))I
T h i s c o m p u t a t i o n a l s o p r o v e s t h e n e c e s s i t y part of t h e next a s s e r t i o n : (3)
A linear functional
image s e t
x
f E L. o = w XI
function
$I
I f t h i s condition
9
where
,
given
z E EA
which
f (x) = z.
For l e t
x and
8
L
to
belongs
lQ(fl)
is
f (x)
the
exists for
satisfied,
all then
E B ( E A ) i s s u c h t h a t f o r any
@ ( z ) = @ ( f ) for. any f
E L
8 be a s i n ( 2 ) f o r a g i v e n 0 .
i s w e l l - defined: indeed, i f
fl(x) = f 2 ( x ) , then
on
1 @ ( f )1 5 1 )
such t h a t
E X
0
w(F) i f , and o n l y i f , t h e r e
-
fl
,f2
E L
and
0 ( f 2 )/ = I~(f1-f2)jIIIf1(X)-f2(x)l/=O;
for
The
38
COMPACT
- OPEN
on t h e o t h e r hand, g i v e n any that
f ( x ) = z , because
f u n c t i o n a l on f E L fore
.
X
/@ (z)
,
i
there is an
z E Ex
f ( x ) = z , then
= IO(f(x))
For t h i s
@
,
I 5
I:f ( x )
what i s
11
for, i f
w XrQ
t h e proof of
iI
=
w
z
)[
which p r o v e s
Xr@
:F
+
and
X
that
f ( x ) , f E L, it is Q = w
( 3 ) i s complete. IT
and
z E Ex
It i s such t h a t f o r a l l
?
( f ) = @ ( f ( x ) ) = CP ( f ) . Hence
To t h e m a p s
such
@ ( z ) = O ( f ( x ) ) , and t h e r e -
or, since L is e s s e n t i a l , f o r a l l
true that
f E L
L is e s s e n t i a l . @ i s o b v i o u s l y a linear
A c t u a l l y , @ E B(E.');
a r e such t h a t
@ E B(EA).
z E Ex
E
TOPOLOGY
w :F
Xr@
E w ( F ) and
-
* B(L') corresponds
t h e map
(4)
IT
To see t h i s l e t (X
, wX,+)
for all that
= (y
,w
f E L, w
@ = $
w
x
i s a n i n j e c t i v e map.
(x , $ 1 a n d ( y , $ 1 b e e l e m e n t s of F s u c h t h a t 1 . Then x = y ; h e n c e w = w , i. e . ,
Yr$
Xr@
( f ) = @ ( f ( x ) )= w
(because Providing
Xr$
X,$ XrJ, ( f ) = $ ( f ( x ) ) implying
L(x) = Ex). X x g ( L ' ) w i t h t h e product topology,
i s t h e n a c o m p a c t H a u s d o r f f s p a c e . The f o l l o w i n g a s s e r t i o n
it
is
t r u e a s a c o n s e q u e n c e ( i n p a r t i c u l a r ) o f t h e s e m i c o n t i n u i t y assumption.
(5)
The image
(TI
x
w)
(F) i s a c o m p a c t s u b s e t of t h e
compact H a u s d o r f f s p a c e I t i s e n o u g h t.0 p r o v e t h a t To t h i s e n d l e t
(xj
, wX j r Q j
(IT x
w ) (F)
X x B(L').
is closed i n
be a n e t i n
X x B(L').
w ) (F) c o n v e r g e n t
(IT x
t o ( y , 0) E X x B ( L ' ) . Since X x B ( L ' ) has t h e product topolo+ Q in B(L'). gy, it f o l l o w s t h a t x j + y i n X and t h a t w X j r @ j
However , c o n v e r g e n c e i n f o r each
B ( L ' ) means weak c o n v e r g e n c e , therefore,
f E L,
r e s u l t s . Using ( 2 ) ,
the inequality
iwxj,9j
(f)
I
/I
f(xj)
II
COMPACT
follows. Since
xj
+
y
-
1:
and
39
OPEN TOPOLOGY
f
1)
i s upper semicontinuous
for
f E L, it follows t h a t
each
=
f E L.
for all
+
where
/@(f)l
/ I f ( Y ) II
(3) now i m p l i e s t h a t
0 E w ( F ) ; a c t u a l l y , CJ = w Yr@
B(E') with + ( z ) = @ ( f ) f o r z E E a n d f E L with Y Y f ( y ) = z . Hence ( y , 0 ) = ( y , w ) E w ( F ) proving t h a t w (F) is E
closed i n
Y,+
X x B(L').
(4),
By
,$1
w is an i n j e c t i v e map
x
TI
so t h a t t h e
cor-
(x ,wx,+ ) E i s t o p o l o g i z e d by t r a n s p o r t i n g t o i t t h e s t r u c t u r e o f a compack
resDondence ( x
Hausdorff space of
2
associate a l l (x
,+ )
for all
E F.
for all
w ) ( F ) v i a t h i s b i j e c t i o r , . With each f EL,
d e f i n e d by
"fx,+)
= wxr+(f) = @ (f(x)) for
2
A s s u m e t h a t it h a s been proved t h a t
and c o n s i d e r
-
r :L
+
C(F;M)
Banach Theorem, t h e e q u a l i t y
i s o b t a i n e d f o r any
C(F;K)
E
such t h a t r ( f ) =
1
TI
-1
x E X
( { x ) ) = {x}
x
and any
+
+
/If
( X I : ! = sup{ I (f (x))l; Ez(EA))
f E L; b u t , c l e a r l y ,
r is a representation
iS(EA). Therefore
L e x c e p t t h a t t h e a s s e r t i o n "r( f ) =
of
w ) (F) i s b i j e c t i v e . F
f E L. The map i s o b v i o u s l y l i n e a r . A s a c o n s e q u e n c e o f
t h e Hahn
since
x
(TI
+
x
(TI
:F + M
f E L
F
E
f
E C (F;M) for a l l f E
s t i l l r e m a i n s t o b e p r o v e d . The t r u t h o f t h e a s s e r t i o n
L"
follows
from t h e f o l l o w i n g remarks. By t h e d e f i n i t i o n o f t h e t o p o l o g y o f TI
x
w
i s continuous. F i x
sf(x
,0)
tinuous. Finally, plete.
-
,
the
map
f E L. The e v a l u a t i o n map
B ( L ' ) ;hence, sf : E
i s c o n t i n u o u s i n t h e g i v e n topology of such t h a t
F
= O(f) f o r a l l
f = r ( f ) = sf o
(x (TI
,0) x
w)
E E x
E(L')
x
-+ IK is con-
E(L')
a n d t h e p r o o f i s com-
40
COMPACT
§ 12.
-
OPEN TOPOLOGY
THE APPROXIMATION PROPERTY: A l o c a l l y convex s p a c e
has t h e approximation prop-
E
A C E l a precompact s u b s e t , p E c s ( E ) , and
e r t y i f given
t h e r e i s a c o n t i n u o u s l i n e a r mapping T
,
into E
x
for all
11
with
i . e . an element
T
// 5
> 0,
o f f i n i t e rank from
E
such t h a t p ( x
8 E,
-
T(x)) <
E
i s normed a n d t h e a b o v e T may b e chosen
When E
E A.
T E E'
E
1, w e s a y t h a t
E
h a s t h e m e t r i c approximation prop-
erty. I n t h i s s e c t i o n , X i s a completely r e g u l a r
Hausdorff
s p a c e and W i s a v e c t o r s u b s p a c e o f C ( X ) c o n t a i n i n g C(X; [ 0 , 1 1 / ) .
W C C(X
Let
THEOREM 1 . 4 4 .
be a v e c t o r subspace
)
containing
C(X;jO,q).Then W w i t h t h e compact - open t o p o l o g y has t h e approxim a t i o n p r o p e r t y . If X i s a c o m p a c t sparae, then W with the uniform topology has the m e t r i c approximation property. PROOF:
Let
topology. L e t
let
E
K = X
set
be a precompact s u b s e t i n t h e compact-open
I3C W
K C X
be a compact s u b s e t and
> 0
be given. I f
and
p =
F C B
-
1
.
x
is a
Since
B
p =
g E F
p(f(x)
-
g(x)) <
Since
F
i s f i n i t e , Vx i s a n o p e n n e i g h b o r h o o d o f
E
/ 3
for all
x
E K.
I< C Vx
(1)
. a l l Q 2 , ... , @ n E
u ... u
1 Choose
V
For each
with
x E K, d e f i n e
x
in X .
{x, , x 2
, ... , x n ] c
C(X;R)
such t h a t
compactness, t h e r e i s a f i n i t e s e t that
and take
i s precompact, t h e r e i s a F i n i t e
f E B, t h e r e i s
such t h a t , given
i:
I
compact Hausdorff s p a c e
By such'
K
xn
ai(x) = 0, i f
x $ vX,
(i=lr2,...,n):
1
5
(2)
0 5 ai(x)
(3)
@,(x)
+
(4)
@,(x)
+ @,(XI
L e t us define
1, f o r a l l
a2(x)
+ ... + + ... +
T :C(X )
x E
x
( i= 1 , 2 , . . . , n ) ;
@ , ( x ) = 1, f o r a l l
x
E K,
@,(x)
x
E X.
+ C(X )
1, f o r a l l
by
COMPACT
for all
Clearly,
f E W.
claim that
T
T
- OPEN
TOPOLOGY
41
i s a f i n i t e rank l i n e a r operator.We
i s c o n t i n u o u s , and i f
X
i s compact
I n d e e d , w e have
for a l l
f E W,
for all
f E W.
by u s i n g ( 2 ) a n d ( 3 ) a b o v e .
Since
@1
-
g(x)) <
Q2
r
...
r
@n E C ( X ; r O
f E B. T h e r e e x i s t s
L e t now
p ( f (x)
r
E
/ 3
for a l l
x E K.
g
, 11) E F
C
W r
T f E Wr
such t h a t
Hence
n
for a l l
x
E K.
To e v a l u a t e t h e r e m a i n i n g sum on t h e r i g h t - h a n d
s i d e, l e t
and
For
i E Iy
,
p(g(x)
-
g(xi)) <
E
/ 3
and f o r
42 i E
COMPACT
jX
,ai(x)
=
-
O P E N TCPgL3GY
0 , by (1) above. Hence
n
for all
x E K . Thus
p(f
(x) -
(Tf)
(x))
r;,
f o r all x t: K ,
end-
i n g the proof. COROLLARY 1 . 4 5 .
If
X
i s a c o m p a c t i f a u s d o r f f space, C ( X )
t h e metric? a p p r o x i m a t i o n p r o p e r t y .
has
COMPACT - OPEN TOPOLOGY
43
APPENDIX
NON-LOCALLY CONVEX SPACES In this appendix we show how to extend some of the results of Chapter 1 to the case of vector-valued functions with values in a topological vector space E which is not locally convex. In this case, instead of the compact-open topology K one considers in C(X;E) a weaker topoloqy, namely the topoloqy K~ of uniform convergence on compact subsets which have finite covering dimension. A neighborhood basis of 0 for the topology K~ is given by the sets of the form N
=
(f
E
C(X;E); f(K) C U)
when K ranges over the compact subsets of X which have finite covering dimension, and U ranges over a basis of neighborhoods of 0 in E. As an example let us prove the following result due to A.H. Shuchat. THEOREM 1
C(X) 63 E is K f - d e n s e i n C(X;E)
.
PROOF Let f E C(X;E) be given. Let K C X be a compact subset of X which has finite covering dimension, say n, and let W c E be a neighborhood of 0 in E. Choose U c E an open balancedneiaborhood of 0 in E such that
u +...+ u c w (where'the sum is (n+l)- fold). ConEach x E K lies in the open set f-'(f (x) + U ) . sider the open covering of the compact set K consisting of -1 {f (f(x) + U) ; x E K). This open covering has a.refinement of Let order at most n + 1 that is also an open coverinq of K. -1 V1,. ,Vm be this refinement where Vi = f (f (xi) + U) , i = l,...,m, and xl,...,x m E K. Let ql, ...,gm be a continuous partition of the unity
..
44
COMPACT
-
OPEN TOPOLOGY
subordinate to this covering, i.e. each qi and
E
C(X), 0
5 qi
1,
..,m;
(1) gi(x) = 0 if x p! Vi for all i = 1,. m < 1 for all x E X; C gi(x) (2) i=l m (3) C gi(x) = 1 for all x E K. i=l
(This is possible because we have assumed X to be completely regular). m Let g = C gi 0 f (xi). Then g E C(X) 0 E and we claim i=1 that (f-g)(K) c W. Indeed, if x E K, then f(x) Let
rX
m
q(x)
=
x
E
g(x)
=
= (1 < i < m;
f(x)
E U,
-
-
C
i=l
qi(x) ( f ( x ) -
f(Xi))
Vi}. By (11, we see that C
i=IX
gi(X)(f(X)
-
f(Xi)).
On the other hand, for each i E Ix we have f(x)-f(xi) and since Ix has at most n + 1 elements, we see that f(x)
-
g(x)
E
u +...+ u c w,
since U is balanced, and this ends the proof. For further results on approximation in the non-locally convex case see the paper of A.H. Shuchat, “Approximation of vector-valued continuous functions”, Proc. Amer. Math. S O C . 31 (1972)I 97-103.
C O M P A C T - O P E N TOPOLOGY
REFERENCES FOR CHAPTER 1.
Number i n b r a c k e t s r e f e r t o t h e B i b l i o g r a p h y
ARENS
11 1 ]
ARENS a n d XELLEY BIERSTEDT BISHOP
[ 8
[5
[2 ]
1
I
BRIEM, LAURSEN and PEDERSEN
[lo
BROSOWSKI and DEUTSCH BUCK CHALICE
113
1 [ 15 ]
DUNFORD and SCHWARTZ
132
1
[ 371, [ 3 8 ]
MACHADO and PROLLA NACHBIN
SINGER STONE STROBELE
[ 3 9 1, [ 4 1 ]
43 ]
PROLLA and MACHADO RUDIN
[ 20 ]
[ 2 6 1 ,.[ 2 7 ]
GLICKSBERG
MACHADO
1
[12 ]
CUNNINGHAP4 and ROY
JEWETT
[ 9 ]
[5411,
[ 5511,
160
[ 62 ] [63
1
[ 52 ] 156
]
45
C H A P T E R
2
THE THEOREM O F DIEUDONNE
Hausdorff L e t X and X 2 be t w o c o m p l e t e l y r e g u l a r 1 spaces. W e s h a l l denote by C(Xl) Q C ( X 2 ) t h e v e c t o r subspace of C ( X ) , w h e r e X i s t h e product space of X1 and X 2 ,
consisting
of
a l l f i n i t e sums of f u n c t i o n s of t h e f o r m (X,Y) w h e r e f E C(Xl)
PROOF
f (XI
g(y)
and g E C ( X 2 ) .
The v e c t o r s u b s p a c e
THEOREM 2 . 1
i n C(X;E)
+
.
L e t A = C(X1)
(C(X1)
Q C ( X 2 ) ) Q E is
8 C ( X 2 ) and l e t W = A Q E .
It
can
e a s i l y v e r i f i e d t h a t A i s a s e l f - a d j o i n t subalgebra of
dense
be
C(X),
c o n t a i n i n g t h e c o n s t a n t s . S i n c e both X1 and X 2 a r e completely r e g u l a r Hausdosff spaces, A i s s e p a r a t i n g over X and W(x) = E , f o r each x E X. By C o r o l l a r y 1 . 1 3 , W is dense i n C ( X ; E ) . COROLLARY 2 . 2
C(X1)
0 C ( X 2 ) is dense i n C ( X ) .
W e can g e n e r a l i z e T h e o r e m 2.1 t o any n u m b e r of
fac-
tors. THEOREM 2 . 3
Let X = n(Xi;
i E I ) be an a r b i t r a r y p p o d u c t
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 s . The v e c t o r s u b s p a c e C ( X ; E ) g e n e r a t e d by a l : mappings of t h e form
where J
c I is a non-empty f i n i t e s u b s e t ; gi
i E J; and vJ E E;
PROOF
E C(Xi),
for
of of
all
is dense i n C(X;E).
Analogous t o t h a t of T h e o r e m 2 . 1 , C o r o l l a r y 2 . 2 ( s e e [181)
T h e o r e m 2 . 3 are due t o Dieudonng
.
and
T H E T H E O R E M OF DIEUDONNE
47
Let now E and F be two locally convex Hausdorff spaces (non-zero) and let T be a topology on the tensor product E Q F such that the canonical bilinear map of E x F into ( E 0 F , T ) is continuous. The upper bound of all such topologies is a locally convex topology, called the p r o j e c t i v e tensor product topology on E 8 F; it is the finest locally convex topobqy on E Q F for which the canonical bilinear map is continuous. Given two completely regular Hausdorff spaces X and Y , let f E C ( X ; E ) and 9 E C ( Y ; F ) . If E t3 F has a topology such that the canonical bilinear map is continuous, then (X,Y) belongs to C ( X THEOREM 2 . 4
x
-+
f(x)
Y; E Q F )
If E
9(Y) = (f
Q
x
PROOF
Q F has a topology such t h a t t h e canonical
i s continuous, then C(X;E) Y ; E Q F).
the form f
Q
Q
C(Y;F)
is
bi-
dense i n
denotes the vector subspace of F) consisting of all finite sums of mappings 9, where f E C ( X ; E ) and 9 E C ( Y ; F ) . Since
C(X;E)
C ( X x Y; E Q
9) (X,Y)
.
l i n e a r map C(X
Q
@ C(Y;F)
of
t(u Q v) = (tu) Q v = u Q tv for all t
E
M, u
E E
and v
E
F, one easily verifies
that
( C ( X ) Q C ( Y ) ) Q ( E Q F) is contained in C ( X ; E ) Q C ( Y ; F ) , the result follows from Theorem 2.1.
and
The above vector-valued version of the Dieudonn6 Theorem can be generalized to tensor products E F of locally convex spaces E and F which are topological modules over some locally convex topological algebra A . REMARK
DEFINITION 2.5 L e t A be a l o c a l l y c o n v e x t o p o l o g i c a l a l g e b r a , and l e t M and N b e two l o c a l l y c o n v e x s p a c e s w h i c h a r e t o p o l o g i c a l m o d u l e s o v e r A. T h e n M QA N i s d e f i n e d t o be t h e q u o t i e n t l o c a l l y c o n v e x s p a c e (M Q N) ID, where M Q N i s t h e t e n s o r p r o projective d u c t o f t h e v e c t o r s p a c e s M and N endowed w i t h t h e t e n s o r p r o d u c t t o p o l o g y , and D i s t h e c l o s e d l i n e a r s u b s p a c e of M Q N spanned by e l e m e n t s o f t h e f o r m (ax Q y-x 8 ay), a E A,
48
THE
THEOREM
O F DIEUDONNE
xEM,yEN. 9 denotes the If f : X + M and 9 : Y + N, then f map (x,y) + f (x) q(y) from X x Y into M N. If f and q are continuous, f QA q is also continuous. Moreover, if f E C(X,M) and g E C (Y,N), then f QA g belongs to the space C(X x Y,M QA N). We will denote by C(X,M) QA C(Y,N) the vector subspace of N) consisting of all finite sums of mappinqs of C(X x Y, M g, where f E C(X,M), q E C(Y,N). the form f
DEFINITION 2 . 7
L e t A be a t o p o l o g i c a l a l g e b r a . A t o p o l o g i c a l
module o v e r A, a l s o s a i d a t o p o l o g i c a l A-module, i s a t o p o l o g i c a l v e c t o r s p a c e B w h i c h i s a ( l e f t or r i g h t ) module o v e r A i n t h e u s u a l a l g e b r a i c s e n s e , s u c h t h a t t h e b i l i n e a r map A x B + B (whose v a l u e a t (a,b) we w i l t d e n o t e by ab) i s c o n t i n u o u s .
If both A and B are locally convex spaces, the bilinear map A x B + B is continuous if, and’only if, qiven any continuous seminorm p on B, there exist a continuous seminorm p2 on B and a constant k > 0 such continuous seminorm q1 on A , a <’ kql(a) p2(b) for all a E A , b E B. that p (ab) EXAMPLES 2.8 (a) Every topological vector space is a topological module over the scalar field. (b) Every topological alqebra with jointly continuous multiplication is a topological module over itself. DEFINITION 2.9
A net
{xi) o f e l e m e n t s o f a t o p o l o g i c a l
alge-
b r a A s u c h t h a t xia + a f o r e v e r y a E A, i s c a l l e d an a p p r o x i mate l e f t u n i t . S i m i l a r l y one d e f i n e s a p p r o x i m a t e r i g h t and two-sided u n i t s .
An approximate left (or right) unit is said
to
be
THE THEOREM OF
DIEUDONNE
49
bounded if the net {xi} is bounded. If A is locally convex,{xi} is bounded if, and only if, sup {q(xi); i E I) < m for each continuous seminorm q on A. If moreover there exists a constant K < + m such that sup {q(xi); i E I) < K for all continuous seminorms q which belong to a family of seminorms which determines the topology of A, then {x,} is said to be uniformly bounded. 1
REMARK 2.10 Let R be any ring and let M be a left R-module. If R has a unit element e, and if em = m for all m E M, then M is said to be a unital module. This motivates the followinq. DEFINITION 2.11 L e t A be a t o p o l o g i c a l algebra w i t h an app r o x i m a t e l e f t u n i t {ai) and l e t B b e a ( l e f t o r r i g h t ) t o p o l o g i c a l m o d u l e o v e r A . If aib + b f o r a l l b E B, t h e n B i s s a i d t o b e a n a p p r o x i m a t e l e f t - u n i t a l m o d u l e . S i m i l a r l y one d e f i n e s app r o x i m a t e r i g h t - u n i t a t m o d u l e s . If {ai} i s a n a p p r o x i m a t e t wo- s i d e d u n i t and aib + b for a l l b € B, t h e n B i s s a i d t o b e a n approximate u n i t a l module. DEFINITION 2.12 L e t A b e a t o p o l o g i c a l a l g e b r a and l e t B b e a ( l e f t o r r i g h t ) t o p o l o g i c a l m o d u l e o v e r A. We s a y t h a t B i s a n e s s e n t i a l A-module i f t h e v e c t o r s p a c e s v a n n e d by AB = {ab; a E A, b E B) i s d e n s e i n B. THEOREM 2.13
L e t A be a l o c a l l y convex t o p o l o g i c a l
algebra
w i t h a b o u n d e d a p p r o x i m a t e l e f t u n i t {xi), and l e t B b e a l o c a l l y c o n v e x s p a c e w h i c h i s a ( l e f t o r r i g h t ) t o p o l o g i c a l A-module. 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 :
(a) B (b) B PROOF
i s a n e s s e n t i a l A -m odul e.
i s an approximate l e f t - u n i t a l module.
It is evident that (b) implies (a). Let bo E B. Given E > 0 and p a continuous seminorm on B, there exist a continuous seminorm q on A, a continuous seminorm p ' on B and a constant k > 0 such that p(ab) < kq(a) p'(b) for all a E A, b € B. Since {xi} is bounded, there exists a constant c > 0 such that q(xi) 5 c for all i E I. Since B is an essential A-module,there exists al an E A and bl ,...,bn E B such that
,...,
T H E THEOREM OF DIEUDONNE
50
and
Choose now j,
E
I such that i > j, implies
-
q(ak
Xiak) < E/(3nP' (bk)
< k < n. Then for i > j, we have for all 1 -
P(bo
-
Xibo) 5 p(b, - C akbk) + p(Cakbk-xi(Xakbk)) + p(xi(C akbk)
-
xibo) <
< E/3 + Cp ( (ak-xiak)bk) + q (xi)p' (Cakbk-bo)k < ~ / 3+ Cq(ak-xiak)p' (bk)
+
< ~ / 3 n
-
~ / 3 n+ ~ / 3=
This ends the proof that (a)=>
+ c
.
€13~
E.
.
(b)
REMARK 2.14 For many topolosical modules the properties (a) and (b) of Theorem 2.13 are equivalent to the stronser property (c) AB = tab; a
E
A, b
E
B)
=
B.
When (c) is valid we say that the A-module B has the factorization property. Clearly any unital module has the factorization property. COHEN proved that every Banach alqebra with a bounded approximate unit has the factorization property, a result that was extended by HEWITT to bounded approximate left-unital Banach modules. THEOREM 2.14 Let A be a l o c a l l y c o n v e x t o p o l o g i c a l algebra and l e t M b e a l o c a l l y c o n v e x s p a c e w h i c h i s a n e s s e n t i a l t o p o l o g i c a t m o d u l e o v e r A. T h e n C ( X , A ) 8 C(Y,M) i s d e n s e i n C(X x Y, M). Both A and M are M-modules and A eK M is the PROOF linear span of AM in M, which is dense in M, since M is essential.
T H E THEOREI;~ O F
REFERENCES FOR CHAPTER 2 .
DIEUDONNE PROLLA STONE
[18]
[SO]
[62]
,
[70]
DIEUDONNE
51
C H A P T E R
3
EXTENSION THEOREMS FOR VECTOR-VALUED FUNCTIONS
In this chapter we shall prove several extension theorems for vector-valued functions defined on compact subsets of completely regular spaces. If Y c X is a closed subset of a completely regular Hausdorff space, then Y is also a completely regular Hausdorff space with the relative topology. If E is a locally convex space, let Ty be the restriction map Ty : C(X;E) + C(Y;E). This map is obviously linear, and since every compact subset of Y is a compact subset of X, the map Ty is continuous. Let Cb(X;E)!Y be the image of Cb(X;E) under Ty. THEOREM 3.1
The v e c t o r s u b s p a c e Cb(X;E) !Y i s d e n s e i n C(Y;E)
.
PROOF Let A = Cb(X) IY. Obviously, A is a self-adjoint subalqebra of C(Y) , containing the constants. Since X is completely regular, A is separating over Y, and W = Cb(X;E)(Y C C(Y;E) is such that W(x) = E , for each x E Y. Now W is obviously an A-module. By Corollary 1.13, W is dense in C(Y;E), as claimed. The following result is fundamental in establishing extension theorems. Our proof follows closely De La Fuente [16:, which is also the source of Definition 3 . 3 below. LEMMA 3 . 2
L e t X b e a compact H a u s d o r f f s p a c e . F o r any
empty c l o s e d s u b s e t Y
C
non-
X, t h e l i n e a r mapping
f r o m C(X;E) i n t o C(Y;E) i s a t o p o l o g i c a l homomorphism, f o r e a c h l o c a l l y c o n v e x s p a c e E.
PROOF It is enough to prove that for some neighborhood base F of 0 in C(X;E) , Ty(U) is a relatively open subset of C(X;E) IY, for all U E F. Let us consider the neighborhood base F of 0 in C(X;E) consistinq of all subsets of the form
53
E X T E N S I O N THEOREMS
U = {g E C ( X ; E ) ;
where p E c s ( E ) and
p(g(X)) <
EI
X E
XI
> 0 . The s u b s e t W of C ( Y ; E ) d e f i n e d by
E
W = {h E C ( Y ; E ) ;
p(h(x)) <
EI
x
E YI
i s t h e n an open neighborhood o f 0 i n C ( Y ; E ) . T x ( U ) = W n [C(X;E) IY]
C(X;E) IY.
, whence
The i n c l u s i o n T x ( U )
W e claim t h a t
T x ( U ) i s r e l a t i v e l y open
c
v e r s e l y , l e t h E W n [ C ( X ; E ) IY]
W
n [ C ( X ; E ) IY]
. Let
i s o b v i o u s . Con-
9 E C ( X ; E ) be such
q ( x ) = h ( x ) f o r a l l x E Y. L e t F = { t E X ; p ( g ( t ) )
F c X i s c l o s e d and d i s j o i n t from Y.
in
I f F = pI,
that
1. €1.
Then
then g E U,
I f F # Jf, t h e r e e x i s t s $ E C ( X ) , 0 <
therefore h E Ty(U).
and
<1,
I#
$ ( X I = 1 f o r a l l x E Y , and $ ( t ) = 0 f o r a l l t E F. L e t $9 E C ( X ; E ) .
Then f ( x ) = g ( x ) = h ( x ) f o r a l l x E Y ,
T y ( f ) . W e claim t h a t f E U.
so p ( f ( x ) ) = 0 <
E.
L e t x E X.
i.e.
h =
I f x E F, t h e n f ( x ) = 0 ,
I f x f! F, t h e n p ( f ( x ) ) = p ( $ ( x ) g ( x ) ) =
$ ( x ) p ( g ( x ) )5 p ( g ( x ) ) <
E.
Thus 9 E U , and h E T y ( U ) .
L e t C b e a class of l o c a l l y convex
DEFINITION 3.3
f =
Hausdorff
s p a c e s . Iv'e s a y t h a t C has p r o p e r t y EC i f :
(1)
e v e r y e l e m e n t of C i s c o m p l e t e ;
(2)
f o r e v e r y compact Hausdorff s p a c e X,
and
for
e v e r y E E C, C ( X ; E ) b e l o n g s t o C; (3)
f o r e v e r y E E C, space F
EXAMPLES.
C E,
and e v e r y c l o s e d v e c t o r
sub-
t h e q u o t i e n t E/F b e l o n g s t o C.
( a ) The class of a l l Banach s p a c e s ;
(b)
t h e class of a l l F r g c h e t s p a c e s . THEOREM 3.4
(De La Fuente
[16!).
L e t C be a class of
convex Hausdorff s p a c e s s a t i s f y i n g p r o p e r t y Ec.
locally
L e t X be a com-
p a c t Hausdorff s p a c e and l e t Y C X be a non-empty c l o s e d subset. Then C ( X ; E ) ( Y = C ( Y ; E )
PROOF
By Theorem 3.1,
closed i n C(Y;E).
f o r a l l E E C.
a l l w e have t o prove i s t h a t C ( X ; E ) I Y i s S i n c e E E C and X
L e t N b e t h e k e r n e l o f Ty.
i s compact, C ( X ; E ) b e l o n g s t o C. The l i n e a r mapping Ty continuous, N i s a closed subspace of C ( X ; E ) .
By c o n d i t i o n
being (3)
54
E X T E N S I O N THEOREMS
of Definition 3.3, C(X;E)/N belongs to C, and therefore by (1) of same Definition, C(X;E)/N is complete. By Lemma 3.2, Ty is a topological homomorphism. Hence C (X;E)/N and Ty (C(X:E) ) = C(X;E) IY are topologically linearly isomorphic. Thus C(X:E) ( Y is complete too, and therefore closed in C(Y:E). This ends the proof. COROLLARY 3.5
L e t C b e a c l a s s of l o c a l l y c o n v e x
Hausdorff
s p a c e s s a t i s f y i n g p r o p e r t y Ec. L e t X b e a c o m p l e t e l y H a u s d o r f f s p a c e and l e t Y
Cb(X:E)lY
c X
b e a non-empty
regular
compact s u b s e t . T h m
= C(Y;E), f o r aZZ E E C.
PROOF The space X is contained in its Stone-cech compactification gX, and Y c X beinq compact is closed in UX. By Theorem 3.4, C(BX;E) IY = C(Y:E). If f E C(Y:E), let h E C(PX:E) be such that h(x) = f (x) for all x E Y. Let now g = hlX. Then gECb(X:E) and Ty(g) = f, i.e. Cb(X:E) IY = C(Y:E). can be REMARK. The extension q E Cb(X;E) in Corollary 3.5 chosen so that 1 lql I x = I (fll y , when E is a Banach space. Indeed, let r = I'ftly.Define I$ : E + E by @(t) = t, if I!tlI < r: and $(t) = rt/l It1 I , if 1 It1 I > r. Let u E Cb(X:E) be an exten- =\If( ly sion of f. Let h = 6 o q. Then Ilh(x)'l = l\I$(q(x))'l
THEOREM 3.6
L e t Y b e a c l o s e d non-empty
s u b s e t of
H a u s d o r f f s p a c e X, l e t E b e a Banach s p a c e , and l e t
a
f
normal E
C(Y:E)
b e a c o m p a c t m a p p i n g . T h e r e e x i s t s a c o m p a c t m a p p i n g g E C(X:E) s u c h t h a t gIY = f, and g(X) i s c o n t a i n e d i n t h e c l o s e d
convex
h u l l of f(Y).
PROOF From the properties of the Stone-Eech compactification, it is known that BY identifies with the closure of Y in f3X (see to Stone [61]) and that a mappinq f E C(Y;E) has an extension f BY if, and only if, f (Y) is precompact. Hence we can extend to BY. Call pf this extension pf E C(f3Y:E). By Theorem 3.4 w e
55
E XT E N S I 0 N T H E 0 R E 1.1s
exists h
E C(f3X;E)
s u c h t h a t h!RY = p f . L e t
q'
= hlX.
Then
g' E C ( X ; E ) a n d g' IY = f . L e t K b e t h e c l o s e d convex h u l l
of
f ( Y ) . S i n c e E i s a Banach s p a c e , K i s compact a n d , t h e r e f o r e , a
r e t r a c t of E (see Dugundji [ 1 9 ] ) . L e t r b e a r e t r a c t i o n of o n t o K.
Then g = r o g' b e l o n g s t o C ( X ; E )
such t h a t g l Y = f , and g(X) C K.
is a compact
E
mapping
56
EXTENS I O N THEOREMS
REFERENCES FOR CHAPTER 3 .
[ 161
DE LA FUENTE DUGUNDJI
[I9 ]
PROLLA a n d MACHADO STONE
[61]
,
[62]
[52]
C H A P T E R
4
POLYNOMIAL ALGEBRAS
5
1
BASIC DEFINITIONS AND LEMMAS
The important notion of polynomial alqebra was introduced by Pekzyiiski [47] in 1957, usinq multilinear transformations. An equivalent definition usinq polynomials was introduced definitions by Wulbert (unpublished) (see Prenter [49! ) Both are more restrictive than the one given here, because they make use of all multilinear transformations and polynomials, respectively, whereas we only ask for invariance under composition with those of finite type, and do not assm that the polynomial algebra contains the constants. Moreover, the previous work on polynomial algebras was restricted to compact spaces X and Banach
.
spaces E. A third equivalent definitions was introduced by Blatter [4], who considered Co(X;E), for locally compact X and Banach space E. We introduced our definition in [52!, for completely regular spaces X and locally convex spaces E, where we proved the equivalence between the several possible definitions of polynomial algebras. Independently, De La Fuente :16] studis3 polynomial algebras too. LEMMA 4.1 L e t W c C(X;E) b e a v e c t o r s u b s p a c e ( r e s p . closed v e c t o r s u b s p a c e ) . The f o l l o o i n g p r o p e r t i e s a r e e q u i v a l e n t . (1) W i s 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 c o n t i n u o u s l i n e a r maps of f i n i t e rank u E E' 8- E. (2) For e v e r y $I E E', v E E, f E W, t h e n x + $I(f( x ) )v b e l o n g s t o W. (3)
A = {$I o f; $I E F', f E W) is a v e c t o r subspace ( r e s p . a c l o s e d v e c t o r s u b s p a c e ) of C(X), suck that ABECW.
POLYNOMIAL ALGEBRAS
.
PROOF
(1) obviously implies (2) Assume (2). The set A is clearly invariant under multiplication by any scalar A E IK. Let @ o f and $ o g be qiven in A . If @ = 0, I$ o f + J, o g = J, o g E A. If @ # 0, choose v E E, such that @(v) = 1. By ( 2 1 , x + @(f(x)) v and x + J,(q(x))v belong to W. Let h E W be the function defined by x + [@(f (XI) + J,(g(x))lv. Then @ o h E A, and @(h(x)) = [@(f(x)) + J,(g(x))!@(v) = [ $ o f + J, o g: (x) for all x E E. Therefore A is a vector subis space of C(X). Obviously, (2) implies A Q E c W, since W W is closed, and that f E a vector subspace. Suppose now that Choose a pair $ E E', v E E, with @(v) = 1. Let q = f Q v.Given K C X a compact subset, E > 0, and p a continuous seminorm onE, there is J, o h E A such that
x.
[f(x) - $(h(x))l
< ~ / ( +1 p(v))
for all x E K. Hence p(g(x) - J,(h(x))v) < t , for all X E K.Since $(h(x))v belongs to W, it follows that g E = W. Since x f = @ o g , we see that f E A , i.e. A is closed too. Assume ( 3 ) . Let f E W and u E E' Q E. Suppose u = C @i 8 vi, with @i E El, vi E E, i = l,...,n. Then u o f = C hi Q vi, where hi = @i o f E A , for each i = 1, ,n. Hence,
w
+
...
u o f which
E
A Q) E. By ( 3 1 , proves (1).
COROLLARY 4 . 2
A Q
E C W, and consequently u o f
E
W,
L e t E be a l o c a l l y c o n v e x H a u s d o r f f s p a c e
the approximation p r o p e r t y . Let W
C
with C(X;E) be a v e c t o r s u b s p a c e
i n v a r i a n t u n d e r c o m p o s i t i o n w i t h e l e m e n t s of El A = {@o
g
E
w if,
f; @
E
E', f
and o n l y i f ,
E
W). Then f o r e v e r y g @ o
g E A,
w
for every @
E
QP E. Let C(X;E) we have E E'.
PROOF is The statement g E * @ o g E x,for every @ E El, always true, even when E does not have the approximation property. Conversely, assume that E has the approximation property, and let g E C(X;E) be such that @ o g E x,for all @ E El. Let K C X, a compact subset, E > 0 and p E cs(E) be given.Since g(K) is a compact subset of E, and E has the approximation profor all perty, there is u E E' Q E such that p(t-u(t)) < E
59
POLYNOMIAL ALGEBRAS
t E q(K). Let u = C +i Then p(q(x)
-
Q
vi, with I$i
E
E', vi
E
E, i = 1,2,...,n.
n 1
i= 1
+i(g(x))vi) <
E
for all x E K. Hence g belongs to the closure of Q E C A €3 E, q belongs to A Q E. By Lemma 4.1, A Consequently, q E W, QED.
Q Q
E. Since E c W.
The above Corollary 4.2 is the most useful tool in proving vector-valued versions of Theorems known for scalar-valued functions, when the range space E has the approximation property. A s an example, let us prove Merqelyan's Theorem for vector-valued functions in this case. We use the following notation. If K c C is a compact subset, A(K;E) is the closed subspace of C(K;E) of all functions holomorphic in the interior of K. Let W = T(C;E) (K. It is easily verified that W is invariant under composition with elements u E E' Q E. Moreover, if we assume that E # {O}, then REMARK.
q(C;C)lK = {I$ o f;+
E
E', f E W}.
Suppose now that C\K is connected, and let q E A For each I$ E E', + o q E A(K;C). By Merqelyan's Theorem, belongs to closure of T ( C ; C ) in C(K;C) By Corollary belongs then to the closure of W in C(K;E). This proves Mergelyan's Theorem for E-valued functions, when E has the approximation property. The proof for general E due to Bierstedt [5! uses the fact that A(K;C) has the approximation property, when K c C has a connected complement. To give another illustration of Corollary 4.2, let E be a Banach space over grand let U C E be a non-void open subset. If F is another Banach space over C, €I(U;F) denotes the space of all holomorphic mappinqs from U to F. If F = C, we write simply II(U). We recall that f: U + F is holomorphic in if f possesses a FrGchet derivative at each point E, E U,i.e. U, at each E, E U, there exists Df(E,) E L(E;F) such that
IK
.
P0 L Y N0MI A L A LGEB RA S
60
See Nachbin [ 4 4 ] for a reference on properties of holomorphic mappings. In particular f has a power series development and the Cauchy formulae are valid. For the first derivative, we get for any p > 0 and x E E such that F, + X x E U for every X E C, Ihl
€,
5
r
0:
+ Ax
where K
E
5
If K c U is compact, we can find a p > 0 U for all x E K, X E C, 1x1 2 P . Hence
such
that
is the compact subset of U, KS =
THEOREM 4.3
fc +
AX; x
E
K, X
E
C, I X ' =
( A r o n and S c h o t t e n l o h e r [ 3 ! ) .
p).
Let F be a
complex
Banach s p a c e . T h e f o l l o w i n g p r o p e r t i e s a r e e q u i v a l e n t :
(a) F 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 . (b) For e v e r y c o m p l e x Banach s p a c e E and f o r every n o n - v o i d o p e n s u b s e t U C E, t h e s p a c e H (U) 8 F i s d e n s e i n tI(U;F) i n t h e c o m p a c t - o p e n t o p o l o g y . (c) H ( F ) Q F i s d e n s e i n II(F;F) i n t h e c o m p a c t - o p e n t O P O l O m .
(d) T h e i d e n t i t y map idF p a c t - o p e n c l o s u r e of
:
F
+
F belongs t o the F i n I1 (F;F)
.
11 (F) Q
.
.
com-
PROOF. (a) * (b) Let W = I1 (U) Q F c C(U;F) Obviously, W is invariant under composition with elements of F' 0 F. Let A = { $ o f; 4 E F', f E W). Then A = I1 (U). Let now q E II(U;F). For every 4 E F', 4 o g is holomorphic in U, whence 4 o q E A . By CorolLary 4.2, g belonqs to the closure of W = H(U) Q F in the compact-open topoloqy. (b) =. (c). Take U = E = F. (c) * (d) Obvious. (d) * (a). Let E > 0 and K c F compact be qlven. By Cauchy's formula, the seminorm
.
p(f)
=
SUP {1'Df(O)x'l;x
E
K)
61
POLYNOflIAL ALGEBRAS
defined in €I(F;F), is continuous in the compact-open topology. By (d), there exists g E H(F) 8 F such that p(q-idF) < E . Let u = Dq(0). Then u E F' 8 F and 1 lu(x) - X I I < E for all x E K, which proves (a). DEFINITION 4 . 4
L e t E and F b e t w o n o n - z e r o
s p a c e s . For e a c h i n t e g e r n > 1, ?:(E;F)
l o c a l l y convex
d e n o t e s t h e v e c t o r sub-
s p a c e of C(E;F) g e n e r a t e d b y t h e s e t of a l l maps of t h e x * [@(x)Jnv, w h e r e @ E E' and v E F. The e l e m e n t s of
form
a r e c a l l e d n-homogeneous f r o m E t o F.
type
9 :(E;F)
c o n t i n u o u s p o l y n o m i a l s of f i n i t e
The vector subspace of C(E;F) generated by the union of all ?:(E;F), n > 1, and the constant maps, is denoted by (E;F), and its elements are called c o n t i n u o u s p o l y n o m i a l s of
Ff
~P(E)
E t o F. when F = M , we write simply arise. (6;F) is Notice also that, when E = C, the space the set of all complex polynomials with coefficients in F, that is the set of all functions of the form m Z * C an zn n=0
f i n i t e type from
and
9,(El , if no confusion may
Tf
where m E N , an E F, n = 0,1,..., m. In this case Tf(C:F) = T(C;F).
we
write
L e t E and F b e tlJo n o n - z e r o locally convex n s p a c e s . For each i n t e g e r n 1, ( E;F) d e n o t e s t h e v e c t o r subn s p a c e of C(E ;F) g e n e r a t e d b y t h e s e t of a l l maps of t h e form
DEFINITION 4 . 5
(x1t--.txn)* @
Zf
(X 1 ... @,(x1)v, 'n df(
wh'ere @l,-.. r@n E E' and vEF.
T h e e l e m e n t s of E;F) a r e c a l l e d n - l i n e a r c o n t i n u o u s of f i n i t e t y p e f r o m E" i n t o F.
LEMMA 4 . 6
( (521 )
Let W
C
maps
C(X;E) b e a v e c t o r s u b s p a c e . The f o l -
lowing p r o p e r t i e s are e q u i v a l e n t .
(1) F o r e a c h i n t e g e r n 2 1, g i v e n ql,..., qn E W and T E 8f(nE;E), t h e f u n c t i o n b e l o n g s t o W. x * T(ql (x), ,gn (XI (2) F o r e a c h i n t e g e r n 2 1, g i v e n q E W and
...
POLYNOMIAL ALGEBRAS
62
p o q b e l o n g s t o W.
p E p:(E:E),
(3) (4)
f E W} i s a s u b a l g e b r a
A = { @o f ; 4 E E l ,
of
C ( X ) s u c h t h a t A Q E C W. W i s i n v a r i a n t under composition w i t h continuo u s l i n e a r maps o f f i n i t e r a n k , and t h e r e exi s t s a c o n t i n u o u s map P : E x E + E and a v e c t o r vo E E s u c h t h a t
( a ) P(vo,vo) # 0; ( b ) P(avo,bvo) = a b . P ( v o , v o ) , f o r a l l a , b C- M :
( c ) g i v e n f ,q E W , b e l o n g s t o W. PROOF
the function x
+
P ( f ( x ) ,q(x))
(1) o b v i o u s l y i m p l i e s ( 2 ) . A s s u m e ( 2 ) . Taking n = 1, w e see t h a t W
p r o p e r t y ( 2 ) o f Lemma 4.1. such t h a t A 8 E 4($
0
c W. L e t f)($
0
satisfies
Hence A i s a v e c t o r s u b s p a c e o f C ( X )
d, o f and JI o 9 be g i v e n i n A.
9) =
[$I
0
f + IlJ
0
s:*
-
!4
0
f
-
Since 2 JI 0 q] ,
a l l t h a t remains t o prove i s t h a t ( @ o f ) 2 b e l o n a s t o A , f o r any $ E E l and f E W. I f 4 = 0 , t h e r e is n o t h i n q t o prove. I f 4 # 0, all choose v E E s u c h t h a t @ ( v )= 1. L e t p ( t ) = [ $ ( t ) J 2 v f o r t E E. Then p E ( E : E ) , and by ( 2 1 , p o f E W.Let 9 = p o f .
9;
2 2 Then 4 ( g ( x ) ) = $ ( p ( f ( x ) ) ) = @ ( [ $ ( f ( x ) ) ]v ) = [ $ ( f ( x ) ) ! = 2 ( 4 o f ) ( X I , f o r a l l x E X , i . e . (I$o f ) 2 E A , and A i s a n
al-
gebra. A s s u m e ( 3 ) . By Lemma 4.1, W i s i n v a r i a n t under comp o s i t i o n w i t h c o n t i n u o u s l i n e a r maps o f f i n i t e r a n k u E E ' 0 E. Choose a p a i r 4 E E ' and vo E E w i t h $ ( v o ) = 1. Define p:ExE+E by P ( s , t ) = $ ( s ) @ ( t ) v of o r a l l s , t E E . Then P i s and s a t i s f i e s ( a ) and ( b )
. Let
continuous
f ,q E W , S i n c e I$ o f and
4 o g
( 3 ) i m p l i e s t h a t (I$o f ) ( 4 o 9 ) E A , and h = all ( 4 o f ) ( 4 o 9) Q vo E W. However, h ( x ) = P ( f ( x ) , g ( x ) ) f o r x E X , which e n d s t h e proof o f ( 4 ) . F i n a l l y , assume ( 4 ) . L e t n > 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 <
E
for all x
E
such that
E with I \ x ! I5 n.
POLYNOIIIAL ALGEBRAS
78
REFERENCES FOR CHAPTER 4 ARON and SCHOTTENLOHER BLATTER
[4]
[5]
BIERSTEDT
[16]
DE LA FUENTE FIGIEL
[23:
JACOBSON JOHNSON LANG
[3]
[ 31 ] [33J
[36;
MACHADO and PROLLA
[39]
NACHBIN
;44]
NACHBIN,
MACHADO a n d PROLLA
PELCZY~SKI [47]
PXENTER
[48]
, 11491
PROLLA a n d MACHADO RESTREPO SCHAFER
[53] [58]
[52]
11461
C H A P T E R
5
WEIGHTED APPROXIMATION
9 1 DEFINITION
OF NACHBIN SPACES
Let X be a Hausdorff space. A family V of upper directed semicontinuous positive functions on X is said to be if given v,w E V, there exists a X > 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))<
G(W)*, there exist bl,
G(W) such that w = blgl LEMMA 5.14
+
+...+
...,bm
E
h +E.
B and glI...,gmE
bmgm.
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 s a -
t i s f i e d . G i v e n v E V, p E cs(E), b E B, g E G(W) and 6 > 0 , t h e m
i s w E W such t h a t IIw
-
bglIv,p < 6 .
WEIGHTED APPROXIMATION
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)) <
all x E X. Define r E Cb(IRn;IR) by settinq r(t) = a(t ll...ltn) for all t = (tl,...,tn,...,tN) E IRN. By hypothesis E oN; N is hence Cb(IRN;IR) is contained in Cwm(JR ;IR) and ?(RN) N dense in Cwm(lR ;R ) Given 6 > 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
WEIGHTED APPROXIMATION
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<
d. Let
u
C
x
be the open set {t E X; v(t)p(f(t) - q(t)) < d). Then the complement of U in X is compact, and S u C U. By compactness, there
..
. ..
u (X\Sn) ,where exist T~ <. < T < u such that X\U C (X\S1) U n Si = S , with T = 1. However, since S n C C S1, it follows that i Sn c U, a contradiction to (b), because T~ < u.
5
3
...
SUFFICIENT CONDITIONS FOR SHARP LOCALIZABILITY
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) THEOREM 5.17 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) 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; ( 2 ) g i a e n a n y v E V, a E G(A), q E G ( W ) , and p s cs(E), t h e r e e x i s t s Y E rl s u c h t h a t v(x)p(q(x)) 5 Y( lab) I ) for a l l x
E
E
X.
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 CV,(X;E).
PROOF
Given any v
E
V
al ,
.. .,an
E
G (A), q
E
G(W) ,
p E cs(E) , there are yi E I's such that v(x)p(g(x)
n for all x E X, i = l,.. ,n. Define w on R by w(t) = [yl(tl). yn(tn for all t = (t
..
I''~
5
and yi( !ai(XI!)
- - .rtn) -
Then
Rn by Lemma 1, 5 27, [ 4 3 ] . Obviously, w(t) = w(1tl) for all t E Rn. Hence, w E R z . By Theorem 5.16, W is sharply localizable under A in CVm(X;E). w E
THEOREM 5.18
(Analytic criterion)
Suppose t h a t t h e r e
exist
W E I GH T E D A P P R O X I FIAT I 0 N
89
s e t s of g e n e r a t o r s G ( A ) and G ( W ) s u c h t h a t :
(1) 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 A; (2) g i v e n any v 6 V, a E G ( A ) , q E G ( W ) , and p E cs(E) , t h e r e a r e c o n s t a n t s M > 0 and m > 0 -m'a(x)I f o r all s u c h t h a t v(x)p(q (x)) 5 M e x E x. 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 CVm(X;E). -mltl defined for all t E IF, The function y(t) = M e belonqs to I's by Lemma 2, 5 28 of [43]. It remains to apply Theorem 5.17 above. PROOF
THEOREM 5.19
(Quasi-analytic criterion)
Suppose t h a t
there
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 ) s u c h t h a t :
(1) 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 A ; (2) g i v e n any v E V, a E G ( A ) , q E G ( W ) ,
m
=
1 la
dIVrPf o r
m
=
0,1,2
and
,... .
T h e n W i s s h u r p l y l o c a l i z a b l e u n d e r A i n CVm(X;E).
PROOF Define y on E? as in the proof of Theorem 9, 1 4 6 1 , then apply Theorem 5.17, above. THEOREM 5.20
(Bounded c a s e )
and
S u p p o s z t h a t t h e r e g 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 ) 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; (2) g i u e n a n y v E V, a E G ( A ) , q E G(W), and p E cs(E), t h e f u n c t i o n a i s b o u n d e d o n t h e S U P port o f v p ( q ) . 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 CVm(X;E).
PROOF Let v E V, a E G ( A ) , p E cs(E) and q E G ( W ) be q1VeX-I. Let m > sup { !a(x)I ; x E S } , where S is the support of the function vp(q); and let M > 1 Iql IVrp. If y is the characteristic function of the interval [-m,m! c IR times the constant M, then S y E rl and v(x)p(q(x)) 2 y(la(x) 1 ) for all x E X. It remains to
Id E I GH T E D A P P R O X I CIA T I O N
90
apply Theorem 5.17. REMARK The above Theorem 5.20 generalizes Theorem 4, 9 2 of Kleinstuck [35], which in turn was a generalization of Theorem 4.5 of Prolla [51] and of the result of Summers 164:. COROLLARY 5.21
L e t W C CVm(X;E) h e a n A - m o d u l e .
Suppose
that
e v e r y a E A is b o u n d e d o n t h e s u p p o r t of e v e r y v E V. 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 CVm(X;E).
PROOF The set A is a strong set support of v contains the support continuous seminorm p E cs(E) and rem 5.20 with G(A) = A and G ( W ) =
of qenerators for A . Sincetk of x -* v(x)p(q(x) ) for any any q E W, we may apply Theo: W.
COROLLARY 5.22 ( K l e i n s t u c k [35] I . Assume t h e h y p o t h e s i s C o r o l l a r y 5.21. T h e n f o r e v e r y f E CVm(X;E), f b e l o n g s t o c l o s u r e of W i n CVm(X;E) i f , a n d o n l y i f , g i v e n a n y v E V, cs(E), E > 0 , and K E there' e x i s t s q E W such t h a t v(x)p(f (x) - g(x)) < E f o r a l l x E K.
x,
of
the
p
F
PROOF By Corollary 5.21, W is sharply localizable under A in CVm(X;E). Since P = rd,, W is sharply gA-localizable, i.e., P given f E CVm(X;E) , v E V, p E cs(E) , there is some maximal antisymmetric set K E such that
xA
IVtp;
inf {I'f-g!
q
E
W } = inf
(1
lflK
g'KI I
VIP
; g
E
W].
This formula generalizes that obtained by Glicksberq in the case of bishop'.^ Theorem (see [26]) , and from it there follows the desired conclusion.
5
4
COMPLETENESS OF NACHBIN SPACES
F(X,E) is the vector space of all mappinqs from X into E, and B(X,E) is the vector subspace of all mappinqs € from X into E such that f(X) is a bounded subset of E, and Bo(X,E) is the vector subspace of B(X,E) consistinq of those bounded mappinqs f from X into E that vanish at infinity, i.e., those f E B(X,E) such that, qiven any continuous seminorm p on E and any E > 0, there is a compact subset K c X such that
WE I GH T E D A P P R O X I [..!AT I O N
91
p(f(x)) < E for every x E X outside of K. The vector subspace C(X,E) f? B(X,E) is Cb(X,E), and C(X,E) n Bo(X,E) is Co(X,E). Finally K(X,E) will denote the subspace of C(X,E) consistinq of those functions that are identically equal to 0 in E outside of some compact subset of X. The correspondinq spaces for E = IR or C are written omittinq E. If V is a directed set of weiqhts on X, the vector spaces of all f E F(X,E) such that vf E B(X,E), for any v E V, is denoted by FVb(X,E). On FVb(X,E) we shall consider the locally convex topology determined by the family of seminorms I f 1 'v,p = sup {v(x)p(f (x)); x E X) when v ranqes over V and p ranges over the set of all continuous seminorms on E. This topoloqy will be denoted by wV, and the space PVb(X,E)endowedwith wv is called a weighted space of vector-valued functions.It has a basis of closed absolutely convex neiqhborhoods of the origin of the form Nv = {f E FVb(X,E); 1 If1I v t p < l}. The vector rP subspace of FVb(X,E) consistinq of those f f F(X,E) such that vf E Bo(X,E), for any v E V, is denoted by FVm(X,E) and it is endowed with the topoloqy induced by FVb(X,E). FVm(XIE) is a closed subspace of FVb(X,E). We shall always assume that FVb(X,E) is a Hausdorff space. This is clearly the case if V is everywhere different from zero on X, i.e., if for every x E X there is a v E V such that v(x) > 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
<
E.
$(XI
K,
h e n c e v ( @ ( x ) ) p ( f ( @ ( x )<) )E .
This
implies
that
W E I G tI T E D A P P R 0 X I 14A T I 0 N
93
u(x)p(f(Q(x))) < E . If x is not in the support of u then u(x)=O and again u(x)p(f(Q(x))) < E . Hence u(f o $ 1 E Bo(X,E). Since u E U was arbitrary, f o 4 E FV-(X,E). We have thus proved the following. L e t U b e a d i r e c t e d of w e i g h t s
PROPOSITION 5.27 Q : X
on
X
and
X b e a mapping s u c h t h a t f o r e v e r y compact s u b s e t K C X , -1 (K) w i t h t h e s u p p o r t o f a n y u E U i s comp a c t . T h e n , f o r e v e r y d i r e c t e d s e t of w e i g h t s V o n X s u c h t h a t U < V o 4, t h e m a p p i n g f + f o Q i s a c o n t i n u o u s l i n e a r mapping f r o m FVm(X,E) i n t o FUm(X,E). -+
the intersection of Q
I f U and V a r e d i r e c t e d s e t s o f w e i g h t s o n X
PROPOSITION 5 . 2 8 with U < V , then
(1)
FVb(XrE)
C
FUb(XrE):
(2)
FVm(XrE)
C
FUm(XrE);
(3)
t h e t o p o l o g y i n d u c e d o n FVb(X,E) by wu i s weake r t h a n wv.
PROOF
In Proposition 5.27 take Q equal to the identity map on
X. COROLLARY 5.29
X with
U =
I f U and V a r e t w o d i r e c t e d s e t s o f w e i g h t s on
V, t h e n FUb (X,E)
=
FVb(X,E) and FUm(XIE) = FVm(XrE)
as topological vector spaces.
Let V be a directed set of weiqhts on X such that the weights v E V vanish at infinity and xc(X) < V.Then B(X,E)C some FVm(X,E) and the topology induced on B(X,E) by wV enjoys of the properties of the strict topoloqy R . Namely, we have the following result. L e t V be a d i r e c t e d s e t o f w e i g h t s o n X s u c h t h a t xc(X) < V and a l l w e i g h t s v E V v a n i s h a t i n f i n i t y . Then on B(X,E):
THEOREM 5.30
(1)
K
(2)
e v e r y a - b o u n d e d s u b s e t i s wV-bounded
5
wv
5
a:
wV-bounded s u b s e t
i s K-bounded;
and e v e r y
WEiGHTED APPROXIMATION
94
(3) wv and
K
a g r e e o n n-bounded
subsets.
+
PROOF Since V < K (X), (1) follows from Proposition 5.28, while (2) is an immediate consequence of (1). To prove (3) let S be a a-bounded subset of B(X,E) and let f E B(X,E) be in the K-ClOSUre of S . Let p be a continuous seminorm on E, v E V and E > 0. Let M > 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
<
VIP -
1 If-ql I
K,P
< ~ / 2+ (M < E/2
+ E/2
Hence f is in the y,-closure
+
.I
VI I
I +
1 If-91'
P
'
. I v ! X-K
f I I 1 . 1 Iv'!X-K
P
= E.
of S . 0.E.D.
PROPOSITION 5.31 L e t U b e a d i r e c t e d s e t o f w e i g h t s o n X and Q, h e a f i l t e r o v e r FUm(X,E). A s u f f i c i e n t c o n d i t i o n f o r 0 t o h e c o n v e r g e n t i s t h a t 0 b e a Cauchy f i l t e r w h i c h c o n v e r g e s w i s e . I f U > 0,
point-
t h i s condition i s also necessary.
PROOF When U > 0 the condition is obviously necessary, since the topology of pointwise converqence is then weaker than wu. Conversely, let 0 be a Cauchy filter over the space FUm(X,E) which converges pointwise to a function fo. Let N be a closed neighborhood of the origin in E and u E U. There exists a set H E Q, such that u(x) (f (x) - g(x) j E N for all f and g in H and x E X. For any point x E X, we have then u(x) (fo(x)-q(x) 1 E N for all g E H, since 0 converges pointwise to f0 and N is closed. Therefore fo E FUm(X,E) and it is the limit of Q, in the space FUm (X,E) THEOREM 5.32
. If E i s c o m p l e t e and U > 0 , t h e n FUm(XrE) i s com-
plete.
PROOF Let Q, be a Cauchy filter over the space FUJX,E). By Proposition 5.31, it suffices to prove that Q, converges pointwise. Given x E X, let u € U be such that u(x) > 0. Given E > 0
95
WE I G H T E D A P P R O X I M A T I O N
and p a continuous seminorm on E, there exists H E cb such that u(t)p(f(t) - q(t)) < EU(X), for all f , q E H and t E X. In particular, p(f(x) - q(x)) < E for all f,a E H. Therefore @ ( X I is a Cauchy filter and thus convercres in the space E, since we have assumed E to be complete. COROLLARY 5.33 If E i s c o m p l e t e t h e n t h e s p a c e s B(X,E) and Bo(X,E) b o t h e q u i p p e d w i t h t h e t o p o l o g y of u n i f o r m c o n v e r g e n c e a r e cornp l e t e .
When E is complete and V > 0 on X, the space FVm(X,E) is complete and the Nachbin space CVm(XrE) is therefore complete if and only if it is closed in FVa(X,E). PROPOSITION 5.34
I f U and V a r e t w o d i r e c t e d s e t s of
weights
on X b > i t h U < V, t h e n CUm(XrE) c l o s e d i n FUm(XrE) i m p l i e s t h a t
CVm(XrE) i s c l o s e d i n FVm(XrE). PROOF Let f E FVm(X,E) belonq to the closure of CVm(XrE) in FVm(X,E). Then f is the limit in FVm(X,E) of a filter Q in CVm(X,E). From Proposition 5.28, f is also the limit of Q inthe weaker topoloqy wu. Since CVm(XrE) c CUm(X,E) and CUm(XrE) is closed in FUm(X,E) , it follows that f E CUm(X,E) ,i.e.,fEC(X,E). Hence, f E CVm(XrE), and CVm(XrE) is closed in FVm(XrE).
THEOREM 5.35
S u p p o s e t h a t E i s c o m p l e t e , and U and V a r e
d i r e c t e d s e t s o f w e i g h t s on X w i t h U < V. T h e n , i f
two
V > 0 on X CVm(XrE)
and CUm(XrE) i s c l o s e d i n FUm(X,E) , t h e N a - h h i n s p a c e i s complete.
PROOF By Theorem 5.32 the space FVm(XrE) is complete, since V >. 0 on X. By Proposition 5.34 CVm(XrE) is closed in FVm(XrE), since CUm(XrE) is closed in FUm(X,E) by the hypothesis made.
If X i s a l o c a l l y c o m p a c t Hau s do rf f topos p a c e and E i s c o m p l e t e , Cb(X,E) e q u i p p e d w i t h t h e s t r i c t
COROLLARY 5.36 ( B u c k ) logy i s complete.
PROOF The strict topoloqy on Cb(X,E) is obtained by taking + + V = Co(X). Now V > 0 on X, and if we take U = K ( X ) , then CUm(X,E) is C(X,E), which is closed in F(X,E) equipped with the
\.I E I GHT E D A P P R O X I M A T I 0 N
96
compact open topology (Bourbaki, Topoloqie G&&rale, 10, 5 1). Obviously K+(X) C C:(X). THOEREM 5.37
Chapitre
S u p p o s e t h a t E i s c o m p l e t e and U and V a r e
two
< V. If V > 0 on X and t h e d i r e c t e d s e t s o f w e i g h t s on X w i t h U i s quasi-complete, then the Nachbin
N a c h b i n s p a c e CUm(X,E)
s p a c e CVm(X,E) i s q u a s i - c o m p l e t e .
PROOF Let A C CVm(XIE) be a closed and bounded subset: Let @ be a Cauchy filter in A. By Theorem 5.32, the space FVm(X,E) is complete. Hence, there exists f E FVm(XIE), such that f is the of limit of 6 in wv. By Proposition 5.28, f is also the limit @ in the weaker topoloqy LL+,. Since A is wU-bounded,f belongs to CUm(XIE), because CUm(XIE) is quasi-complete by hyp0thesis.Hf E C(X,E) , i.e., € E CVm(XIE). The set A is closed and therefore f € A, i.e., A is complete.
5 5
DUAL SPACES OF NACHBIN SPACES
Throughout this paragraph X will be a locally compact Hausdorff space. In this case, for any directed set of weights V on X I the space K(X,E) is densely contained in the Nachbin space CVm(X,E). In fact, even K(X) 8 E is densely contained in CVm(X,E). Let E; denote the topological dual of E endowedwith the weak *-topology a(E',E). An Ei-valued bounded Radon measure u on X is by definition a continuous linear mappinq u from K(X) into EL, when K(X) is endowed with the topoloqy of uniform convergence on X I given by the sup-norm 1 \ @ ) I m = sup{l@(x)l; x EX} for $I E K(X). Any continuous linear functional L on C(X,E) defines an El-valued bounded Radon measure u on X I if we define W u(@) for each @ E K(X) by the formula = L ( 6 8 y)
(1)
for all y E E. Conversely, followinq A. Grothendieck, Produits tensoriels topoloqiques et espaces nucleaires , Memoirs Amer. Math. S O C . NO 16 (1955), an Ei-valued bounded Randon measure u on X is called i n t e g r a l if the linear form L definedover K(X) 8 E by
I.!
E I GH T E D L(C
:I.
97
AP P R 0 X I MAT I 0 N
(8 y i )
>: ' . Y i , U ( Q i )
=
>
(2)
i s c o n t i n u o u s i n t h e t o p o l o g y i n d u c e d by C ( X , E ) , i n w h i c h case 0
it can be uniquely continuously extended t o C o ( X , E ) ,
K(X)
@ E i s dense i n C
0
(X,E).
I n order t o characterize the dual
o f C ( X , E ) as a s e t o f E l - v a l u e d bounded 0
since
W
Radon m e a s u r e s
w e h a v e f o f i n d n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s measures t o be i n t e g r a l . I f L E C o ( X , E ) '
for
such
to
l e t us r e t u r n
the
( 1 ) . The transpose
E l - v a l u e d b o u n d e d Radon m e a s u r e u d e f i n e d y by W
o f u i s t h e n a l i n e a r map f r o m E i n t o M b ( X ) ,
u'
on X ,
t h e s p a c e o f all y F E t h e r e cor-
b o u n d e d Radon m e a s u r e s o n X . H e n c e , . f o r e v e r y responds a unique r e g u l a r Borel measure < u ' ( y ), x B > ,
II such t h a t 1) ( B ) = Y Y f o r a l l B o r e l s u b s e t s B of X . S i n c e L i s c o n t i n u o u s
t h e r e e x i s t s a c o n t i n u o u s seminorm p on E and a c o n s t a n t such t h a t
IL(f)
1 5
I IP
kl If
/ < y , u ( g )>
I
=
for all f
IL($ 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. ) :<
i k, I , we h a v e
An E i - v a l u e d b o u n d e d Radon m e a s u r e u o n X t h e corresponding set function
IJ
such t h a t
s a t i s f i e s ( 3 ) f o r sonr c o n t i n u a s
WE I G H T E D A P P R O X I M A T I O N
98
seminorm p on E a n d some c o n s t a n t k > 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
for all @ E K(X)
.
0
u> =
<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
~
,
WEIGHTED APPROXIMATION
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
<
J: l J ( I J J i ) P ( X i ) = 1 1 ( C
=
lJ(p
0
f)
=
I I f 1 ll.
ICiP(Xi))
WE I G H T E D A P P R O X I M A T I O N
101
Hence T i s b o u n d e d o n a d e n s e s u b s p a c e o f L1 ( X , i i , E ) P and w e can e x t e n d i t c o n t i n u o u s l y t o L1(X,li,E ) and s t i l l have P IT(f)
for all f
E L1(X,\~,E
P
I 5 IIf1
I1
1. In particular, i f 8 xi)
f = v(C Qi
E
v(K(X) 8 E )
then
ai
ILO:
8xi)I
': I l l 1 1 I - I I V
= /T(f)I
(c
p
5
(bi 8
I l f l [ l = u ( v p o (C @ i s x i ) ) <
I ILu
Xi)
I
=
1-1
11 @
[iJI
8
Iv,p-
Xil
T h e r e f o r e L i s c o n t i n u o u s o n K ( X ) €3 E w i t h t h e t o p o l o g y induced b y CV,, ( X , E ) a n d c a n be u n i q u e l y c o n t i n u o u s l y e x t e n d e d
to
CVm(X,E).
C o n v e r s e l y , l e t L be a 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 D e f i n e t o n K ( X ) by
on C V _ ( X , E ) .
< x , t ( @ ) >= L ( @ 8 x ) for all x
E
E . O b v i o u s l y , t ( @E) E * .
S i n c e L i s c o n t i n u o u s *re
e x i s t s v t V and a c o n t i n u o u s seminorm p o n E s u c h t h a t IL(f)
I 5 I I f 1 Iv,p
f o r a l l f F CVm(X,E). Therefore, i f f = @ 8 x then I < x , t ( @ ) > l= I L ( @ 8 x ) /
5 ll@
T h i s p r o v e s t h a t t ( @E) E ' .
€3 x I I v , p <
I
Iv
@ I lm.P(X)
On t h e o t h e r h a n d , i f K C X
is
compact s u b s e t and @ E K ( X ) h a s s u p p o r t c o n t a i n e d i n K ,
[
[ V
$ 1 l m 51 IvI I K . I 1 @ 1 , I
Radon m e a s u r e o n X . function l/v. px
ux
a
then
w h i c h s h o w s t h a t t is a n E L - v a l u e d
L e t w be t h e p o s i t i v e e x t e n d e d r e a l - v a l u e d
T h e n w i s lower s e m i c o n t i n u o u s . L e t x E E ,
and
x 0 t t h e c o r r e s p o n d i n g n u m e r i c a l Eadon n e a s u r e o n X . L e t - u l - p 2 + i ( p 3 - p 4 ) be t h e minimal decomposition o f iix =
w i t h pi
0 < $
0
5
(i = 1,2,3,4).
w , w e have
pi(@)
5
sup
Notice t h a t f o r any 9 E K ( X ) ,
1 [ v $ 1 I m 2 1, { I ( X o t ) ($11;
and 0
5
I
/ $ 8 xi
5 a,
Iv,p.zp ( x ) . S i n c e
$ E K(X)
it follows t h a t p i ( @ ) < p ( x ) , f o r any 0 < @ < w , as $ 5 @ plies then @ < w , and t h e r e f o r e 1 ( X o t) ($1 I < I I $ 8 XI I v , p p ( x ) . Hence
with
in-
5
WE I G H T E D A P P R O X I MAT I 0N
102
5 4 'w,
sup {rii($); 0
=
rl;(w)
and w f L1(X,ui),
i = 1,2,3,4.
4
p(x),
K(X)I(
Thus w E L 1 ( X , x o t )
for all x EE,
i . e . , w is i n t e g r a b l e with r e s p e c t t o t. Define u($)
9
k.
for, each
by
K(X)
=
\'X,U(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 $, x
'1,
t>l =
11
I IQl
<
i t f o l l o w s t h a t u ( @ ) t_ E ' ,
w 4 d(x
t)I
11
I,,.J8 P ( X ) ,
a n d i n f a c t u i s a n EWl - v a l u e d
Moreover t h e i n e q u a l i t y
measure on X.
5 J8
IPX,U((P)>/
P(X) I
Id I w
implies t h a t t h e corresponding set function B (3) with k =
Radon
J8, i . e . , u
+
u(B)
satisfies
t: % ( X , E ' ) .
I t follows t h a t
x for all
x
t K(X),
ip
t> =
(I
E
O,X o
E, i.e.,
have s e e n t h a t @ i s one-to-one
t = v u. D e f i n e O ( L ) = v u . and o n t o VMb(X,E')
,
W e
and t h a t
i s a s e t o f E l - v a l u e d Radon m e a s u r e s on X .
VMb(X,E')
Notice that
W
i s c l o s e d u n d e r m u l t i p l i c a t i o n b y s c a l a r s . Let v ' u ' , By t h e f i r s t p a r t o f o u r p r o o f , t h e r e e x i s t
VMb(X,E')
v"u"
t
L',L"
i n CVa,(X,E)'
L e t O(L'
U>
VMb(X,E').
+
L")
=
such t h a t O(L') = v ' u '
a n d O(L")
= v"~".
v u . T h e n f o r e v e r y I$ E K ( X ) a n d x E E ,
we
have cx,v u($)> = (L'
L")
(Q 8 x ) = L ' ( $ 8 x ) + L"
<x,v' u' (()I>
=
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<(X), it Since lg(x)@(x), -
WE I GH T E 0 A P P R 0X I M A T I 0N
follows t h a t and
1 I
where
e E Mb(X,E').
F d e n o t e s t h e s u p p o r t of
Let and T =
lie
lip
=
q l u l p and s i n c e u
Ie'P
on t h e s u p p o r t o f
e
and
P
its characteris-
XF
.
S i m i l a r l y i f !le 1 1 = 1, t h e n P P 11-11 So w e may a s s u m e 0
.
/ I p-1e a n d T T = VT . C l e a r y S b e l o n g s -1 (1 - 1 1 e l i p ) (L - R ) , where I/
u =
1 ~ 1
0, then
o n t h e s u p p o r t of
g = 0
g = 1
-
lloreover,
h a v e t h e same s u p p o r t :
tic function. I f
and
105
Do , while VrP v e , i m p l i e s T E W'.
W '
to
R =
On
t h e o t h e r hand, w e have
I1
T
Ilp
=
(1 -
=
(1 -
= 1.
Hence
T
E D
0
P
11
1.e. L = (1-
and c o n s e q u e n t l y
e
1; P ) T + 1 1
e
1 1 PS .
T F :D
rP
Since
L = T = S. From t h i s i t f o l l o w s t h a t
therefore
q
L
. Eut is
an
P extreme
point,
, , ell v u = ve = g v u ,
i s c o n s t a n t o n t h e s u p p o r t of
d e f i n i t i o n t h e s u p p o r t of L .
b
vu= (l-liel~) v ~ + l ~ VU, el
P
vu,
which
is
and by
IJE I GHT E D A P P R O X I M A T I O N
I06
PROOF:
[51].
The p r o o f i s a n a l o c j o u s t o t h a t of Theorem Notice t h a t i f
s u p p o r t of
L
L = vu, where
u
is chosen
is equal to t h e support of
3.2,
so
Prolla
that
the
u , then L defines a
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 CVm(K,E) f o r a n y c l o s e d sub-
set
K of
X
w h i c h c o n t a i n s t h e s u p p o r t of
u.
W E I G H T E D A P P R 0X I M A T I 0 N
107
APPEND IX
FUNDAMENTAL WE IGHTS The proof of Theorem 5.19 relies on showinq that a certain function y on IF? is a fundamental weiqht (see Definition 5.5) in the sense of S. Bernstein. This is done by an appeal to the Denjoy-Carleman Theorem on quasi-analytic classes. This is the reason for the name "quasi-analytic criterion" qiven to Theorem 5.19. Conversely, weiqhted approximation techniques can be used to solve the problem of characterizinq quasi-analytic clas-
ses of functions. In this appendix we present a very simple proof, due to G. Zapata, of Merqelyan's theorem which characterizes the fundamental weights on the real line. This result was then used by Zapata to show that Hadamard's problem on the characterization of quasi-analytic classes of functions is equivalent to S. Bernstein's problem on the characterization of fundamental weiqhts. Usinq this equivalence, one qets a solution of Hadamard's problem by the sole means of weiqhted approximation theory. As pointed out by Zapata, this seems to be an interesting approach to the qeneralized quasi-analytic problem of Mandelbrojt. (See S. Mandelbrojt , " S k i e s adhgrentes, rggularisation des suites, applications", Paris, Gauthier-VillarsIl952).
5
1
MERGELYAN'S THEOREM
Let w be a w e i g h t on I R , i.e. an upper semicontinuous positive function defined on IR. Recall from Definition 5.lthat the Nachbin space Cwm(IR) is the vector space of all continuous and complex valued functions f on IR such that wf vanishes at infinity. The topology of Cwm(lR) is determined by the seminorm f * sup {w(x)lf(x)I; x
E
I R ) = 1lfll,
weight w on IR is said to be r a p i d 2 y d e c r e a s i n g i n f i n i t y when the set of complex valued polynomials A
p(lR)
at
on
WE I
108
GH T E D A P P R O X I M A T I O N
IR is contained in C w m ( l R ) . (See Definition 5.5). The weiqht w is called f u n d a m e n t a l if (IF?) is densely contained in C W ~ ( I R ) , i.e. for every E > 0 and every f E C W m ( l R ) there exists a polynomial p E (IR) such that w(x)
for all x
If
(XI
- p(x) [
<
E
IR. The problem of determining necessary and conditions for to be fundamental was first stated E
sufficient hy S. Bernstein in 1924.(See S. Bernstein, "Le problgme de l'approximation des fonctions continues sur tout l'axe rgel et l'une de ses applications", Bulletin de la Socigtg Mathgmatique de F'rance, tome 52 (1924), 399-410). For weights w and v on IR , if w < v and v is a fundamental weight, then is a fundamental weight too. (See Proposition 1, 5 24, Nachbin [43]). Also, if the weight w is bounded on IR, then REMARK 1
C0(IR)
c
CWm(lR).
In this case C o ( l R ) is dense in C w m ( I R ) , because K ( I R and R is locally compact. (See Proposition 2, 5 22, [43 1,
c
Co(IR)
Nachhin
.
Notice also that the inclusion map is continuous,when Co(IR 1 carries the uniform topoloqy 0 . Indeed, ,
IlfIIw=
IlwflI =
IIWll.IlfII
for all f E C o ( R 1 , where we have set
I Ih!I for all h: IR REMARK 2
+
= sup Clh(x)!; x E IR
1
4: which is bounded on IR
.
Let D = { z E C; Im(z) # O}. For z
E
D, let
1
z (t) = t-z for all t E I R . Let A = C b z , qz] be the complex algebra ated by the functions gz and 9,. Notice that A C C o ( l R ) .
claim that A is dense in C o ( l R ) .
Indeed, the
alaebra
qenerWe A is
109
WEIGHTED APPROXIMATION
separating, everywhere different from zero on IR, and is selfadjoint. Since Co( IR) C Cb(lR) , we may apply Theorem 5 . 2 0 (or Corollary 1.9 to a one-point compactification of I R ) t o conclude
.
that A is dense in Co(lR) As a corollary, if u is bounded,then A is dense in C w m ( m ) , by Remark 1. REMARK 3:
Let
a,b o(t)
for all
=
E
El, a # 0, and let
at + b
t F l9.. For any f : R T(f) = f o o
Also,
the restriction of T
a,
let
.
to Cwm(m) in an isometry of Cwm(R)
. the weight onto C(w o C J ) ~ ( R )Hence, only if w o o is fundamental.
DEFINITION 2.
w
is fundamental if,and
L e t w b e a w c i g h t o n IR, F o r e v e r y t E I R ,
In 1954, using the function
!I
w*
let
introduced by himself,
Mergelyan was able to give a necessary and sufficient condition for w
to be fundamental, thus solving Bernstein's problem
for
the real line. His proof, as well as a survey of earlier results on Bernstein's problem, is available in English in a
translation
of R. P. Boas, Jr. (See S. N. Mergelyan, "Weighted approximation by polynomials", American !lethematical Society Translations,Sesies 2, v o l u m 1 0 (1958), p p - 5 9 - 1 0 6 ) .
WE I G H T E D A P P R O X I M A T 1 O N
110
(a) Nii'i;L:ity.
PROOF:
Let
(0
b e a f u n d a m e n t a l w e i g h t o n IR.
z F D, and l e t
i
IR i s b o u n d e d ,
Co (IR) C Cum (IR) , by R m r k 1. Therefore g2
> 0 be given.
Let
P €F(JR) b e s u c h t h a t
Let
K = i n f { (1
+
,
1x1 )
Let
S i n c e any f u n d amen tal w e i g h t on
: x
gz(x)i
€
IR 1
.
F- Cwm(IR).
Then t h e p o l y n o -
m i a l d e f i n e d by K
Q(x) = for a l l
x
(1
-
(X
- ~ ) P ( x ) )
IR, is such t h a t
E
,I 5
iIlil*Q
1. I n d e e d , f o r a n y x
E
IR,
w e have
P41,19c(z) 2 ~Q(z)= K
Therefore
Mb)*( 2 )
+
=
/
.
F.
Since
E
> 0 was arbitrary,
.
'U
(b) : ; u y f I ' P i i . n i . y . (see Z a p a t a [ 6 8 ; ) , w h i c h
W e w i l l p r e s e n t Z a p a t a ' s proof
r e s t s o n t h e f o l l o w i n g t w o lemmas.
LEMMA 1. :oirii
z
PROOF:
I E
/>
v
/
D. 7:) ri
Let
(7
v
IJ
z g ! l i o n IR
a s e m i - n o r m on
W .
P t ?(IR),
11
,I
Assume t h a t
= ] I v
P # 0, such t h a t
v has a f i n i t e set as i t s support,
now t h a t say m
.
1)
!
W e have
>
+
m
f')P
1 since
m
=
11 =
whence
0
T(R)s u c h t h a t \'P . Then 1 - 1 1 i s
P F \iP'l
i s n o t a norm. Then t h e r e
vP
i s a norm. Assume t h a t m
MV(z) =
it'iut
be t h e v e c t o r s p a c e o f a l l
W
is bounded. F o r a n y s u c h P , p u t 1 ' P exists
,eui'ii
i . ; r n p i d i y d+(*r2i.aL;int/ a 1 i n f Y n 7 ' i y .
IJ
0. This implies W =
5)(IR).
that Assume
h a s f i n i t e dimension,
would imply
MV (z)= O . Notice
I4 E I GH T E D A P P R O X I FIAT I 0 N
111
P F iJ and n=degree ( P ) imply tn E w, whence tm- 1 11, ..., 1 is a basis for W and the mapping a i : W that
by
Qi(ao +
... +
Also, the set
am-l tm-’
T dis
=
the
set
C given
a . is continuous for i=O ,...,m-1.
closed and bounded, hence
our assumption. From che compactness of a positive constant C such that for all
F)) is compact
Qi(FV), there P
E
by
exists
p,) ,
tm- 1 , we have ( a i ]
...
+ a C for all i=O, ...,m-1. m- 1 m- 1 Thus P(z)l C C zIi for all P E?,: that 1s M V ( z ) < + m , i=O contradicting the hypothesis. Hence W has infinite dimension. Since tn F implies tm E VI for all o 5 m 5 n, it follows
P = a0 +
that
W
.
To finish the proof, notice that v is rapidly decreasing at infinity if, and only if, V P is bounded on IR for every P Eg(IR). (See Definition 1, 5 24, Nachbin [ 4 3 ] ) .
PROOF:
= ?(R)
Let
ti =
‘m.Since
it is er;ough to prove that
W is a vector subspace of Cwm(B), n qz(qz) m F W for all n,m E IN. First,
we will prove by induction that
In fact, for n = 0 this is clear. Assume that it is true for some n and let P F: p(IR).Since Q = gz(P - P ( z ) ) E (IR)and n+ 1 n qz ( P - P ( z ) = qz Q , we have
Note that if g is a continuous and bounded complex valued function on IR, then the mapping f gf from C w ~ _ ( i R ) into itself is continuous. So
Since
g:+’
E
: IJ = g: 9
m), the above remark and
the induction
WEIGHTED APPROXIMATION
112
n+ 1 hypothesis imply g:+l E W. Then, from g"+'P = g (P - P(z)) + Z n+l and ( 3 ) , we get g:+l + P(z)gz P E W. Thus (2) is proved. Nom EW tice that h E W implies h E W whence, from ( 2 ) , (q2Im = gz ~
for all
m
E N
n - m and then, from (4), g 2 (gz)
,
E
:g e(lR)c
for all
n,m E N . Using (2) and the fact that W n - m E W for all n,m E IN. conclude that g2(gz)
z E
is closed, we
Proof of S u f f i c i ~ n c y :We have 14w*(z) = + for some D. From Lemna 1, it follows that (1 + t2)y(IR)c Cw:(W) whence
y(IR)C Cwm(lR).
Let
Q
53,.
E
p =- 9 2
(Q
Q(2) Then
EF(IR)and
P
for all g2
t E
E
IR.
3 1 -.
be such that
Letting
1
w*(t)
Q(z)
Q(z)
# 0, and
put
- Q(2)). Hence,w(t)/gz(t)-Pc+)I 5
g 2 - P = g,Q/Q(z).
+ I t ; )1 gZ(t)
- ;Q(z)l-'(l
SO
v)
f
~
m,
g(t)
5
constant.
IQ(2)l-l
we have that /lo(g2- P) I
From Lemma 2, it follows that CI: [ g2
,g2 J
0.
C R S .
Since w is bounded, by Remark 2 we have that CI: [ gz , g2 is - --I @?) is dense in Cwm(IR). dense in (IR), and so we conclude that
9
REMARK 4: From the proof of the necessity of the condition in Theorem 1, it follows that MU,(z) = + m for every z f D, whenever w is a fundamental weight.
5
2.
FUNDN4ENTAL WEIGHTS AND QUASI-ANALYTIC CLASSES OF FUNCTIONS
In the following, M will denote a sequence (M,) , n E N, of positive real numbers. We will denote by C(M) the class of all complex valued indefinitely d fferentiable functions f on IR such that there exist positive constants C and c (depending on f ) for which If(n (t)1 The class
C(M
5 c cn Mn
for all
is called q u o s i - n n a l y
t i c
t
E
R, n
E
N
.
if the following condition
WEIGHTED APPROXIMATION holds:
f
E
s
C(r4) vanishes identically, if there exists f(n) (s) = 0
such that
113
for all
n t: N
IR
E
.
Hadamard's problem consists in finding necessary and sufficient conditions on a given sequence M = (!.In) in order C (M)
that the class REMARK 5:
be quasi-analytic.
Let IJ be as in Remark 3. Then C(M) o We write y
REMARK 6:
M
(1
C C(14).
for the weight on IR given by
We have c j ) ( I R )
c
C(Y!,~), (IR). Further, either Y M
an upper- semicontinuous function with compact support lim sup n'l,P
+
n
m )
or it is a continuous function that
is
(when never
vanishes. In the first case, y M is fundamental by the Wierstrass polynomiai approximation theorem.
PROOF:
The necessity of the condition follows from the
of Lemma 2, 5 2 9 , Nachbin [431/
proof
.
Let us prove the sufficiency. Assume that
the class
C (!,I) is not quasi - analytic. From Remark 5 , there exists f E C(M) such that
(1)
f(n) ( 6 )
=
0
f o r all
n €IN
and
fI[O,+
")+O.
Let U = tz
E C , Re(z)
01,
V = t z E C, Re(z) > 1 }
and put
Integrating by parts and using induction, we get, in
(1)I
view
of
WEIGHTED APPROXIMATION
114
Z"F(Z)
(2)
=
i)
(-
(t)e-iz dt for all
Sirice
f
E C(I1)
,
z
U, n
E
N .
E
there exist constants C and c such
that
1
(3)
c
f(n) (t)l 5
for all
cn M~
t
E
m ,
z
E
n
N.
E
From this and ( 2 1 , it follows that (4)
,
F(z)
i
c
n -n c PIn/ z
for all
V,
n
E
N .
Since F is a holonor hie function not identically zero,we have for some
F(a) # 0
Let
h~
E
V.
be the weight given by
ti) / F ( 1 +
~ ( t= ) (1 + Fix
a
P
E
yw* and
it) 1 1 ~
+
it1-l
for all
t
E
IR.
let
G ( z ) = P(i
- iz)F(z)z-1
for all
z
E
U.
Then G is holomorphic in V and continuous on V.Letting t € & since l G ( 1 + it), = /w*(t)P(t)I , we have that IG1 5 1 on 2V . A l s o , G is bounded on V from (4). Since because P F a F V, it follows from the maximum modulus theorem that -1 i G ( n ) I 5 1, whence ;P(i - i u i < jF(a)l . Notice that, P being arbitrary, Ilw,(i - in) + m . Since i - i a E D,Remark
yo*
4 implies that that
w
for a l l
,
t
F
IR
is not fundamental. Furthermore, (4) implies
n
E N ;
whence
W E I G H T E D APPROXIMATION
115
From this and Remarks 1 and 3 , we conclude that damental since COROLLARY 1:
is not fundamental.
w
m
Lct
@
~ J C
a cornplcx v a i u c > 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
WEIGHTED APPROXIMATION
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
WEIGHTED APPROXIMATION
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
-
...,
...,
-
...
{ I If wil I ; i = 1, 2 , n l and wi - wx €or x = xi, i = 1, 2 , . .n. Consider t h e corresponding f u n c t i o n s a l , i n C b ( X ; I R ) . Let w = a l W l + + a n + a wn E W. W e claim t h a t I If i < C + E . Indeed, given M = max
...,
.
...
-
n
x E X , w e have
x
Now, i f
-
wi x E
(XI
ux
e
t h e n ai ( x ) < 6 , and t h e r e f o r e
Ux
i 6
I
II
f
- wil
I 5
6
M.
ai ( x )
On t h e o t h e r
then i
ai(x) (c Hence,
c + E/2)
+
~ / 2 ) .
< c
+
E
II
hand,
f
(x)
-
if
C,(X;E) T h i s shows t h a t d c c
+
123
W I T H T H E UNIFORM TOPOLOGY
E.
Since
E >
d <
0 was arbitrary,
c,
and t h e p r o o f i s s e e n t o b e c o m p l e t e . W e s h a l l n e x t consider t h e q u e s t i o n o f extreme f u n c -
t i o n a l s . Our a i m i s t o p r o v e t h e t h e o r e m o f Brosowski, J k u t s c h and Morris (Lemma 3.3,[10]) g e n e r a l i z i n g i t t o vector fibrat i o n s s a t i s f y i n g (1), ( 2 ) and ( 3 ) o f Remark 6.13, i.e. to Nachbin s p a c e s of a s p e c i a l k i n d . L e t t h e n L be a Nachbin s p a c e o f cross-sections s a t i s f v i n g ( 1 1 , ( 2 ) and ( 3 ) o f Remark 6.13. L e t B ' be the b all u n i t b a l l o f L ' , and f o r e a c h x E X, l e t B i be t h e u n i t 6x:
o f E L . The mapping
Ek
6,W
+
(f)
L'
d e f i n e d by
= +(f(x))
mapf o r a l l 4 E EA and f E L i s t h e n a c o n t i n u o u s l i n e a r p i n g of norm 5 1, a n d t h e r e f o r e maps B; i n t o B ' . The map 6 x i s one-to-one i f { f ( x ) ; f E L} = E x . I f L i s e s s e n t i a l (Def i n i t i o n 1.31, by
image o f EA =>
5
1 0 , C h a p t e r 1) t h i s h a p p e n s for a l l x E X.The cSX
JI ( f ) = 0 , Wf
i s c o n t a i n e d i n t h e set
{JI
E L';
f(x) = 0
E L) = Ax.
LEMMA 6.15. L e t L be a Nachbin s p a c e w h i c h i s an e s s e n t i a l K(X) -module, s a t i c r f y i n g (1) - ( 3 ) o f Remark 6.13. The mapp i n g cSX i s a l i n e a r i s o m e t r y o f $ o n t o Ax f o r e a c h x E X . PROOF.
W e saw a l r e a d y t h a t
p i n g of norm
5
1, from
E;
6x is a continuous l i n e a r
into.Ax.
JI E Ax. F o r e a c h and p u t E ~ ( J I (v) ) i s w e l l d e f i n e d on E x , a n d i t i s W e claim t h a t E ~ ( @ )E E i . L e t E > 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$(9 f )
<
I l$l I
UNIFORM TOPOLOGY
I
.(1+ € 1
= IIJ,II.(1 + el
~ ~ ( $E 1 E;
Hence
Therefore
I -<
I
o t h e r ? and of norm
and I ~ 1. S i n c e
5
fl I I If(x) I I
< 1191 1 - 1 I9
E ~ ( $ I) I 5 I 6 x and
1, w e see t h a t
hX
*
IIVII I for
a l l J, E Ax. are i n v e r s e s o f each 6 x and e x are l i n e a r
isometries. LEMMA 6.16. Let L be an e s s e n t i a l Nachbin s p a c e s a t i s f y i n g ( 3 ) o f Remark 6.13. L e t Q = U { 6 x ( B ' x ) ; x E XI. c o n d i t i o n s (1)
-
(a)
Then
Q is weak*-closed.
( b ) G(Q)= B ' .
PROOF.
i n Q. E B;
.
( a ) Suppose J, E L' is t h e w e a k * - l i m i t o f a n e t t e a } Each JI, i s o f t h e form $; 6 x (+,I, where xa E X and of Y = X u {w} be t h e one p o h t compactification Let
a W e may assume { x a } c o n v e r g e s t o some p o i n t x E X I w e p r o c e e d as i n t h e p r o o f o f Lemma 1.33
X.
1) to show t h a t
JI E Q . I f
x E Y.
(5 lot
x = w t we have f o r any
If
Chapter
f E L
l$(f
because
x *
-< I If ( x ) I 1
l i m sup
1 - 0
vanishes
Hence J, = 0 E Q. ( b ) The p r o o f of p a r t ( b ) of Lemma 1.33
(5
10,Chapter
1) carries o v e r w i t h o u t m o d i f i c a t i o n .
L e t L be an e s s e n t i a l Nachbin s p a c e J which i s a K(X)-module, s a t i s f y i n g c o n d i t i o n s (1) (3) o f Remark 6.13 then E ( B ' ) = U{6,( E ( B 2 ) ; x E X t Ex # (0)).
THEOREM 6.17.
-
DO]).
( B r o s o w s k i , Deutsch,. Morri8 L e t X be a non-zero a l o c a l l y compact H a u s d o r f f space and l e t E be and B i the normed s p a c e . L e t B' be t h e u n i t b a l l o f Co(X;E) u n i t b a l l of E ' . T ' h e n E(B') = U {6,( E ( B g E ) ) ; x E X}
COROLLARY 6.18.
Co( X;E)
W I T H THE UNIFORM TOPOLOGY
Apply theorem 6.17 t o t h e Nachbin space
PROOF.
L e t X be a locally compact Hnusdorff s p a c e .
COROLLARY 6.19.
L e t B'
be t h e unit baZ2 of
A6x
: Co(X;lX)
M
+
;
x
I.
Then
E X,
A E IK , I X I = 1 1 ,
i s d e f i n e d by
= Af(x)
Ab,(f)
for all
Co(X)
= {Afix
E(B')
where
L = Co(X;E).
f E Co(X;M).
PROOF OF THEOREM 6.17. 440,
pg. 6x($)
f o r some
0
(Bk 1 -
E E
Q.
E(B')
Q
E
By Lemma 5 , Dunford and Schwartz [207, T h e r e f o r e any JI E E ( € 3 ' ) i s of t h e form
BA, x
E X.
JI
Conversely, l e t Ex # { O } .
JI
6 x i s an i s o m e t r y ,
E(BA ) ) for some x E X Q E BA. S i n c e ~ ~ ( = $ Q1
E 6x(
$ 1 f o r some
$ = 6x(
Then
Since
#
,
(Ax A B'), because
E~ i s an i s o m e t r y . We c l a i m t h a t $ E E(B'). A s s u m e , by t h a t $I E E (B') ; -Then = ( 1 $ ~ + $ ~ ) for / 2 some JIl,
E E
,
9,. L e t
5
f(x) = 0,
f E L with
contradiction, $,
1 and 1
E B' and JI, > 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
W I T H T H E S T R I C T TOPOLOGY
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 <:
(4) i f E i s c o m p l e t e , t h e n (Cb(X;E),P) is c o m p l e t e .
(1) Let 6 E Co+(X) and p f E Cb(X;E) we have
PROOF
I !fl where 1141
I
E
cs(E) be qiven.
For
any
5 ! !41 I x
= SUP { $(x)l ; x E XI and
This shows that 8 is smaller than u . On the other hand, qiven a compact subset K c X, choose t# E CA(X such that $(x) = 1 on K. Then 1 I f l IK,p 5 I I f l 4 IP for any f E Cb(X;E) and p E cs (El. Therefore, K is smaller than f3. < u , any uniformly bounded set is stric(2) Since B tly bounded. Conversely if the set S c Cb(X;E) is strictly bounded but were not uniformly bounded, choose fn E S and xnE X such that p(fn(xn)) > 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) <
E.
If x 1 KI then we have $(X)p(f(X)
-
g(X)) < 2M
. E / ~ M=
EI
since both f and g belong to S. Hence I 1 f-g 1 I 5 E , and 6 IP belongs to the (BIS)-closure of T I i.e. g E T and therefore is (K/S)-closed.
g
T
Let {fa} be a net which is B-Cauchy. By (11, (4) {fa} is then K-Cauchy. Since the space C(X;E) is K-complete, whenever X is locally compact and E is copplete (see Bourbaki, Topologie GCnGrale, Chap. X), { f a ) converqes in the topoloqy K + to a mapping f E C(X;E) Let $ E Co(X) be qiven. Let p E cs(E) and E > 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)
<
Cb(X;E)
> 0. Let
Cb(Y;E); $(y)p(h(y))
where 0 = $I o u. We claim that J,
in
E
<
+ Co(Y).
Indeed, let
V6 = {y E Y; @(u(y)) 2 6 ) . If K6 = {t E X; @(t) > 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
W I T H T H E S T R I C T TOPOLOGY
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
W I T H T H E S T R I C T TOPOLOGY
, 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
NONARCHIMEDEAN APPROXIMATION THEORY
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
NONARCHIMEDEAN APPROXIMATION THEORY
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
<
,hese i s - a sequence
{xn}
in
and t h e r e f o r e xn + xn
-
XI
r;lax(d(yo, xn)
,
d(xn
= IYo
XI
Br(x),
t h e n f o r any
-
r
XI)
< a x n , x)
for
n E N
d(xn , X I
s u f f i c i e n t l y l a r g e , as
< r
,
and
d(xn
, yo)
+
0.
Now
yo E B r ( x ) .
T h i s shows t h a t it f o l l o w s t h a t t h e s e t
-
Br(x)
B,(x)
i s c l o s e d . S i m i l a r l y , from (6)
i s open. Re c a llin g t h a t a t o p o -
l o g i c a l space i s s a i d t o be of dimension 0 i f t h e r e i s a
basis
o f open s e t s formed by c l o s e d s e t s , w e have p r o v e d : PROPOSITION 9 . 6 .
sion
Every nonarchimedean v a l u e d f i c l d i s o f dimen-
0 .
COROLLARY 9 . 7 .
Every nonarchimedean v a l u e d f i e Z d i s t o t a l l y dis-
connected. PROOF.
Let
x a n d y b e d i s t i n c t p o i n t s i n a nonarchimedeanval[ 1 . L e t r < d ( x , y ) , A = R r ( x ) and
ued f i e l d ( F , I
B = ( t E F ;
t
f!
B,(x)}
.
156
NONARCHIREDEAN APPROXIMATION THEORY
since
i s c l o s e d , B i s open. T h e r e f o r e A and B are and open; moreover, F = A U B w i t h x E A
Br(x)
d i s j o i n t , non-empty, and
y E B. T h i s shows t h a t
THEOREM 9 . 8 .
0
sion
Let
i f , and o n l y i f , C b ( X ; F ) i s s e p a r a t i n g o v e r
PROOF.
Let
I)
I
be a nonarchimedean v a l u e d f i e l d . L e t
c
U
@ , ( t= ) 1, if
t F U;
@ , ( t= ) 0,
t @ U;
if
f (XI
#
y E X, with
.
x f y
open
x E U , y @ U.
The
C b ( X ; F) s u c h t h a t
C ( X ; F) is s e p a r a t i n g over b There e x i s t s € E C b ( X ; F)
U = { t E X : ( f (x)
Let
2.
and y b e t w o
X
.
with
f (y).
U is clopen, x
8
x
let
and
# @,(y).
C o n v e r s e l y , assume
,
0 ,
by
i s t h e n a n e l e m e n t of @,(XI
x
such t h a t
X,
@udefined
F - characteristic function
t E F,
f o r any
There e x i s t s a clopen s e t , i . e . an
X .
s e t which i s a l s o c l o s e d ,
for all
X,
I).
I
( F ,
b e a H a u s d o r f f s p a c e of d i m e n s i o n
X
distinct points i n
Let
i s of dimen-
be a Hausdorff s p a c e . Then X
X
n o n a r c h i m e d e a n VaZued f i e l d
(F ,
i s t o t a l l y disconnected.
F
U
E
-
f ( t )1
<
I f (x)
-
f ( y ) I}.
Then
-
is a
y $ U.
and
KAPLANSKY'S THEOREM. If
X
i s a compact H a u s d o r f f s p a c e a n d
n o n a r c h i m e d e a n v a l u e d f i e l d w e endow o f u n i f o r m c o n v e r g e n c e on
f
-+
1)
f
)I
(F,]
1)
C ( X ; F) with t h e topology
g i v e n by t h e s u p - n o r m
X ,
= sup
{lf(x)l
;
x E X I .
The f o l l o w i n g s e p a r a t i n g v e r s i o n of t h e Stone-Vkierstrass theorem i s due t o I . Kaplansky THEOREM 9 . 9 .
and l e t
X
Let
(F, I
*
1)
[72].
be a nonarchimedean v a l u e d
bc! a c o m p a c t H a u s d o r f f s p a c e .
L e t A C C ( X : F)
field, be
a
157
NONARCHIMEDEAN APPROXIMATION THEORY
.
u n i t a r y subalgebra which i s s e p a r a t i n g o v e r X
Then A i s u n i -
formly dense i n C ( X : F ) . f E C ( X ; F). S i n c e X
Let
PROOF.
p a c t . Now, f o r a n y open b a l l s
f ( X ) C F i s com-
i s compact,
> 0 , f ( X ) i s c o n t a i n e d i n t h e u n i o n of a l l
E
BE ( f ( t ) ) when ,
t E X ,
i.e.
f ( X ) C U ( B E ( f ( t ) ) ;t E X I . By c o m p a c t n e s s , t h e r e e x i s t s a f i n i t e s e t
, t2,
{tl
...
tn
1
I
such t h a t
f(x) c Let
B1 = B E ( f ( t l ) ) a n d f o r
Then t h e s e t s
i = 2r3,...rn,
Bi a r e a p a i r w i s e d i s j o i n t c l o p e n c o v e r o f f ( X ) .
Therefore, t h e sets-
= f - l (Bi)
U.
1
w i s e d i s j o i n t clopen cover of some put then
i = 1,2,. +k =
+
. .I n
and
7
i = 1,2,.
.., n ,
f o r m a pair-
Let
t E X . Then t E Ui
for
for a l l
j # i . Hence,
we
X .
t $ CJ.
,
if
I t h e c h a r a c t e r i s t i c f u n c t i o n o f Uk , k = 1 , 2 , . . . , n , 'k
+ i ( t )= 1 a n d
implies
u... u ~ ~ ( f ( t ~ ) ) .
aE(f(tl))
0 . (t) = 0
for a l l
3
If ( t ) - f ( t i )I <
n
c
g =
k=l
f(tkMk
.
A
in
E
, where
.
T h e r e f o r e , a l l w e have t o prove i s t h a t t h e c l o s u r e of
t E Ui
Now
If ( t )- g ( t )I <
and t h e n
E
j # i
g belongs t o
C ( X ; F ) . T o show t h i s , i t i s s u f f i c i e n t t o
s belong t o t h e c l o s u r e of A , i.e. t h a t i f U i s a clopen set i n X , then @ u b e l o n g s t o t h e c l o s u r e of A i n C ( X : F ) . L e t 0 < E < 1.
prove t h a t t h e
Fix
+k
x E X,
x
j? U .
For each
t E U,
A i s a s e p a r a t i n g u n i t a r y subalgebra of
gt E A
such t h a t
gt(t) = 0
m a , t h e r e e x i s t s a polynomial i s zero s u c h t h a t compact s e t
and pt:
p t ( l ) = 1 and
g t ( X ) C F.
Then
since
C ( X : e ),
x # t , and there
is
g t ( x ) = 1. Eiy K a p l a n s k y ' s WmF
+
F , whose c o n s t a n t
term
Ipt(s)I 5 1 for a l l s i n the o gt belongs t o A,
- pt ft -
158
NONARCHIMEDEAN APPROXIMATION THEOhY
( t ) = 0 , f t ( x ) = 1, a n d ; I f t 1 1 = s u p { I f t ( y ) 1 ; y E X 1: 1. By t c o n t i n u i t y of f t t h e r e e x i s t s a n o p e n n e i g h b o r h o o d Nt of t such f
that Ift(y)I < exists
for all
E
t l , . ..,tn E U
Consider
such t h a t
fx = f
/ I f x 1; 5 1,
and I f x ( y ) l < A
Since U
U C N
...
f t2
belongs t o
.
y E Nt
.
ft
...
U Nt
n fx(x)=l,
f x E A,
y E U.
Now
hx = 1
1 E A. Noreover, h x ( x ) = 0
too, since
there
n
for all
E
U
tl Then
is compact,
-
fx
and
11 - h x ( y ) 1 < E f o r a l l y E U . By c o n t i n u i t y o f h, t h e r e exi s t s a n o p e n n e i g h b o r h o o d Wx o f x s u c h t h a t I h x ( y ) I < E for a l l y E Wx Since X \ U i s c o m p a c t , t h e r e e x i s t s xl, ... , xm
.
in
X\U
Consider
such t h a t
h = hx
h
1
-
x2
...
-
hx
m
.
Then
h E A . Moreover, i f
..
,m, and / h, (y) I < y E Wx f o r SOLE i = 1 , 2 , . i i / I h x , 'I / < 1 f o r all j = 1 , 2 , ...,m . Hence 3
y I? U
, then
(a)
ih(y)/ <
E
for a l l
i = 1,2,.
.. , m .
W e c l a i m t h a t / 1 - hx ( y )
1
for all k = 1,2,...,m.
-
This i s clear f o r
*
hx ( y ) I < E f o r i * hx ( y ) I < E k
...
k = 1. A s s u m e t h e claim
1 < k = j < m. Then
for
<max(/l-hx,
(y)/
, il-h,
because
-
5
/ h x , (y)I 1+1
hx ( y ) 1
*
...
*
(y)
1
3+1
11
while
y @ U.
On t h e o t h e r h a n d , i f y E U, t h e n 11
all
E,
...
hx ( Y ) / ) j
1. By t h e i n d u c t i o n h y p o t h e s i s , hx ( y ) I <
j
E.
Hence, 11- hx ( y ) *..:h
1
(y)I < E l xj+l
NONARCHIMEDEAN APPROXIMATION THEORY
+ 1.
and t h e c l a i m i s t r u e f o r j In particular,
11
(b)
-
This proves t h e claim.
k = m
for
h(y.)i <
E
,
, we
get
for a l l
y E U.
From ( a ) a n d ( b ) , w e see t h a t belongs t o t h e c l o s u r e of
in
A
159
C ( X ; F)
-
,
h
11
<
E
,i.e.
%J
€ o r any c l o p e n U C X ,
and t h i s e n d s t h e p r o o f . The K a p l a n s k y ' s Lemma r e f e r r e d t o i n t h e a b o v e i s t h e following.
REMARK.
Let
LEMMA 9 . 1 0 .
let
,
x E F
,I
*
1)
be a nonarchimedean valued
x # 0 . G i v e n any compact s u b s e t
a polynomial p(x) = 1
(F
p : F
and
5
for all
1
[ 75
]
such
that
t E K.
F o r a p r o o f see Lemma 1, K a p l a n s k y B e c k e n s t e i n , a n d Bachman
f i e l d and
K C F , there e x i s t s
whose c o n s t a n t t e r m i s z e r o
F
-+
Ip(t)l
proof
[172] ; o r
Narici,
.
The f i r s t a u t h o r t o p r o v e a S t o n e - W e i e r s t r a s s T h e o -
r e m f o r n o n a r c h i m e d e a n v a l u e d f i e l d s w a s Dieudonn6, who [70]
proved
f o r t h e f i e l d o f p - a d i c numbers.Kaplansky's
such a r e s u l t i n Theorem w a s e x t e n d e d t o t h e g e n e r a l case o f a r b i t r a r y K r u l l uations, i.e. not necessarily r e a l
- valued
Val-
valuations, by Chernoff,
R a s a l a a n d W a t e r h o u s e . They p r o v e d t h e f o l l o w i n g . THEOREM 9 . 1 1 .
tion, except
Let
F
be a f i e l d w i t h an a r b i t r a r y K r u l l v a l u a -
C w i t h i t s usual absolute value. Let
be a u n i t a r y s u b a l g e b r a which i s s e p a r a t i n g o v e r uniformly dense i n
X .
A C C(X;F)
Then A
is
C(X ; F).
F o r a p r o o f see C h e r n o f f , R a s a l a a n d W a t e r h o u s e [ 6 9 ] . S i n c e w e s h a l l n o t t r e a t t h e c a s e of n o t n e c e s s a r i l y real-valued v a l u a t i o n s , w e s h a l l n o t u s e theorem 9 . 1 1 i n t h e s e q u e l . L e t u s d e s c r i b e a q u o t i e n t c o n s t r u c t i o n which p e r m i t s
t o d e r i v e f r o m Theorem 9 . 9 ,
i.e.
f r o m t h e s e p a r a t i n g case,
a
g e n e r a l v e r s i o n o f t h e S t o n e - Weierstrass Theorem, namely a v e r s i o n d e s c r i b i n g t h e c l o s u r e o f a u n i t a r y s u b a l g e b r a of whenever
X
i s a compact Hausdorff s p a c e .
A s i n t h e c a s e of
M - valued f u n c t i o n s , given
C(X;F),
160
NONARCHIMEDEAM APPROXIMATION THEORY
A C C(X ; F) , we denote by X / A the equivalence relation defined on X a follows: if x , y E XI then we say that x y (modulo X / A ) if, and only if, f (x) = f ( v ) for all f E A. Let Y by the quotient topological space of X modulo X / A and let IT be the quotient map of X onto Y ; IT is continuous and for each x E X , y = IT(X) is the equivalence class [x] of x modulo X/A. Hence, for each f E A, there is a unique g : Y -t F such that f(x) = g(T(x)) for all x in X . We claim that g is continuous. Indeed, for every open subset G C F, the set f-’(G) is open in X , and f-’(G) = IT l(g-l(G)). By the definition of the quotient topology of Y , this means that g-l(G) is an open subset of Y . Let us define B c C(Y; F) by setting B={gEC(Y;F);f=go.rr
,fEA).
It follows that B is a subalgebra (resp. a unitary subalgebra) of C ( Y ; F ) whenever A is a subalgebra (resp. a unitary subalgebra) of C(X;F). Notice the important fact that B is separating over Y . By Theorem 9.8, and the fact that %(Y;F) = C(Y,Fk as Y is compact, it follows that Y is of dimension 0 . THEOREM 9.12. L e t X b e 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 ( F , * 1 ) be a n o n a r c h i m e d e a n v a l u e d f i e l d . L e t A C C(X;F) be a u n i t a r y s u b a l g e b r a , and l e t f E C(X;F). T h e n f b e l o n g s to t h e u n i f o r m c l o s u r e of A i n C(X;F) i f , and o n l y i f , f i s c o n s t a n t o n e a c h e q u i u a Z e n c e c l a s s o f X modulo X/A. PROOF. Clearly each f E C(X;F) which belongs to the uniform closure of A is constant on each equivalence class of X modulo X/A. Conversely, let f E C(X;F) be constant on each equivalence class of X modulo X/A. Let Y , IT and B as before. There exists g : Y + F such that f = g O I T . As in the proof that B is contained in C(Y;F) , it is easy to see that g belongs to C(Y;F). Now, since B is a separating unitary subalgebra of C(Y;F), by Theorem 9.9, B is dense in C(Y;F). Therefore g belongs to the uniform closure of B in C(Y;F). Since the mappiq h I+ h o TI is an isometry of C(Y;F) into C(X;F) , it fOUowS
161
NONARCHIMEDEAN APPROXIMATION THEORY
that
belongs t o t h e uniform c l o s u r e of
f
in
A
be u n i t a r y can
The h y p o t h e s i s t h a t t h e a l g e b r a A very annoying
s o m e t i m e s , so l e t u s remove i t .
THEOREM 9 . 1 3 .
Let
X
and
F
C(X;F).
b e a s i n Theorem 9 . 1 2 . L e t P C C ( X ; F )
b e a s u b a l g e b r a , and l e t
f E C(X;F). Then
uniform closure o f
C ( X ; F ) i f , and o n l y i f ,
A
in
be
f
belongs
Co
the
the following
conditions hold: given x
(1)
g E A
,y
x ,y
E X
that
11
f
f E C(X;F)
-
g
11
<
E
g(x) # g ( y ) ; f ( x ) # 0 , there exists g E A
with
be i n t h e uniform c l o s u r e of
f (x) # f (y). Let
with
g(x) = g(y) for
.
f ( x ) # f ( y ) , t h e r e exists
with
g(x) # 0.
such t h a t Let
X
such t h a t
given x E X
(2)
PROOF.
E
g E A.
-
= If ( x )
E
g E A
Let
A .
Let
f ( y ) I > 0. Assume
be such t h a t
Then
a c o n t r a d i c t i o n . This proves Conversely, l e t
(1).A n a l o g o u s l y , o n e p r o v e s ( 2 ) .
f E C(X;F) be a f u n c t i o n
satisfying
c o n d i t i o n s (1) a n d ( 2 ) . CASE I .
There e x i s t s a p o i n t
a l l functions g
in A .
x
in
such t h a t
X
for
By c o n d i t i o n ( 2 ) , w e h a v e f (x) = 0 t o o .
B C C ( X ; F ) b e t h e s u b a l g e b r a g e n e r a t e d by
Let
g(x) = 0
s t a n t s . The e q u i v a l e n c e r e l a t i o n s
X
/
A
A
X/ B
and
and t h e c o n a r e the
-,
a n d by c o n d i t i o n (1), f i s t h e n c o n s t a n t on each equivalence class of
X
modulo
X
/ B. B y Theorem 9 . 1 2 ,
c l o s u r e of t h e u n i t a r y subalgebra given. There e x i s t s
f
belongs t o t h e
B C C(X;F). L e t
g E A and c o n s t a n t
X
E
F
E
uniform > 0
such t h a t
be
162
NONARCHIMEDEAN APPROXIMATION THEORY
for all
t E X.
I
t = x, we obtain
Making
A
I
<
E
.
Since
F
i s nonarchimedean, I f ( t )- g ( t ) l = I f ( t )- g ( t )
for a l l
t E X.
Hence,
form c l o s u r e o f CASE 11.
-
Ilf
I/
g
The a l g e b r a h
<
5
A + XI
E
,
and
b e l o n g s t o the uni-
f
A .
h a s n o common zeros. By P r o p o s i t i o n 2 ,
A
C h e r n o f f , Rasala a n d VJaterhouse a function
-
[69]
v a n i s h i n g nowhere on
,
the algebra
Now
X .
contains
A
l / h belongs t o
-
C(X;F) and i s c o n s t a n t on e a c h e q u i v a l e n c e c l a s s modulo By Theorem 9 . 1 2 ,
X / B l / h belongs t o t h e uniform c l o s u r e B of B i n
C(X;F). Since A
is a B
C(X;F) i s a B - m o d u l e .
- module,
Ti
t h e uniform c l o s u r e
1 = h ( l / h ) E 'fi. T h i s p r o v e s t h a t
Hence
is a u n i t a r y subalgebra of
C(X;F). Since
X/A
and
a r e t h e same e q u i v a l e n c e r e l a t i o n , by c o n d i t i o n (l), f s t a n t on e a c h e q u i v a l e n c e c l a s s modulo
3.
X/'fi
i s con-
By Theorem 9 . 1 2 , f
X/A.
T i , namely
belongs t o t h e uniform c l o s u r e of
5
of A i n
itself.
NORMED SPACES. Let
(F
t o r space over
1)
,I
b e a v a l u e d f i e l d , and l e t E b e a vec-
F .
11
R mapping
DEFINITION 9 . 1 4 .
11
:
E
+
IR i s c a Z Z e d a norm o n
E if
x
/I
2
0,
x
I/
=
o
(1) (2)
11 jI
(3)
llhx
(4)
jI
If
11
11
x + y
11
1
x E E; if, a n d o n ~ yif
for uzz
-
I / x 11, / I 5 IIx II + Ii
=
A
I
i s a norm o v e r E
x = 0;
for uzz
y
,
I/,
A E F,
I
E E;
f o r aZZ x l Y E E .
we say t h a t (E
a normed s p a c e o v e r t h e v a l u e d f i e l d ( F ,
x
*
1).
, 11
11)
is
163
NOPARCHIMEDEAN APPROXIMATION THEORY
DEFINITION 9 . 1 5
ued f i e l d
Let
(Ell\
I).
lje
(F;
x ,y
i f , f o r a22
E
II
(5)
11)
b e a normed s p a c e o v e r t h e vaZ-
11
/I
say t h a t
i s a nonarchimedeannomi
E, w e have
II
x + Y
I/
max(II x
5
I n t h i s case, w e s a y t h a t
II
I
11)-
Y
11)
,1 1
(E
is a nonarchi
,I
m e d e a n n o r m e d s p a c e over t h e v a l u e d f i e l d ( F
*
.
1)
Prom now o n we s h a 2 2 a s s u m e t h a t to.}.
CONVENTION 9 . 1 6 .
a22
-
normed
s p a c e s c o n s i d e r e d a r e not r e d u c e d t o REMARKS 9 . 1 7 . L e t (E
(a)
,I /
*
1; )
b e a n o n a r c h i m e d e a n normed space.
.
E # (0 Hence t h e r e e x ]lx ) ) > 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
NONARCHIMEDEAN APPROXIMATION THEORY
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
NONARCHIMEDEAN APPROXIMATION THEORY
NOW, for each 1 <- i <- n, either else x @ Ki and then
Let M = max
{I(
6M < c +
E
.
IIai - gi
11
< 6.
for all
x
E
wi
11
.. .,n}
i = 1,
;
For each
Let us define
x
E
X
E
w
Ki
and the
gi(x) = 0; or
and choose 6 > 0
i = l,...,n,
X,
Indeed, for any
x
=
such that
there is ai E A such that n C a . w . . Then w E W and i=l 1 1
we have
L e t E , A a n d W b e a s in Theorem 9 . 1 9 . F o r e a c h C(X;E), t h e r e e x i s t s a n e q u i v a Z e n c e c Z a s s S C X moduZo X/A
THEOREM 3.21.
f
E
such t h a t
PROOF.
Let Y and
IT
as before. For each
g
E
W, the
is upper semicontinuous on Y , by Lemma 1, Machado
and
fucntion
Prolla
168
NONARCHIMEDEAN APPROXIMATION THEORY
too, and t h e r e f o r e
is upper semicontinuous on Y supremum o n then
Y .
By Theorem 9 . 1 9 ,
y F Y be t h e p o i n t w h e r e
S = n-l(y).
attains
t h i s supremum i s
d(f,W).
its
Let
i s a t t a i n e d and l e t
d(f;W)
Then
as d e s i r e d . COROLLARY 9.22.
that
A
Let
i s separating over
and o n l y i f
in
for any E
W
b e a s i n T h e o r e m 9 . 1 9 . Assume
Then W i s d e n s e i n
X .
W(x) = { g ( x ) ; g
More g e n e r a l l y , f ( x ) E w(x)
and
E , A
E W}
f E C(X;E),
f o r each
f E
i f J
C(X;E)
i f
for e a c h x E X .
is d e n s e i n E ,
and
only
i f J
x E X.
REMARK 9 . 2 3 . W e s a w i n t h e p r o o f o f Theorem 9 . 1 9 t h a t w e may assume w i t h o u t l o s s of g e n e r a l i t y t h a t A i s u n i t a r y . U s i n g t k i s f a c t , i t i s c l e a r t h a t Theorem 9 . 1 3 w a s n o t n e d e e d h e r e . On t h e o t h e r h a n d , Theorem 9 . 2 0 i m p l i e s Theorem 9 . 1 3 , w i t h o u t n e e d f o r P r o p o s i t i o n 2 o f C h e r n o f f , Rasala a n d W a t e r h o u s e
let
[69].
Indeed,
f E C ( X ; F ) be a f u n c t i o n s a t i s f y i n g c o n d i t i o n s (1) a n d ( 2 )
of Theorem 9 . 1 3 .
The a r g u m e n t s of P r o p o s i t o n 1 . 2 , C h a p t e r 1 , a r e
M s u b s t i t u t e d f o r F . Hence, flS i s i n t h e uniform AIS i n C ( S ; F ) f o r e a c h e q u i v a l e n c e c l a s s S C X X / A . S i n c e A i s a m o d u l e over i t s e l f , f b e l o n g s t o t h e
valid with c l o s u r e of modulo
u n i f o r m c l o s u r e of
A
in
C ( X ; F ) , b y Theoren; 9 - 2 0 .
w e can prove Kaplansky's r e s u l t o n i d e a l s i n f u n c t i o n a l g e b r a s . L e t E be a n o n a r c h i d e a n n o d U s i n g C o r o l l a r y 9.22
n o n - a s s o c i a t i v e a l g e b r a w i t h u n i t (= u n i t a r y ) over a ( n e c e s s a r i l y ) nonarchimedean f i e l d
F ; t h a t is,
E
is a not necessarily
associatFve l i n e a r a l g e b r a w i t h u n i t e over F e q u i p p e d w i t h
a
n o n a r c h i n e d e a n norm s a t i s f y i n g
C o n d i t i o n (1) i m p l i e s t h a t m u l t i p l i c a t i o n
is
jointly
NONARCHIHEDEAN APPROXIMATION THEORY
continuous: I f
169
i s any compact H a u s d o r f f s p a c e , C ( X ; E )
X
with
p o i n t w i s e o p e r a t i o n s a n d s u p norm becomes a n o n a r c h i d e a n no& algebra with u n i t t o o (over t h e s a m e f i e l d
F).
arises of c h a r a c t e r i z i n g t h e c l o s e d r i g h t I C C(X;E).
Suppose t h a t f o r e v e r y
l e f t ) ideal
x E X
left)
ideals
a c l o s e d r i g h t (resp.
i s g i v e n , and l e t u s d e f i n e
Ix C E
for a l l
I = {f E c(x;E); f ( x ) E
:.lanifestly,
N o w t h e problem
(resp.
I is a closed r i g h t
x
XI
in
.
(resp. l e f t ) ideal i n
M e s h a l l prove t h a t any c l o s e d r i g h t
C(X;E).
( r e s p . l e f t ) i d e a l i n C(X;E)
h a s t h e above form. Namely w e have t h e f o l l o w i n g . THEOREM 9 . 2 4 .
Let
(Kaplansky)
Hausdorff space. Let
be a
X
0 - dimensional
be a n o n a r c h i m e d e a n normed a l g e b r a w i t h
E
u n i t e o v e r a ( n e c e s s a r i l y nonarchimedean) valued f i s l f I C C(X;E)
l e f t ) ideal i n
E.Then
f ( x ) E Ix
,
For e v e r y x i n X
foi.all
E.
W e claim t h a t
I
C(::;F)
g E C(X;F) be g i v e n . D e f i n e
e i s t h e u n i t of
Since
f h E I,
E .
g f
If
I
in X I .
E
i s j o i n t l y continu-
i s a r i g h t (resp. l e f t ) ideal i n
I(x)
is a
x
I ( x ) is c l e a r l y a r i g h t (resp. left)
i d e a l i n E . Since t h e m u l t i p l i c a t i o n i n
o u s , t h e c l o s u r e Ix o f
i s a closed right
Ix
and
E,
I = ( f EC(X;E)j PROOF:
F . Let
b e a c l o s e d r i g h t ( r e s p . l e f t ) i d e a l . For each x E X ,
l e t Ix be t h e c l o s u r e of’ I ( x ) i n Iresp.
compact
-module.
h E C(X;E)
I n d e e d , l e t f E I and
t o be
x
+
g ( x ) e , where
i s a r i g h t i d e a l , t h e n f o r a l l x E XI
belongs t o
I .
(The c a s e o f a l e f t i d e a l i s
t r e a t e d s i m i l a r l y . ) I t remains t o apply Corollary 9 . 2 2 separating algebra
C (X;F)
and t h e c l o s e d
C (X;F)
COROLLARY 9.25.
Under t h e h y p o t h e s i s of Theorem 9 . 2 4
that the algebra
E
i s s i m p l e . T h e n any two - s i d e d
to
- module
the I
.
assume,
closed ideal
consist:: of aZZ functions vanishing o n a c l o s e d s u b s e t of
X ,
NONARCHIMEDEAN APPROXIMATION THEORY
170
and t h e i d e a l i s maximal if and o n l y if t h e c l o s e d s e t i s a singleton. W e f i r s t recall t h a t t h e u n i t a r y algebra
PROOF:
E
is said
to
and
E.
b e s i m p l e i f it h a s no t w o - s i d e d i d e a l s o t h e r t h a n be a c l o s e d s u b s e t of
N C X
Let
Z(N)
= { f E C(X;E);
Conversely, i f
f E Z(N)
N
,
is closed i n
is a closed
I
f ( x ) # 0 . Now
and
X
x E X
f E Z(N)
,
two
I C Z(N).
I(x)
= 0 , a n d so
x ji? N .
,
E
i n N)
-
sided
so
x E N.
ideal
for a l l
in
11.
f E
Conversely, l e t f $ I.
By
Theorem
Thercfore ,
f ( x ) ji? I,.
such t h a t
so
i d e a l i n t h e simple algebra plies
x
C(X;E).
and assume bv c o n t r a d i c t i o n t h a t
there is some
9.24,
for a l l
N = ( x E X; f ( x ) = 0
c(x;E), let us define
Clearly,
Clearly, the subset
X .
f(x) = 0
i s a closed two- sided i d e a l of
0
However, Ix i s a t w o - s i d e d Ix = { O } .
Now
0
Ix =
This contradicEion
1 im-
shows t h a t
f E I. L e t L = C ( X ; E ) , x E X.
M = { f E L,
Let
closed two- sided i d e a l i n L Let
.
f ( x ) = 0).
Clearly,
Then M
"l(x) = 0 .
f ( x ) # f ( y ) . Such a n
f E C(X;F) b e s u c h t h a t
a
y # x.
Let
f
is
e x i s t s be-
i s 0 - d i m e n s i o n a l . I n f a c t w e may assume f ( x ) = 0 and f ( y ) = 1. T h e r e f o r e f E M. L e t a E C . Then a f E Mand a f ( y ) = a .
cause X Thus
M(y) = E , f o r a l l
x # y. L e t
i d e a l c o n t a i n i n g M . Then
is a t w o or
-
sided ideal i n
I(y) = E E
I = C(X;E).
Therefore M
Since E
E X.
C (X;E)
Since E
be a two-sided
y # x.
Now
I(x)
is simple, e i t h e r I(x) = 0 I = M or
t h i s means t h a t e i t h e r
i s maximal.
Conversely, i f M ideal in
x
.
I ( x ) = E . By Theorem 9 . 2 4 ,
I C C(X;E)
for all
i s a maximal p r o p e r c l o s e d two-sided
, t h e n by Theorem 9 . 2 4 ,
M(x) # E
for
some
i s s i m p l e , M(x) = 0 . T h e r e f o r e M C {f E C ( X ; E ) ;
f (x) = 0).
S i n c e t h e r i g h t - hand s i d e o f t h i s i n c l u s i o n i s a p r o p e r c l o s e d i d e a l and M
i s maximal,
Pl = { f
E C(X;E);
f (x) = 01
.
171
NONARCHIMEDEAN APPROXIMATION THEORY § 5.
VECTOR FIBRATIONS.
A s in Chapter 1 one can extend the results of the preceding 5 4 to the case of vector fibrations. By a v e c t o r f i b r a t i o n we mean a pair (X;(Ex; x E XI), where i( is a Hausdorff topological space and (Ex ; x E X) is a family of vector spaces over the same scalar field F . The product set TI Ex is made into a vector space over F in the usu-
XE x a1 way and its elements are called the c r o s s - s e c t i o n s given vector fibration.
of
the
vector space W of cross- sections, i. e. a vector W C Ii Ex , is said to be a m o d u l e o v e r a s u b a l g e b r a
A
subspace
XE x A C C(X;F) , or an A - m o d u l e , if for any f E A arid g = (g(x);x
in F7
, we have fg
= (f
(x)g(x); x
E
X)
E
E
X)
W.
We shall restrict our attention to vector spaces of cross- sections satisfying the following condi-
L C Il Ex XEX tions :
(1)
X
is a compact Hausdorff space;
(2)
each Ex is a nonarchimedean normed space over the same valued field (F, I 1 ) ; the norm in each
-
Ex will be denoted by (3)
t
++
11
t
I! i
for each f E L the function x H IIf (x)1 1 upper semicontinuous on the space X .
is
From (1) - ( 3 ) it follows that L is a nonarchimedean normed space over (F; I I ) , if we define
-
L. A l l the results of 5 4 can be extended to such n o n a r c h i m e d e a n normed s p a c e s of c r o s s - s e c t i o n s , and the proofs are exactly the sane. In particular, Theorem 9.19 reads as follows. for all
f
E
THEOREM 9.26.
Let
an A - m o d u l e ,
where
L b e a n o n a r c h i m e d e a n normed s p a c e of crosss e c t i o n s s a t i s f y i n g c o n d i t i o n s ( 1 ) - ( 3 ) a b o v e . Assume that L is A
i s a subaZgebra of
C(X;F).
For
every
172
NONARCHIMEDEAN APPROXIMATION THEORY
A - submodule
f E L, w h e r e PA d e n o t e s t h e s e t of a l l equivalence
for a12
ses
modulo
S C X
COROLLARY 9.27.
each
f E L,
X/A
such t h a t
PROOF:
we h a v e
W C L;
cZas-
X/A.
Let
L
,A
W
and
For
be a s i n Theorem 9 . 2 6 .
t h e r e e x i s t s an e q u i v a l e n c e c l a s s
moduZo
S C X
Apply Lemma 1, Machado and Prolla ;39].
Let L ,A and W b e a s i n T h e o r e m 9 . 2 6 . AsA i s s e p a r a t i n g o v e r X . Then W i s d e n s e i n L i f , and o n l y i f W ( X ) = {g(x); g E W } i s d e n s e i n L ( x ) c E x for e a c h x E X. More g e n e r a l l y , for a n y f E L, f b e l o n g s t o t h e c l o s u r e of W i n L i f and o n l y i f f ( x ) b e l o n g s t o t h e closure of W ( X ) i n L ( x ) for e a c h x E X.
COROLLARY 9.28.
sume t h a t
To state the analogous result of Theorem 9.24 for vector fibrations, assume that besides (1)- (3) the following further conditions are satisfied: (4)
each E x is a not necessarily associative linear algebra with unit ex over the same nonarchimedean valued field ( F , I I ) ; moreover each Ex is equipped with a nonarchimedean norm satisfying
(4.1) (4.2)
11 11
(5)
e = (ex
DEFINITION 9.29.
11
uvll 5 ex
If
I!
u
I/
-
11
v
11,
for all
u
,v
F Ex
= 1 ;
x
E X)
belongs to L
.
(1)- ( 5 ) a r e s a t i s f i e d we s a y t h a t
L
a n o n a r c h i m e d e a n normed u n i t a r y ~ Z g e b r aof c r o s s - s e c t i o n s . d e e d , i f we d e f i n e o p e r a t i o n s c o o r d i n a t e w i s e i n becomes a n o t n e c e s s a r i l y a s s o c i a t i v e
n
x EX
Ex
i s In-
, this
l i n e a r a l g e b r a over F , w i t h
173
NONARCHIWEDEAN APPROXIMATION THEORY
u n i t e = ( e x ; x E X), L i s a s u b a l g e b r a c o n t a i n i n g e, and t h e norm o f L s a t i s f i e s :
I,
(a)
If
I i :I /
fg
f
Ij
11
*
11
g
for all
f
the
r
g E L
i s complete w i t h r e s p e c t t o t h e metric
L
by t h e norm o f
w e say t h a t
L
L is a
unit
induced
nonarchimsdean
unitary
Banach a l g e b r a of c r o s s - s e c t i o n s . Suppose t h a t L i s a unitary nonarchimedean normed a l -
DEFINITION 9 . 3 0 .
g e b r a of c r o s s
- sections over
s p a c e . F o l l o w i n g Murphy f E L
i s in W
Let
X .
locally a t a point
t h e r e i s a neighborhood
U
c
Vl
L
be a vector s u b -
one s a y s t h a t a c r o s s
[74;,
of x
x
X,
E
- section
i f f o r each
i n X and
an
E
> 0
g E W
element
such t h a t
for all
y E U . The v e c t o r s u b s p a c e
-
W
i s said t o
be
local
i f
W of W i n L c o n t a i n s a l l t h e c r o s s -sett i o n s w h i c h a r e i n W l o c a l l y a t a21 p o i n t s o f X . L e t A(W) = the uniform closure
J.#
= {f E L; N
A(W) 3
w,
w
is in
f
l o c a l l y a t a l l p o i n t s of
W i s l o c a l i f and o n l y i f
and
Let
DEFINITION 9 .31.
L
g e b r a of c r o s s - s e c t i o n s
1:
f
/ I I 1,
Let
U C X
of
cross-section
over
X . We s a y t h a t
= ex
au(x) = 0
DEFINITION 9.32.
L i s full i f
L
Let
Clearly,
x , y E X
L is
separating
there i s
f E L
f(y) = e . Y be any s u b s e t of X . The c h a r a c t e r i s t i c
f(x) = 0,
U, d e n o t e d by
$,(XI
A(W)
XI . C G.
b e a u n i t a r y n o n a r c h i m e d e a n normed a l -
i f f o r a l l p a i r s of d i s t i n c t p o i n t s
such t h a t
N
L
and
aU
i s d e f i n e d by
if
X E U
if
x $
u.
be a s i n D e f i n i t i o n 9 . 3 1 .
We s a y
contains the characteristic cross-sections
that of
174
NONARCHIMEDEAN APPROXIMATION THEORY
a l l clopen subsets o f
If
X .
i s 0 - d i m e n s i o n a l , t h e n e v e r y l o c a l u n i t a r y sub-
X
a l g e b r a o f a f u l l a l g e b r a i s s e D a r a t i n g . The c o n v e r s e is btrphy's Stone
- Weierstrass
Theorem (see [ 7 4
j1.
We state if
l e s W C L. R e c a l l t h a t a v e c t o r s u b s p a c e right
W
( r e s p . l e f t ) m o d u l e o v e r an a l g e b r a
(resp. over A
If
WA C W ) .
,
c
L
A C L
for A
is
called.
a
if
AW
M
i s a r i g h t module o r a l e f t
W C L
w e say simply t h a t W i s an
b o t h a r i g h t a n d a l e f t module o v e r bimoduZe. Clearly, a subalgebra
A - module. A ,
A C L
- moduC
module
W C L is i s a n A-
When
we say t h a t W
i s a bimodule o v e r
it-
self. THEOREM 9 . 3 3 .
Let
be a f u l l u n i t a r y nonarchimedean
L
normedal-
g e b r a o f cross - s e c t i o n s o v e r a 0 - d i m e n s i o n a l compact H a u s d o r f f space X
.
let W C L is l o c a l .
PROOF:
Let
be a separating unitary subalgebra,
A C L
b e a v e c t o r s u b s p a c e w h i c h is a n A - m o d u l e .
Assume
i s a r i g h t module o v e r A . L e t
W
f E
and
Then
r(W) a n d
0 < E < 1 b e g i v e n . F o r e a c h x E X , t h e r e i s s o m e gx E W s o m e c l o p e n n e i g h b o r h o o d o f x , s a y Ux s u c h t h a t
for all
{ x l I x2
y E Ux
.
, ... , x n }
1
U1
= Ux
Ui
,
1
= UX . 1
Then, t h e of
X .
Let
ai
a
finite
and
set
such t h a t
C X
x c ux u ... u u, Let
X I there is
By c o m p a c t n e s s o f
W
Ui's
and f o r
-
n i = 2,3,
...,n
TIx,
j
J
f o r m a p a i r w i s e d i s j o i n t c l o p e n cover
be t h e c h a r a c t e r i s t cross- s e c t i o n of
Ui
,
i = 1,2,...,n.
A s s u m e t h a t w e have proved t h a t t h e uniform c l o s u r e
Ti
NONARCH I ME DEAN APPROXIMATION THEORY of
A
belongs t o
-
AW C
Ti
$i E
i s f u l l . Then
175
...,n .
for all i = 1,2,
Hence $ J ~ c J , , 1
x
C A M
%, f o r a l l
C
i = 1,2,...,n.
Hence
g d e f i n e d by
g = 4
9
x1
belongs t o g , a n d given ;If(y) where
i = 1,2,
Therefore
;If
-
4n9x
+
n
y E X,
g(y)
...,n - g 11
...
+
li
I/
=
f(y)
-
gx ( y )
I1
i s t h e unique index
i
a n d t h i s shows t h a t
< E
<
E
i
such t h a t
y E Ui.
f E 3, i.e. W is
local.
is f u l l . Let @ u be t h e we c h a r a c t e r i s t i c c r o s s - s e c t i o n of a clopen s u b s e t U C X. I t remains t o prove t h a t
claim that y
E U
+u
E
A. L e t
0 <
< 1 be g i v e n . L e t
E
is separating, there is
be g i v e n . S i n c e A
x
9Y
U E A
and such
that
Since
t
I-+
a c l o p e n neighborhood
/Igy(t) V
11
of
Y
i s u p p e r s e m i c o n t i n u o u s there is
y
in
x
such t h a t
f o r a l l t E V . Now U i s c o n t a i n e d i n t h e u n i o n of a l l V I s I Y Y y E U. By c o m p a c t n e s s . t h e r e i s a f i n i t e set { y 1 , y 2 , . . . , y , } C U such t h a t
u c v Let
Yl
...
...
u *
g
11
v Yn
E A.
Yn
. Yoreover
176
NONARCHIMEDEAN APPROXIMATION THEORY
because
g,
.
By u p p e r s e m i c o n t i n u i t y of
h
0 <
hx - e
< 1. L e t
E
-
hx E A , b e c a u s e
Then
A
is u n i t a r y , and
F o r all
t E U, w e h a v e
borhood
of
Wx
Now
x
x in
such t h a t
X
, x 2 , ... , xm} C
<
E
for a l l t
E Wx.
all
WX'sl
of
m
1
h = hx
1
-
...
hx
-.<
1,
-
h
11
We claim that
-
h(t)
i = 1,2,...,m,
Ilet -
/I
=
m t E X\U
and f o r a l l
<
h(t)
Indeed, i f t
E.
/I
<
E.
If
U,
t E U,
then then
for
each
w e have
(t)
...
*
hx ( t ) 1 1 < k
Indeed, t h i s is t r i v i a l for l < k = j < m .
E A. I4oreover
hx
2
/Ih 11
such t h a t
X\U
... u wx .
x \ u c wx u
$,(t)
11
ilhx(t)
is contained is t h e union
X\U
{x,
11
t h e r e i s a c l o p e n neigh-
By c o m p a c t n e s s , t h e r e i s a f i n i t e s e t
E X\tJ.
Let
X
Let
E
f o r all k = 1 , 2 ,
k = 1. Assume i t S
, I et -
v1 v2
. . . v 3. v 1+1 . I]
t r u e f o r some
s = 1,2,. . . , j + 1.Then
v s = hx ( t ) f o r a n y
w e have
is
...,m.
=
177
NONARCHIMEDEAN APPROXIMATION THEORY
-
-
lIet
Vj+l
Vj+l
+
-
v1 v 2
.. . v 7. v 7. + 1
I/
T h i s p r o v e s o u r c l a i m by i n d u c t i o n . I n p a r t i c u l a r , f o r k = m , w e get ll$u(t)
-
h(t)
11
=
et - h ( t )
<
E.
is f u l l .
This ends t h e proof t h a t COROLLARY 9 . 3 4 .
[I
A s s u m e t h e h y p o t h e s i s of T h e o r e m 9 . 3 3 .
Then
PROOF: C l e a r l y o n e h a s t h e i n c l u s i o n
Conversely, l e t
f E L
W e have t o prove t h a t
f E
i. By
it i s s u f f i c i e n t t o prove t h a t E
> 0 are g i v e n ,
f ( x ) E w(x)
be such t h a t
g E W
there is
f o r a l l x E X.
r(W).NOW,
if
x E X
s u c h t h a t / l g ( x )- f ( x )
upper s e m i c o n t i n u i t y t h i s i s t r u e i n a neighborhood X,
i.e.
; / g ( t )- f ( t ) I / <
THEOREM 9 . 3 5 .
E
so
Theorem 9 . 3 3 , W i s l o c a l , f E
for all
t E U. Hence
( V e c t o r - v a l u e d Kaplarrsky ' s T h e o r e m ) .
I/
and
<
E.
x in
of
U
f E
By
r(W).
Let
X
be
a 0 - d i m e n s i o n a l c o m p a c t f l a u s d o r f f s p a c e , and % e t E b e a nonarc h i m e d e a n normed u n i t a r y a l g e b r a o v e r a v a l u e d f i e l d Let
A C C(X;E)
and l e t
PROOF: x E X
be a separating unitary subalgebra
1'7 C C ( X : E )
be an A
*
1).
- module. Then
C o n s i d e r t h e v e c t o r - f i b r a t i o n g i v e n by and t a k e
(F,I
A C C(X;E),
Ex = E
for a l l
L = C ( X ; E ) . Then, f o r a n y c l o p e n s u b s e t U C X ,
NONARCHIMEDEAN APPROXIMATION THEORY
178
4
the c h a r a c t e r i s t i c vector valued function o u s . Hence
+
i s continu-
E
i s f u l l . I t r e m a i n s t o a p p l y C o r o l l a r y 9.34.
C(X;E)
Let
COROLLARY 9 . 3 6 .
:X
X
and E
b e as i n "heorem 9.35. Let
A C C(X;E)
be a s e p a r a t i n g u n i t a r y s u b a l g e b r a . Then
PROOF:
Consider
and recall
i n Theorem 9 . 3 5 ,
W = A
that
any
a l g e b r a i s a b i m o d u l e over i t s e l f .
Let
COROLLARY 9 . 3 7 .
X
and E
b e as i n Theorem 9.35. Let
be a separating subalgebra containing the constants. uniformly dense i n C ( X ; E ) .
A C C(X;E)
Then A
Since A c o n t a i n s t h e constants, A is u n i t a r y andA(x)=E
PROOF: for all
I t remains t o a p p l y C o r o l l a r y 9.36.
x E X.
( K a p l a n s k y Is T h e o r e m )
THEOREM 9 . 3 8 .
valued f i e l d .
Let
A C C(X;F)
s e p a r a t e s t h e p o i n t s of X
.
Let
X
(F,I 1 ) be be a u n i t a r y
c o m p a c t i l a u s d o r f f s p a c e and l e t
Then A
be a 0 - dimensional
a
nonarchimedean
subalgebra
E = F , it i s enough t o
PROOF:
By C o r o l l a r y 9 . 3 7 ,
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 . 3 1 . be g i v e n i n
such t h a t
X .
taking
f ( x ) # f ( y ) . Since
g ( y ) = 1.
Now
F
g(X)
c F
that
p ( 0 ) = 0 , p ( 1 ) = 1, a n d
Then
h = P Og A
REfARK 9 . 3 9 .
f E A
is a Ejeld, and A c o n t a i n s t h e g E A
such t h a t
g(x) = 0
i s a compact s u b s e t , and t h e r e
f o r e b y 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
Therefore
prove
Let then
By h y p o t h e s i s t h e r e i n a n e l e m e n t
c o n s t a n t s , it follows t h a t t h e r e i s and
which
i s u w i f o r m l y dense i n C(X;F).
that A x # y
i s
Ip(t)l 5 1
p : F
f o r all
-
+
F
such
t
E
g(X).
belongs t o A, , ! h , i 5 1, h ( x ) = 0 , and h ( y ) =1. i s s e p a r a t i n g i n t h e s e n s e o f D e f i n i t i o n 9.31. The r e m a r k s p r e c e d i n n T h e o r e m 9 . 3 3 a n d t h e a r g u -
m e n t s i n t h e p r o o f of Theorem 9 . 3 3 s:iow t h a t , i f i n Theorem 9 . 3 3 ,
X a n d L are as
t h e n f o r any c 7 o s e d c n i t a r y s u b a l q e b r a
A
c
L
NONARCHIMEDEAN APPROXIMATION THEORY
179
t h e following are e q u i v a l e n t A is separating.
(i)
(ii) A is f u l l .
( i i i ) P. i s l o c a l .
L e t u s now c o n s i d e r t h e c h a r a c t e r i z a t i o n
i d e a l s . For t h e c a s e o f
x
Ex = E f o r a i l
E
X , and
of
closed
continuous
f u n c t i o n s , t h i s w a s d o n e i n Theorem 9 . 2 4 a n d Corollary 9.25 above. THEOREM 9 . 4 0 .
essential.
Lzt
Cet
x E X,
each
c
I
b e a s i n Theorem 9 . 3 3 .
L
be a closed right
L
Ix b e t h e c l o s u r e of
let
i s a cZosed r i g h t Iresp. I
PROOF:
=
If
A = L
Take
and
t h a t by Remark 9 . 3 9 , L
Indeed, i f
If
g E I
a right ideal i n means t h a t
f E L
such t h a t
for all
l e f t ) ideal.For Then
Ex.
x E X,
Ix
and
x
in
xI . notice
i s separating.Hence
such t h a t f (x) = a .
g ( x ) = b . Now
belongs t o
L ( x ) . The h y 2 o t h e s i s
L ( x ) = Ex
L
is
I ( x ) is a right ideal i n
a E L(x), there is
b a = g(x) f (x) = (gf) (x)
,
L
i n C o r o l l a r y 9.34, and
W = I
full implies that
b E I ( x ) , there is
and
Ex
for all
Now i t i s e a s y t o show t h a t L(x).
(resp.
I(x).in
l e f t . ) idea2 i n
f ( x ) E Ix
E L;
Assume t h a t
I ( x ) . Thus
that
L
is
gf E I I ( x ) is
essential
proof
and t h i s e n d s t h e
o f Theorem 9 . 4 0 . COROLLARY 9 . 4 1 .
x
f o r each
Let
E X,
Ex
L
b e a s i n Theorem 9 . 4 0 ,
and assume
i s s i m p l e . Then any c l o s e d two -sided ideal
c o n s i s t s o f a l l f k n c t i o n s v a n i s h i n g on a c l o s e d s u b s e t Moreover, form
PROOF:
that
of
X.
e v e r y m a x i m a l c l o s e d t w o .- s i d e d p r o p e r i d e a l i s of t h e
{f E L; f ( x ) = 01 Let
N
f o r some
x E X.
be a c l o s e d s u b s e t of
X .
I t i s e a s y t o see t h a t
180
NONARCHIMEDEAN APPROXIMATION THEORY
Z ( N ) = { f E L;
f(x) = 0
L.
is a closed t w o - s i d e d i d e a l of
Conversely, l e t
.
in L
L
be a c l o s e d t w o - s i d e d
ideal
Define N ( 1 ) = { x E X;
I t i s e a s y t o see t h a t
Conversely, l e t f $ I.
N
.
o t h e r hand,
and c l e a r l y
X
.
f E I ]
I C Z(N).
x
t h e r e i s some
such t h a t
E X
so
f E Z(N),
x
On
N.
the
i s a t w o - sided closed i d e a l i n t h e simple alge-
Ix
f ( x ) $ Ix
Since
Ex.
for a l l
i s closed i n
f ( x ) # 0 . Now
Hence
t h i s implies
f(x) = 0
f E Z ( N ) , a n d assume by c o n t r a d i c t i o n t h a t
By Theorem 9 . 4 0 ,
f ( x ) @ Ix bra
c
I
x E NI
for all
,
Ix
I ( x ) = 0 . Hence
# Ex.
Therefore
0
Ix =
x E N. This contradiction
1 , and shows
f E I.
that
For
x E XI
set
.
I3 = { f E L ; f ( x ) = 0 )
.
M i s a closed
t w o - s i d e d i d e a l i n 'L C l e a r l y , rl(x) = 0 . L e t y E X be a d i s t i n c t p o i n t . Since X is 0 - dimensional, t h e r e i s a clopen neighborhood a E E with
Y
Hence
U
of
= L(y)
y , with
(because L
f ( y ) = a. Hence
g E I4
L e t then
I ( y ) = Ey
x $ U. S i n c e L
and
I C L
is f u l l ,
is essential), there
$u
E L.
If
is s o m e
f E L
, for
all yfx.
g = @"f is such t h a t
g(y) = a.
Therefore
M(y) = Ey
be a two- s i d e d i d e a l c o n t a i n i n g
for all
y # x . Now
. S i n c e EY i s s i m p l e e i t h e r Y t h e f i r s t case, w e have
E
-
I(t)= n ( t ) , By C o r o l l a r y 9 . 3 4 ,
I(x)
PI. Therefore is a t w o - sided ideal i n
I(x) = 0
for all
1 = p4.
I n t h e s e c o n d case, w e have
or
t
E
I(x) = Ex.
X.
In
NONARCHIMEDEAN
APPROXIMATION
By t h e same C o r o l l a r y , I = L . T h e r e f o r e ideal i n
M
i s maximal.
C o n v e r s e l y i f M i s maximal c l o s e d t w o - s i d e d , t h e n M(x) # Ex f o r some x E X . S i n c e Ex
proper
is sim-
L
M
p l e , M(x) = 0 . T h e r e f o r e f (x) = 0 }
{ f E L;
181
THEORY
c
{ f E L;
i s p r o p e r and M
f (x) = 0
i s maximal, w e have
f(x) = 0 1
E l = I f E L;
1. S i n c e
.
This ends t h e proof of Corollary 9.41.
5
6.
SOME APPLICATIONS.
i s a compact Hausdorff s p a c e and E 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 v a l u e d f i e l d ( F , ] 1 ) . In t h i s section
X
E # { O
By c o n v e n t i o n 9 . 1 6 ,
1
.
The v e c t o r s u b s n a c e of
C(X;E)
c o n s i s t i n u o f a l l f i n i t e sums o f f u n c t i o n s of t h e form x + f ( x ) v , where
f E C(X;F)
Clearly,
PROOF:
C(X;F) 8 E
Let
v E E , w i l l b e d e n o t e d by
is a
C(X;F) i s s e p a r a t i n g o v e r
If
X
C(X;F) 8 C(Y;F)
c o m p a c t Hausdorff space.
i s uniformly dense i n
W = C(X;F) 8 E .
Corollary 9.22,
W
Then W
Y
are t w o
C(X;E)
is a
For each
X .
i s dense i n
and
C ( X ; F) 8 E .
C(X;F) -module.
Let X be a 0 - d i m e n s i o n a l
THEOREM 9 . 4 2 .
Then
and
C(X;F) 8 E
.
C(X;F) - m o d u l e ,
x E X,
W(x) = E .
and By
C(X;E).
compact
Hausdorff
spaces,
d e n o t e s t h e v e c t o r s u b s p a c e o f C ( X x Y;F) con-
s i s t i n g o f a l l f i n i t e sums o f f u n c t i o n s o f t h e form
where
f E C(X;F)
and
mensional spaces, then
g E C(Y;F)
subalgebra of
C ( X x Y;F)
THEOREM 9 . 4 3 .
Let
d o p f f s p a c e s . Then C(X x Y;E).
X
.
I f both
X
and
Y
are 0
- di-
C ( X ; F ) 8 C(Y:F) i s a s e p a r a t i n g u n i t a r y
and
. Y
b e two 0 - d i m e n s i o n a l
(C(X;F) Q C(Y;F)) 8 E
c o m p a c t Haus-
i s u n i f o r m Z y dense i n
182
NONARCHIMEDEAN APPROXIMATION THEORY
PROOF:
W = (C(X;F) Q C ( Y ; F ) ) 8 E. W i s
Let
module s u c h t h a t
W(x,y) = E
-
C(X;F) 8 C(Y;F)
f o r every p a i r
( x l y ) E X x Y.The
r e s u l t now f o l l o w s f r o m C o r o l l a r y 9 . 2 2 . When
REPARK:
just
(C(X;F) Q C ( Y ; F ) ) 8 E
E = F, then t h e space
C ( X ; F ) Q C(Y;F)
and one o b t a i n s
Dieudonng's
is
ThGorGme
.
2, 17011
I n C h a p t e r ,3 w e s t u d i e d p o l y n o m i a l a l g e b r a s o f f u n c -
IR
t i o n s with values i n vector spaces over
a.
or
To s t u d y t h e
nonarchimedean a n a l o g u e l e t u s a d o p t t h e f o l l o w i n g A v e c t o r subspace
DEFINITION 9 . 4 4 .
n o m i a l a l g e b r a if of
,
A = { u ( f ) ; u E E'
i s a
f E 111
subalgebra
A 8 E C W.
such t h a t
C(X;F)
is c a l l e d a poly-
IV C C ( X ; E )
L e t u s g i v e a n example o f a polynomial a!gebra.
Let
Pf ( E ; F ) C C ( E ; F ) be t h e a l g e b r a o v e r
F
g e n e r a t e d by t h e t o p o l o g i c a l d u a l
E'
of
E . An e l e m e n t
p E P f ( E ; F ) i s c a l l e d a c o n t i n u o u s polynomiaZof f i n i t e t y p e f r o m E into F , a n d i s o f t h e f r o m
(1)
1 where a
E F,
=
K
(K~,..
( Ul
u =
c
p = ,
.,
K
a K uK
1%
n 14
K ~ ) E
l...,~n)
E
,
(E')
n E N* n
,
, I
K
1
= K
t
spaces
El
W e define
E
E.
and
E2
and
a s t h e v e c t o r subspace of
' # 101.
Then
X
c
t E El
+
El
C(E1
,E2)
p(t)v
gen-
where
A = { u ( p ) ; u E E i ; p EPf(E1,E$).
A Q E 2 C Pf ( E l , E 2 ) .
A = Pf ( E l ; F )
mial algebra. Also, i f
form
L e t now
v E E.
C l e a r l y , A C Pf ( E l ; F ) , a n d (E2)
normed
o v e r t h e same n o n a r c h i m e d e a n v a l u e d f i e l d F .
Pf ( E l , E 2 )
, F)
N,
n
L e t u s now c o n s i d e r t w o n o n a r c h i m e d e a n
e r a t e d by t h e f u n c t i o n s o f t h e p E Pf(E1
.. + K n ,
a n d we d e f i n e K
for a l l
+.
1
Suppose
a n d Pf ( E l ; E 2 ) i s a polynoi s any compact s u b s e t l t h e n
NONARCHIMEDEAN APPROXIMATION THEORY
1
183
i s a p o l y n o m i a l a l g e b r a c o n t a i n e d i n C(X;E2). S C C(X;F) i s any s u b s e t , l e t A C C(X;F)
W = Pf ( E l ; E 2 ) X
More g e n e r a l l y , i f
b e t h e s u b a l g e b r a o v e r F g e n e r a t e d by S.
If
# {O},
E'
,f
A = {u(f); u E E'
W}
E
m i a l a l g e b r a , when
.
i s a polyno-
In particular, C(X;E)
# 0 ( e . g . , when
E'
E = F).
i s s p h e r i c a l l y c o m p l e t e , t h e Hahn
When t h e f i e l d F
Banach Theorem i s v a l i d f o r a n y n o n a r c h i m e d e a n normed over F
[71] 1 ,
(see I n g l e t o n
and a f o r t i o r i ,
#
E'
and t h e n
space
of a l l
where
f E C(X;E)
f o r m c l o s u r e of S C X
modulo
W
C(X;E)
in
,
THEOREM 9 . 4 5 .
E'
f o r each
C(S;E),
Thus, i f
E
W C C(X;E)
LLq(W) t h e s e t
i n t h e uni-
:IS
equivalence
class
denotes t h e uniform c l o s u r e of
*f
t h e Theorem 9 . 2 0 m y be s t a t e d as f E Let
E LA(W).
b e a n o n a r c h i m e d e a n normed s p a c e s u c h that
i s separating over
algebra. Let
we d e n o t e by
A C C(X;F),
in
WIS
E
i s s e p a r a t i n g overE,
E'
such t h a t t h e r e s t r i c t i o n
X/A.
-
{ 0 }.
L e t u s i n t r o d u c e t h e i'ollowing n o t a t i o n . I f
i s a n A-module,
then
case we have
i s a polynomial a l g e b r a . Indeed, i n t h i s
W = A Q E
and l e t
E ,
A = {u(g); u E E'
a
W C C(X;E) be
,g
E
polynomiaZ
W}. Then, for every f
E C(X;E)
the following conditions are equivalent. (1)
f
(2)
given
E
that (3)
(a)
i; x , y E x and I(f(x) - g(x) given is
(b)
9
given
E
PROOF :
(1) (2)
E
=
1;
f(x)
f
-
3
f(y)
I(<
E
W
and I I f ( y ) - g ( y )
11
> 0, E
there i s g
such <
E;
x , y E x, w i t h f ( x ) # f ( y ) , t h e r e W s u c h t h a t g(x) # g ( y ) ; a n d x E X,
such t h a t (4)
F
with
f ( x ) # 0,
there i s g E W
q ( x ) # 0;
E LA(A Q E l .
( 2 ) . Obvious.
(3). L e t
11
> 0.
x , y E X
with
By ( 2 ) t h e r e i s
f ( x ) # f (y). g E PI
Define
such t h a t
184
NONARCHIMEDEAN APPROXIMATION THEORY
If
g ( x ) = q ( y ) , then
-
= :if (x)
E
g(x) + q(y)
-
max( j I f ( x )
-
g(x)ll
r
-
/Ig(y)
a c o n t r a d i c t i o n . T h i s p r o v e s ( a ) . The p r o o f o f (3)
dulo
g
E
.
E r
(b) is similar.
b e a n e q u i v a l e n c e c l a s s mo-
S C X
u ( g ( x ) ) # u ( g ( y ) ) . T h i s i s impos
such t h a t u ( g ) E A.
be i t s c o n s t a n t v a l u e . S
f(y)ll)<
.
u E E'
s i b l e , because over
5
x , y E S . I f f ( x ) # f ( y ) , by ( a ) t h e r e i s g ( x ) # g ( y ) Since E ' is separating over E ,
such t h a t
there is
Let
-
I/
and l e t
X /A,
W
(4).
f(y)
Hence
If
is constant over S . L e t t E E then 0 E A 8 E agrees with f
f
t = O ,
t # 0 , t h e n , by ( b ) t h e r e i s
If
g ( x ) # 0 , where be such t h a t
x
E S
-
such
q E M
that
u E E'
i s c h o s e n a r b i t r a r i l y . L e t now
u ( g ( x ) ) = l . Then t h e f u n c t i o n h = u(g) 8 t
belongs t o
A 8 E
and a g r e e s w i t h
f
over
S . Therefore
f E LA(A 8 E ) . (4)
*
( 1 ) . By Theorem 9 . 2 0 a p p l i e d t o t h e
A 8 E C C(X;E), f C(X;E).
Since
E
and
A 8 E
in
A 8 E C W, t h e p r o o f i s c o m p l e t e .
Let
COROLLARY 9 . 4 6 .
and l e t
belongs t o t h e uniform c l o s u r e of
A-module
X
b e a 0 - d i m e n s i o n a l compact Hausdorff space,
b e a s i n Theorem 9 . 4 5 .
W
The f o l l o w i n g statements
are equivalent.
(1)
W
(2)
W ( x , y ) = { ( g ( x ), g ( y ) ) ; g E
is u n i f o r m l y d e n s e i n
f o r every pair (3)
(a)
If
x, y
C(X;E);
r7}
i s dense i n X
x
X,
E X;
x # y, there i s
g E W
such t h a t
g(x) # g(y);
(4)
(b)
Given
Let
A = {u(g); u E E'
rating over for e v e r y
x E X,
X
and
x E X.
there i s
,g
W(X)
E
g
E
W w i t h g(x) # 0 .
W}. T h e n A i s sepag E V7 l = E
= { g ( x );
185
NONARCHIMEDEAN APPROXIMATION THEORY
(1) =>
PROOF:
(2)
==?
a r e immediate from Theorem 9.45.(3)3(4)
(3)
-
El is separating over
f o l l o w s from t h e h y p o t h e s i s t h a t A 8 E
from
c W.
Finally, A-module
(4)
E
and
(1) by C o r o l l a r y 9 . 2 2 a p p l i e d t o
the
which i s c o n t a i n e d i n M .
A Q E,
(Weierstrass p o l y n o m i a l a ? p r o x i m a t i o n )
COROLLARY 9 . 4 7 .
and E 2 b e t w o n o n a r c h i m e d e a n normed s p a c e s o v e r i s separating over
Ei(i
Pf ( E l ; E 2 ) 1 K
the s e t
=
Let
El
F s u c h t h a t E;
1 , 2 ) . For e v e r y c o m p a c t s u b s e t K C E 1
i s uniformly dense i n
C(K;E2).
T'J = P f ( E l ; E 2 ) 1' K . S i n c e E; i s s e p a r a t i n g o v e r E 2 ' W i s a p o l y n o m i a l a l g e b r a c o n t a i n e d i n C ( K ; E 2 ) . Now W contains
PROOF:
Let
t h e c o n s t a n t s and i t i s s e p a r a t i n g ov e r r a t i n g over
.
El
K , because
Ei i s sepa-
It remains t o a p p l y t h e p r e c e d i n g C o r o l l a r y .
A s another a p p l i c a t i o n of t h e general r e s u l t s
proved
abo.re, l e t u s g i v e a nonarchimedean a n a l o g u e o f B l a t t e r ' s S t o w W e i e r s t r a s s Theorems f o r f i n i t e real a l g e b r a s Let
- dimensional
(see Theorems 1 . 2 2 a n d 1 . 2 4 3f E
be a f i n i t e
-
d i m e n s i o n a l non
non ;4]
-
associative
1.
- associative
(i.e.
n o t n e c e s s a r i l y a s s o c i a t i v e ) l i n e a r a l g e b r a o v e r a c o m p l e t e nonarchimedean n o n - t r i v i a l l y valued f i e l d F
.
Since every
field
p r o v i d e d w i t h a t o p o l o g y i n d u c e d by a n o n - t r i v i a l v a l u a t i o n i s
s t r i c t l y minimal t o p o l o g y on E
(see Nachbin
p5])
,
t h e r e i s a unique Hausdorff
t h a t makes i t a t o p o l o g i c a l v e c t o r s p a c e o v e r F,
and moreover, under t h i s topology, e v e ry l i n e a r T: E + E
is continuous.
W e s h a l l always c o n s i d e r
( S e e Nachbin E
transformation
1 7 5 1 , Theorems 7 and 9 . )
endowed v i t h i t s u n i q u e
t o p o l o g y ' t h a t makes i t a t o p o l o g i c a l v e c t o r s p a c e o v e r topology, c a l l e d a d m i s s i b l e i n If
{el,
... , e n }
[75]
is a b a s i s of
E
,
Hausdorff F . This
c a n b e d e f i n e d a s follows.
over
F
,
then t h e nonarchi-
medean s u p - norm
n whenever
v =
C
viei
i=l
is i n
E
,
d e f i n e s t h e unique a d m i s s i b l e
186
NONARCHIMEDEAN APPROXIMATION THEORY
topology of
E .
I f w e d e f i n e operations pointwise, non-associative if x
v E E and -+
f E C(X;E)
f ( x ) v belong t o
then
E
the
x
is called a
+
vf(x)
a and
W C C(X;E)
i f i s a bimodule o v e r
a b o v e o p e r a t i o n s . An a l g e b r a E
u,v
mappings
A v e c t o r subspace
C(X;E).
c a l l e d a submodule o v e r
becomes
C(X;E)
t o o , as w e l l a b i m o d u l e o v e r E :
algebra over F
zero
-
is
with
the
aZgebra
if
E l
i s c a l l e d s i m p l e i f it i s n o t a z e r o - a l g e b r a a n d h a s no s u b s p a c e s i n v a r i a n t r e l a t i v e for a l l
uv = 0
E E.
The a l g e b r a E
t o t h e r i g h t and l e f t m u l t i p l i c a t i o n s , e x c e p t
E .
and
I t follows t h a t a non- z e r o - a l g e b r a
i s a i r r e d u c i b l e a l g e b r a of t r a n s f o r m a t i o n s . The c e n t r o i d o f E i s t h e s e t o f a l l T
all
of
p l e i f , and o n l y i f , &(El transformations
Let
E.
is called the multiplica-
r i g h t and l e f t m u l t i p l i c a t i o n s . & ( E l t i o n algebra of
0
g e n e r a t e d by t h e s e t
be t h e s u b a l g e b r a of d ( E )
&El
is s i m -
linear linear
w h i c h commute w i t h a l l r i g h t and l e f t
E g ( E )
m u l t i p l i c a t i o n s . C l e a r l y , a l l l i n e a r t r a n s f o r m a t i o n s o f t h e form X I
belong t o t h e cefitroid of
i d e n t i t y map o f just
E .
E
W e say t h a t
,
where
F
and
is t h e
I
i s c e n t r a l i f i t s c e n t r o i d is
E
{ h I ; A E F).
THEOREM 9 . 4 8 .
Let
F
s i m p l e non
-
non - t r i v i a l Z y v a l u e d - d i m z n s 6 o n a l central and
b e a c o m p l e t e and
nonarchimedean f i e l d . L e t
E
be a f i n i t e
a s s o c i a t i v e algebra over
F
s u b a l g e b r a w h i c h is a s u b m o d u l e o v e r
E
. Let . Then,
W C C(X;E)
be
a
for every f E C ( X ; E ) ,
(1) - (.4) o f Theorem 9 . 4 5 a r e e q u i v a l e n t .
conditions
The p r o o f c o n s i s t s i.2 s h o w i n g t h a t , u n d e r t h e a b o v e hy-
PROOF:
W
p o t h e s i s on E l any s u b a l g e b r a over E
C C X;E)
w h i c h i s a submodule
i s a polynomial a l g e b r a . By Theorem 4 , C h a p t e r
&(El
X E
= &(El.
X
,
Jacobson
Hence t h e submodule W
By Lemma 4 . 1 ,
u
E E'
,
we
have
s i n v a r i a n t u n d e r composi-
t i o n w i t h any l i n e a r t r a n s f o r m a t i o n A = {u(f);
[31]
T
,f
e x t e n d e d t o t h e case o f
Ed(E).
E
F
W)
,A
Let
. is a v e c E o r subspace
NONARCHIMEDEAN APPROXIMATION THEORY of
C(X;F)
and
I t r e m a i n s t o p r o v e t h a t A is closed
A 8 E C P7.
under m u l t i p l i c a t i o n . Since pair
u
,vo
in
v(f)
u ( u o v o ) = 1. L e t
belong t o W , b r a of
since
C (X;E)
.
E
such t h a t
E
187
and
is not a z e r o - a l g e b r a , uovo # 0. L e t
u E El
b e i n A . The mappings
w(g)
A 8 E C W . By h y p o t h e s i s ,
W
i s a subalge-
Therefore ,
M . C a l l i t h . Then u ( h ) E A , and u ( h ) u ( u v 1 = 1. Thus W i s p o l y n o m i a l a l g e b r a . 0 0
belongs t o since REMARK.
choose a such t h a t
Notice t h a t a subalgebra
which
P7 C C ( X ; E )
t h e c o n s t a n t s i s a submodule o v e r E
.
= v(f)w(g),
Conversely,
if
contains
W
is
a
u n i t a r y s u b a l g e b r a w h i c h i s a submodule o v e r E l t h e n W contains the constants.
8
7.
BISHOP ' S THEOREM.
i s a f i n i t e e x t e n s i o n of
p l e t e , and K valuation
t
i s comt h e rank one
F b e a nonarchimedean valued f i e l d . I f
Let
+
I
t [ E IR,
of
F ,
then
F c a n b e e x t e n d e d from
i n a u n i q u e way a s a r a n k o n e v a l u a t i o n . I f
F
F
to
K
F is n o t complete,
t h e n i t s v a l u a t i o n c a n be e x t e n d e d t o a r a n k one v a l u a t i o n o f K i n f i n i t e l y many n o n - e q u i v a l e n t ways. DEFINITION 9 . 4 9 .
Let
F be a nonarchimedean valued f i e l d ;
K b e a f i n i t e u l g e b r a i c e x t e n s i o n of
F,
I.et
e n d o w e d w i t h a rank one
F . Let A c C ( X ; K ) be a subalgebra. A subset S C X is c a l l e d A - a n t i s y m m e t r i c ( w i t h r e s p e c t t o F) i f , for e v e r y a E A , a / S b e i n g F - v a l u e d i m p l i e s t h a t a l S is constant. v a l u a t i o n e x t e n d i n g t h a t of
DEFINITION 9 . 5 0 .
A-antisymmetric
Let set
x , y E X. S
We w r i t e
which c o n t a i n s b o t h
x E y
i f t h e r e is an
x and y .
188
NONARCHIMEDEAN APPROXI MATION THEORY
The e q u i v a l e n c e classes modulo t h e e q u i v a l e n c e tion
x E y
spect t o
are c a l l e d maximal A - a n t i s y m m e t r i c
sets
rela-
(with re-
.
F)
The f o l l o w i n g r e s u l t 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
i s a nonis a f i n i t e algebraic extension of
M a c h a d o ' s v e r s i o n o f B i s h o p ' s Theorem archimedean valued f i e l d ; K F, a n d
K
I n it, F
[37j.
i s v a l u e d by o n e e x t e n s i o n t o K of t h e v a l u a t i o n
i s a compact Hausdorff s p a c e and normed s p a c e o v e r K .
F; X
Let
THEOREM 9 . 5 1 .
is a
E
W C C(X;E)
be a s u b a l g e b r a ; Z e t
A CC(X;K)
For e a c h f E 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 a n A-module. t h e r e i s a maximal A - a n t i s y m m e t r i c s e t
of
nonarchimedean
( w i t h r e s p e c t t o F) S C X
such t h a t
COROLLARY 9 . 5 2 . .
Let
Then
f E C(X;E).
f
W
and
A
b e a s i n Theorem 9.52,
b e l o n g s t o t h e c l o s u r e of
e a c h maximal A - a n t i s y m m e t r i c PROOF OF THEOREM 9 . 5 1 . assume
(P
5
d
, S)
f o r any
S C X.
i f ,
F)
S C X.
d = d ( f , W ) . Wecan
clear
for
Let
be t h e s e t o f a l l ordered
D
d = 0,
x
P i s a p a r t i t i o n of
since
i n t o non-empty pairwise
d i s j o i n t and c l o s e d s u b s e t s of (ii) S E P
and
The p a i r ( { X D
the partition Q
,X)
belongs t o
(P , s )
by s e t t i n g
5
D
D
i s a maximal e l e m e n t
so if,
# 0. W e p a r -
D
and o n l y
if,
T C S . The a r g u m e n t s i n
[37])
apply here,
so
h a s a n u p p e r bound. By Z o r n ' s Lemma t h e r e (Q,T)
symmetric (wi th r e s p e c t t o
is
,
(Q , T )
i s f i n e r t h a n P , and
t h a t each c h a i n i n
X ;
d = d(f1S;LViS).
M a c h a d o ' s p r o o f o f B i s h o p ' s Theorem (see
{a E A ; a l T
C(X:E)
such t h a t
(i)
t i a l l y order
let
C ( S ; E ) , for
(with respect t o
f E C ( X ; E ) . Put
d > 0, t h e r e s u l t being
d ( f l S ; WIS) pairs
Let
set
in
W
W/S i n
and o n Z y i f , f / S b e l o n g s t o t h e c l o s u r e o f
and
E D.
F)
F - valued}
.
VJe c l a i m t h a t
Indeed,
. By
let
AT
contradiction
T
is A - a n t i -
be t h e s e t admit t h a t
189
NONARCHIMEDEAN APPROXIMATION THEORY
B = A and
T
IT
c o n t a i n s non
-
constant functions. Since
i s a B-module,
WIT
V C T
lence class
B
c c (T;F) ,
by Theorem 9 . 2 1 w e may f i n d a n e q u i v a -
(modulo
T/B)
such t h a t
d ( f / T ; WIT) = d ( f l V ; W l V ) . d = d ( f I T ; WI T) , a n d
Since tition
P of
V
is proper s u b s e t of
c o n s i s t i n g o f t h e e l e m e n t s of
X
T a n d by t h e e q u i v a l e n c e classes o f f i n e r then
Q
,
t h e maximality of
6
8.
(Q ; T ) < (P , V )
and t h e r e f o r e
T
which c o n t a i n s
,
Q
,
,
t h e par-
distinct
from
T / B
is s t r i c t l y
which
contradicts
modulo
T
T
( Q ,T ) . The maximal A - a n t i s y m m e t r i c i s then such t h a t d = d ( f 1 S ; S) .
set
S,
T I E T Z E EXTENSIOK THEOREM Let
Y
b e a c l o s e d non
s i o n a l compact Hausdorff s p a c e
- empty
s u b s e t of a
X I and l e t
d e a n Banach s p a c e o v e r a v a l u e d f i e l d
E
(F;
I
-
0
- dimen -
be a nonarchime-
1)
(i.e., E
is a
n o n a r c h i m e d e a n normed space o v e r F w h i c h i s c o m p l e t e ) . L e t
b e t h e r e s t r i c t i o n map d e f i n e d as f o l l o w s :
for a l l
f E C(X;E).
T h i s map i s o b v i o u s l y l i n e a r a n d c o n t i n u -
ous, s i n c e
for all
f E C(X;E). Let
C(X;E) [ Y
b e t h e image o f
C(X;E)
under
Ty
in
C (Y;E). THEOREM 9 . 5 3 .
i n
The v e c t o r s u b s p a c e
C(X:E) / Y
is u n i f o r m l y dense
C(Y;E).
PROOF:
L e t us d e f i n e
A = {f
E C(Y;F); f =
glY f o r some g E C(X;F)).
190
NONARCHIMEDEAN APPROXIMATION THEORY
C l e a r l y , A is a u n i t a r y subalgebra of mensional, Clearly, since W
separates the points of
A
is an A
W
C(Y;E),
DEFINITION 9 . 5 4 .
.
i
W = C(X ;E) Y
Let
for all x E XI
c o n t a i n s t h e c o n s t a n t s . By C o r o l l a r y 9.45,Wis
dense i n
uniformly
as c l a i m e d . A t o p o l o g i c a l space
U
e x i s t s two c l o p e n d i s j o i n t s e t s
i s called
X
i f , g i v e n a n y two c l o s e d d i s j o i n t s e t s
and
.
Y
M o r e o v e r , W (x) = E
- module.
i s O-di-
Since X
C(Y;F).
and
A
and
V
in
B
ultranormal
in
X
, there
such t h a t A C U
X
B C V.
LEMYA 9 . 5 5 .
Let
b e a t o p o l o g i c a l s p a c e . Then
X
X
i s ultranor-
ma2 i f , and o n l y i f , g i v e n a p a i r of c l o s e d d i s j o i n t s e t s A and B in X
, and a n o n a r c h i m e d e a n v a l u e d f i e l d f F C(X;F)
an e l e m e n t
If(x)/ 5 1 PROOF:
Let
X
f ( A ) = (0)
such t h a t
x
f o r aZZ
U
nonarchimedean v a l u e d f i e l d . L e t A C U
and
ued c h a r a c t e r i s t i c f u n c t i o n
, because
.
c l o s e d d i s j o i n t sets A f(A) = { O } ,
there
f ( B ) = (11
i s and
+
F
and
U
n
= {O
, 1)
@,(X)
*
I)
f(B)
X
and B and
(F,
I
-
be a
~
i s c l o p e n , t h e F-val-
V
is continuous. Clearly, V = $8.
.
Hence
Since
)X(,+I
i s such t h a t , given
B C V,
[ 5 1
for
a pair
of
i n X I and a nonarchimedean v a l -
t h e r e is an element = { l }
and l e t
a n d V b e two c l o p e n d i s j o i n t
+v:X
A C U
Finally,
,
X
B C V. S i n c e
C o n v e r s e l y , assume ued f i e l d (F,[
I),
be a ultranormal t o p o l o g i c a l space, l e t A and B
sets such t h a t
+V(B) = a l l x F X.
,
-
E X.
b e a p a i r o f c l o s e d ’d i s j o i n t s e t s i n
G V ( A ) = {O}
(F,I
f E C(X;F)
lf(x))
5
1
such
for all
that X
F X.
L e t us d e f i n e
and Then
i s clopen,
U
u
= { t E
x;
/ f ( t ) I < 11
v
=
{t E
x;
I f ( t ) / 11.
U = f-’(B1(O)). S i n c e B1(0) = { v E f ; I v 1 < 1 ) i s c l o p e n . C l e a r l y A C U . On t h e o t h e r h a n d , t h e
NONARCH I ME DEAN APPROXIMATION THEORY
set
i s c l o p e n too, and
S = F\B1(0)
V = f - l ( S ) . Therefore
V
0.
This endsthe
E v e r y compact H a u s d o r f f 0 - d i m e n s i o n a l
space i s ul-
c
is clopen, B
191
n
U
and t h e i n t e r s e c t i o n
V
V =
proof. LEP4MA 9 . 5 6 .
t r a n o r m a I. PROOF: A
X
Let
and
B
be a 0 - d i m e n s i o n a l compact Hausdorff s p a c e . L e t
b e t w o c l o s e d d i s j o i n t sets o n X .
a E A, t h e r e i s a clopen set
each
i n X such t h a t
Va
a E Ua,
b E
Ua
in
va,
Ua
X
Let
b E B.
n va
=
clopen set
0.
By compact-
... , a n E ... u Ua
n e s s o f A , t h e r e a r e f i n i t e l y many a l , a 2 , t h a t A i s c o n t a i n e d i n t h e u n i o n Ub = U u al Ub i s a c l o p e n s e t i n X , w i t h A c ub L e t v
.
For
a
and
=va n
such .Clearly,
A
n
... n v . .
an Vb i s a c l o p e n neighborhood o f B, t h e r e a r e f i n i t e l y many b l , b 2 , , bm E B s u c h t h a t B i s c o n t a i n e d i n t h e u n i o n V = V b U ... u V . Clearly, V is a b 1 b . By c o m p a c t n e s s o f
Then
...
clopen set i n
U
X
with
B C V.
i s a clopen set i n X w i t h
u n v
that
=
THEOREM 9 . 5 7 .
1
n
. .. n
Ubm
.Clearly,
A C U. F i n a l l y , o n e e a s i l y
Let
X
sees
b e a c o m p a c t H a u s d o r f f 0 -dimensional space.
Ty : C ( X ; E )
+
subset
C(Y;E)
Y
C X,
the continuous
linear
i s a t o p o l o g i c a l homomorphism,for
e a c h n o n a r c h i m e d e a n normed s p a c e PROOF:
bm
U = Ub
pr.
For any c l o s e d n o n - e m p t y mapping
1 Let
E .
L e t u s c o n s i d e r t h e neighborhood b a s e o f
0
in
C(X;E)
c o n s i s t i n g of a l l subsets o f t h e f o r m
for
E
> 0.
W e have t o prove t h a t f o r e a c h such N
T ~ ( N ) i s r e l a t i v e l y open i n
C(X;E!
IY
L e t then, f o r each such N ,
,
= T~(c(x;E)).
define
the
image
NONARCHIMEDEAN APPROXIMATION THEORY
192
T h i s i s an o p e n n e i g h b o r h o o d of
in
0
C(Y;E). W e c l a i m t h a t
Ty ( N ) i s r e l a t i v e l y o p e n i n t h e image Ty(C(X;E)) =C(X;E) / Y .
whence
The i n c l u s i o n (i)
T y ( N ) C W fl Ty(C(X;E)) is obvious. h E W fl Ty(C(X;E)). L e t
let that
Conversely,
g E C (X;E) be such
g(x) = h(x) f o r a l l
h = Ty(g). Therefore
x E Y . Define
Then
all
i s closed a n d d i s j o i n t f r o m Y . I n d e e d , h E W
B C X
plies t h a t
I(h(x)
x E Y . If
11
B =
<
for a l l
E,
8,
then
CJ
/Ig(x) /! <
x E Y . Hence
and t h e r e f o r e
E N,
im-
E
for
h E Ty(N).If
# 8 , b y Lemma 9 . 5 5 , t h e r e i s f E C ( X ; F ) s u c h t h a t f ( B ) = {O}, f ( Y ) = {l} a n d I f ( t ) l 5 1 f x a l l t E X . W e c a n a p p l y Lemma
B
9.55,
b e c a u s e b y Lemma 9 . 5 6 ,
is ultranormal. Let
X
x
k E C(X;E), k ( x ) = f ( x ) g ( x ) = h ( x ) , i f We claim that
Therefore that
k E N.
Let
too. T h e r e f o r e
k(t) = 0
Ilk(t)
i.e.
k E N,
11
<
E
t E X.
for ail
From Ty(N)
and l e t
( i )a n d
Y C X
Let
t
If
E X,
then
h=Ty(k). 0 , so
f(t)
t $ B, t h e n w e h a v e
and
k E N.
T h i s shows
X
C Ty(N).
( i i ) ,i t f o l l o w s t h e d e s i r e d
is r e l a t i v e l y open,
THEOREM 9 . 5 8 .
E.
k = f g . Then
Therefore
h E T y ( N ) . T h i s ends t h e p r o o f t h a t
(ii) W n Ty(C(X;E)
and
t E B,
If
/k(t)/ <
E Y.
equality,
Q E D.
b e a c o m p a c t H a u s d o r f f 0 -dimensionuZ space,
be n n o n - e m p t y
n o n a r c h i m c d c a n Banach s p a c e E,
cZosed s u b s e t .
Then,
over valued f i e l d
for
(F;/
I),
each we
193
NONARCHIMEDEAN APPROXIMATION THEORY
have: C ( X ; E ) I Y = C (Y;E)
PROOF: i.e.
a l l w e have t o prove is t h a t
By Theorem 9 . 5 3 ,
i s closed i n
Let
C(Y;E).
.
be t h e k e r n e l o f
K
K = { f E C(X;E); T y ( f ) = 0 }
l i n e a r mapping, t h e k e r n e l
K
.
Since
Ty
C(X ;E)
,
Continuc
IS
in
is a
Ty
C(X;E) Y
i s a closed s u b s p a c e o f t h e nonar-
C ( X ; E ) . Hence t h e quotient space C(X:E) / K
c h i m e d e a n Banach s p a c e
i s a Banach s p a c e t o o , a n d t h e r e f o r e c o m p l e t e . By Theorem 9 . 5 7 ,
t h e mapping
C(X;E) / K
momorphism. Hence
and
Ty(C X;E)
t o p o l o g i c a l l y l i n e a r l y isomorphic. Thus and t h e r e f o r e c l o s e d i n REMARK 9 . 5 9 .
When
i s a t o p o l o g i c a l ho-
Ty
= C(X;E)
C (X;E)
IY
E = F
and F
i s thl
w i t h t h e e x t e n s i o n of
/
- adic *
IP
valuation
from Q t o
t h e n Theorem 9 . 5 8 i s d u e t o J . D i e u d o n n s (see Th&r&ne
REMARK 9 . 6 0 .
When
E = F
and
num-
f i e l d of p - a d i c
b e r s , i . e . t h e c o m p l e t i o n of f i e l d Q w i t h t h e p
,
are
C(Y;E).
d e f i n e d i n Example 9 . 3 , F
IY
i s complete,
F i s a l o c a l l y compact
1,[7O]k (hence
c o m p l e t e ) n o n a r c h i m e d e a n n o n t r i v i a l l v v a l u e d f i e l d , t h e n Theorem 9.58 i s valid without t h e hypothesis t h a t enough t o assume t h a t
X
X
be c o m p a c t .
is ultranormal, and t h e n t h e
It is
conclusion
is t h a t
f o r any c l o s e d s u b s e t Y C X. T h i s v e r s i o n o f t h e T i e t z e extens i o n Theorem i s d u e t o R. L . E l l i s , A n o n a r c h i m e d e a n a n a l o g o f t h e T i e t z e - U r y s o h n e x t e n s i o n Theorem, I n d a g a t i o n e s M a t h . , p.
70,
3 3 2 - 333.
59.
THE COMPACT
DEFINITION 9 . 6 1 . (F,]
1).
-
OPEN TOPOLOGY.
Let
A mapping
(1)
p(x)
E be a v e c t o r space o v e r a valued p :E 0
-+
IR is c a l l e d a s c m i n o r m o n E
f o r aZ2
x
E E;
field 7:s
194
NONARCH IMEDEAN APPROXI MAT1ON THEORY
Let
field
(F,I
b e a v e c t o r space o v e r a n o n a r c h i m e d e a n v a l u e d
E
I).
*
r
Let
b e a f a m i l y o f n o n a r c h i m e d e a n seminorms
on E . W e d e f i n e a t o p o l o g y neighborhoods of
{x E E ; pi(x) 5 where
~
pology
T
by s e t t i n g as
on E
T
a
b a s i s of
t h e s e t s o f t h e form
0
E
i = 1,2,...,nI
I
E rr i = l I 2 , . . . , n , a n d i s d e t e r m i n e d by t h e f a m i l y
i
> 0. W e s a y t h a t t h e to-
E
r
.
Then
(E,T)
is a to-
p o l o g i c a l v e c t o r space over F , i.e. t h e following is t r u e t h e map ( x , y )
(i)
++
-
continuous; ( i i ) t h e map
(A
,x)
+
x Ax
y
of
of
F
E x E
X
E
into
into E
E
is
i s con-
tinuous. DEFINITION 9.62.
valued f i e l d
Let
(F,
1
b e a v e c t o r s p a c e o v e r a nonarchimedean
E
I).
A subset
i s s a i d to b e F - c o n u e r
X C E
i f t h e f o l l o w i n g is t r u e : a x + B y + y z F X
~ y 5 i 1, and
REMARK. E
over
and
a
foraZ1
+ 6 +
x , y , z E X
y = 1
F o r e v e r y n o n a r c h i m e d e a n seminorm p o n a v e c t o r s p a c e (F,
I
-
1)
{x E E; p ( x
t h e f o l l o w i n g sets a r e c o n v e x { x E E; p ( x - x o ) <
I
-
D E F I N I T I O N 9.63.
valued f z e l d
(F,I
xo)
Let *
1).
2 E
E}
I
for all
xo E E , a n d
E
E}
> 0.
b e a v e c t o r s p u c e o v e r a nonarchimedean A topology
T
t h a t makes
(E
,
T)
a
N0NA R CH I NE DE AN AP P R 0 X I MAT I 0 N T H E OR Y t o p o l o g i c a l v e c t o r space o v e r
F
195
i s s a i d t o b e l o c a l l y F-eonvex
i f t h e r e e x i s t s a fundamental s y s t e m o f neighborhoods o f s i s t i n g o f F-convex s e t s . It
r
f o l l o w s t h a t a n y t o p o l o g y d e t e r m i n e d by
o f seminorms o n
E
is l o c a l i y
Let
( S e e Monna
be a Hausdorff space.
X
a
family
F - convex. Conversely, one can
show t h a t t h e c o n v e r s e is a l s o t r u e . DEFINITION 9 . 6 4 .
0 con-
[73] )
.
Let
(F,!
1)
a nonarchimedean v a l u e d f i e l d . For e v e r y compact s u b s e t
be
K C XI
let
f E C(X;F).
for
One e a s i l y v e r i f i e s t h a t norm o n
PK
i s a nmarchimedeansemi-
C ( X ; F ) . The c o m p a c t - o p e n t o p o l o g y o n
is
C(X;F)
l o c a l l y F - c o n v e x t o p o l o g y d e t e r m i n e d by t h e f a m i l y o f
r
= {PK; K C X
compact }
the
seminorms
.
More g e n e r a l l y , i f E 1 s a n o n a r c h i m e d e a n l o c a l l y F c o n v e x s p a c e , whose t o p o l o g y i s d e t e r m i n e d by a f a m i l y r of nonE l szc d e f i n e s a c o r r e s p o n d i n g f a m i l y
a r c h i m e d e a n seminorms o n o f seminorms o n
for
f E C(X;E)
C ( X ; E ) by s e t t i n g
,
p E
r
and
compact- o p e n t o p o l o g y o n
K
c
X
C(X;E).
a compact s u b s e t . T h i s is t h e I n p a r t i c u l a r , when E
n o n a r c h i m e d e a n normed a l g e b r a , w i t h norm
t ++ 11 t
11 ,
is
a
the semi-
norms
on
C(X;E)
have t h e p r o p e r t y (i)
for all
f
,g
PK(fg)
E c(x;E).
5 PK(f) If
E
PK(g)
is unitary, with u n i t
e,
I/
el/ =1,
196
NONARCHIMEDEAN APPROXIMATION THEORY
x
then the constant function u n i t of
,
C(X;E)
-
e , s t i l l d e n o t e d by
e
,
is the K C X , one h a s
a n d f o r e v e r y compact s u b s e t
(ii) p K ( e ) = 1
I n view o f p r o p e r t i e s ( i ) and ( i i ) ,o n e s a y s t h a t t h e PK a r e a l g e b r a s e m i n o r m s . A s a C o r o l l a r y m u l t i p l i c a -
seminorms tion i n
is continuous,
C(X;E)
Therefore
with t h e compact-open topology i s
C(X;E)
termed a n o n a r c h i m e d e a n t o p o l o g i c a l a l g e b r a . I t i s e a s y t o t h a t t h e c l o s u r e of a n a l g e b r a , or o f a r i g h t in
is also a subalgebra or a r i g h t
C(X;E)
C(X;E).
( r e s p . l e f t ) ideal
(resp. l e f t ) idealin
-
The p r o b l e m a r i s e s o f c h a r a c t e r i z i n g t h e compact
closure of a subalqebra or of a r i q h t
open
( r e s p . l e f t ) i d e a l i n C(X;E),
C ( X ; F ) , s i n c e by p r o p e r t y ( 3 ) of D e f i n i -
and i n p a r t i c u l a r i n tion 9.1,
see
any nonarchimedean v a l u e d f i e l d
(F,I
1)
i s a unitary
nonarchimedean normed a l g e b r a o v e r i t s e l f . THEOREM 9.65.
Let
be a Hausdorff s p a c e . L e t
X
be a u n i t a r y
E
n o n a r c h i m e d e a n normed a l g e b r a o v e r a v a l u e d f i e l d Let 31)
I
1).
( i n t h e s e n s e of D e f i n i t i o n 9 .
be a s e p a r a t i n g
A C C(X;E)
u n i t a r y s u b a l g e b r a of
(F,
C(X;E),
t o r subspace which i s an A - m o d u l e .
and l e t Then W
be avec-
W C C(X;E)
is' l o c a l .
B e f o r e p r o v i n g Theorem 9.65 l e t u s d e f i n e what w e man by s a y i n g t h a t . W
is local.
DEFINITION 9 . 6 6 .
Let
X
b e a H a u s d o r f f s p a c e , and l e t
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 v a l u e d f i e l d W CC(X;E).
We s a y t h a t
W
i s l o c a l i f any
l o c a l l y a t a l l p o i n t s of X
in W
(F,I
*
E
be
1).
a Let
f E C(X;E) which i s
i s t h e n i n t h e compact
- open
c l o s u r e of W . Notice t h a t , s i n c e a l l functions
f
in
m,
W l o c a l l y a t a l l points of
l o c a l i f and o n l y i f
{
x 1
i s c o m p a c t , f o r any x E X, W , are i n
C X
t h e compact-open c l o s u r e of
c
A(W)
X =
,
w.
c.
i.e., A(W) 3
w.
Therefore
r=
PROOF OF THEOREM 9.65. and
E
> 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
b e a s i n Theorem 9 . 6 5 .
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
NONARCHIMEDEAN APPROXIMATION THEORY
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)).
b e a s i n Theorem 2 . 7 0 .
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
+ $2 +
For each
(d)
t E X
for all
~
...
+ Q n = l
i = 1,2,...,n
l @ i ( t-) hi ( t )I <
choose E
( i= l r 2 , . . . , n ) ;
on hi
/PQ(gx, )
X . E A
Define
g
,h
E Cb(X;E)
by
,
h =
r
P (g
n g =
Gigx i=l i C
n C h . g i=l 1 xi
W e claim that
(el
P$(f - g ) 5
Indeed, t a k e
E
t E X.
4
Then
-
h)
such t h a t
for all
1
E
-
E X
t E X.
202
NONARCHIMEDEAN APPROXIMATION THEORY
Now f o r t h o s e
i E {1,2,...,n}
t
such t h a t
Kx,
E
we
have
1
I @ i ( t )=l 0 , a n d f o r t h o s e / Q i ( It ) 5 1 while
then
l l @ ( t )( f ( t ) -
( t ) )1 1 <
CJ,
,
s u c h that t E Kx
i E {1,2,...,n}
E
I
. This
shows
i
t h a t i n any c a s e
II @ and t h e r e f o r e
P
@
On t h e o t h e r h a n d ,
Hence
P+(g
-
h)
5
E
.
From ( e ) i t f o l l o w s t h a t
P
I t remains t o n o t i c e t h a t
belongs t h e B
- closure
COROLLARY 9 . 7 3 .
f
E
Let
A
of
W
in
and W b e a s
in E f o r e a c h x -dense. t h e c o n s t a n t s , t h e n W is Since A
h)
h E W,
f ( x ) E w(x)
PROOF:
-
(f
in
-
is separating,
c h a r a c t e r i s t i c f u n c t i o n of
@,(XI
and
=
E
.
t o conclude t h a t
f
X
in
W E X.
U
,
Then
9.70.
i f , a n d only
Cb(X;E)
M o r e o v e r , if W c o n t a i n s
is 0-dimensional.Let @"
U
of
xEX.
x i n X . The
, belongs then t o
Co(X;F)
1 > 0 . Hence t h e c o n d i t i o n i s n e c e s s a r y .
Conversely, l e t in
E
Theorem
Choose t h e n a compact a n d o p e n n e i g h b o r h o o d F
5
Cb(X;E).
zs i n t h e B - c l o s u r e of
Cb(X;E)
if,
$
f o r each
x
E X.
f
E
Cb(X;E)
Thus g i v e r .
x
b e s u c h t h a t f (x)
E X
and
E
E
> 0, there
W(x)
is
203
NONARCHIMEDEAN APPROXIMATION THEORY
( x ) - g(x) / I < c . By c o n t i n u i t y t h i s is still t r u e i n a neighborhood U of x i n X.Thus f E A b ( W ) . By Theo-
g
r e m 9.70, E
X
,
w,
f F If
x
11 f
such t h a t
E W
- c l o s u r e of
the
W
in
Cb(X;E).
W(x) = E
contains t h e constants, then
W
a n d by t h e a b o v e a r g u m e n t ,
COROLLARY 9 . 7 4 .
Let
fiausdorff space. Let d u l e a n d for e a c h
x
be
X
u let
X,
is D -dense.
W
O- dirnc .iz:.ionn?
P I C Cb(X;E) E
Mx
for all
compact
ioro/l,y
be a B -cZosed
Cb(X;F) - m o -
of
be the closure i n E
the
set
M(x) = { f ( x ) ; f t h e n >Ix
PROOF:
E MI;
i s a c l o s e d v e c t o r subspace
Since
of E
and
i s 0 - d i m e n s i o n a l , t h e u n i t a r y a l g e b r a Cb(X;F)
X
i s s e p a r a t i n g , and by t h e p r e c e d i n g c o r o l l a r y , M = { f E C b ( X ; E ) ; f ( x ) E P4x
Since
-
M is B-closed,
v e c t o r s u b s p a c e of
COROLLARY 9 . 7 5 .
E
Lct
PI
= PI.
for all
The f a c t t h a t e a c h
Mx
x
E X
1.
is a closed
is easy to establish.
X
b e as i n C o r o l l a r y 3 . 7 . 1 a n d l e t
E
be a
u n i t a r y n o n a r c h i m e d e a n normed a l g e b r a . Assume t h a t E i s s i m p l e . Then any B,-closed
two- sided ideal i n
f u n c t i o n s v a n i s h i n g o n a c l o s e d s u b s e t of X
mal t w o - s i d e d
6 -closed ideal i n
{f E Cb(X;E)
f(x) = 0 }
PROOF:
If
;
N C X
.
M o r e o v e r , a n y maxi-
i s of t h e f o r m
Cb(X;E)
f o r some p o i n t
consists o f a l l
Cb(X;E)
x
in
X
.
is a closed subset, clearly
I ( N ) = {f E C b ( x ; ~ ) ; f ( x ) = 0 f o r a l l
i s a a - c l o s e d t w o c sided i d e a l . there is
@ E Co(X;F)
with
@(XI
(Recall t h a t given any > 0.)
x
E
N}
x
E X,
204
NONARCHIMEDEAN APPROXIMATION THEORY
If I C Cb(X;E) fine the closed set N
=
{t
E
is a 6-closed two- sided ideal, de-
X; f(t)
=
0
for all
f
E
I1 .
One easily sees that I c I(N). To apply the preceding corollary f E I and we must show that I is a Cb(X;F)-module. Let g E Cb(X;F). Then x H g(x)e belongs to Cb(X;E). Call it h. Clearly g f = hf E I . Assume now that f E I(N), while f g I . By Corollary 9 . 7 4 there exists x E X such that f (x) $ Ix . Hence f (x) # 0. Thus x )? N , because f E I(N). On the other hand, since E is implies simple, either Ix = {O} or Ix = E . Now f(x) ?j! Ix Ix = {O} . Thus I(x) = 0 shows I(N) C I .
, i.e. x
E N
I
a contradiction.
This
For further results on Cb(X;E) with the strict REMARK 9 . 7 6 . topology 6 , and for more general nonarchimedean Nachbin spaces see the Doctoral Dissertation of J O S ~P. Carneiro, Universidade Federal do Rio de Janeiro, 1 9 7 7 . In fact, Theorem 9 . 7 0 is a Corollary of his result on localizability in the bounded case of the nonarchimedean Bernstein - Nachbin problem, dealing with not necessarily separating subalgebras A C Cb(X;F).
NONARCHIMEDEAN APPROXIMATION THEORY
REFERENCES FOR CHAPTER 9. MACHADO and PROLLA
[39]
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i 701
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KAPLANSKY P40NNA MURPHY NACHBIN
[ 72 ]
[73] [74] [75!
NARICI, BECKENSTEIN a n d BACHMAN PROLLA
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205
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SYMBOL INDEX
..................... B .......................... T ' .......................... CA(X;F) .................... C(X;E) ..................... Cb(X;E) .................... Co(X;E) .................... C(X) ....................... Cb(X) ...................... C(Eu;F) .................... CVm(X;E) ................... C V m ( X ) ..................... C(Xl) 8 C(X2) .............. C(X) @ E ................... cS(E) ...................... r 1 ......................... r dl ......................... r s ......................... A(W) ....................... B(W) ....................... A g ( W ) ...................... A o ( W ) ...................... A(K;E)
...................... A W ......................... A x ......................... E' ......................... Al(W)
146 80
EA E
1 E
17
E
....................... 138 Q F .................... 140 t3€ F ................... 141 F ................... 1 4 1 &J€
75
.................... 139 E(S) ..................... 27 1 1 f 1 1 K,p ................ 1 I / f 1 1 ................... 1 1 1 f 1 1 v, ................ 79 G(A) ..................... 82
79
G(W)
1 E 1
79 1
2
79 46 14 1
81 81 81
67 173 128 119 67 67 28 1
E
F
.....................
82
.................... 82 H(U;E) ................... 148 H(U) ..................... 148 H(U) Q F ................. 148 ........................ 80 xA....................... 23 K(X;E) ................... 79 k R ...................... 141 IK ....................... 1 LA(W) .................... 3 z e ( E b ; F) .............. 138 Z f ( n E ; F) .............. 6 1 2 (El ................. 65 mod . S ................... 2 P A ....................... 81 G(W)*
K
214
SYMBOL I N D E X
............................
22
.......................... 23 Q ( E ; F) ................... 61 9; ( E ; F ) ................. 61 q f ( E ; F) .................. 61 m .......................... 1 u ........................... 79 G ........................ 2 2 v ........................... 79 [x] ......................... 16 x : y (m0d.S). .............. 2 wv .......................... 91 .......................... 81 d .......................... 81 R z .............. ............ 8 1 Pp
. . ..
'
INDEX
A
A-module,
Bounded case, 89
3,7,25
A n a l y t i c c r i t e r i o n , 88
Bounded f u n c t i o n ,
A n t i s y m m e t r i c s e t , 23
Bourbaki, N.,
Approximation by polyno-
B r i e m , E.,
m i a l s , 69
17
Brosowski, B . ,
Approximation problem,
Buck, R . C . ,
Bernstein-Nachbin, Approximation property,
80 40, 133,
1
96, 132 1 2 3 , 124
28,
2 0 , 21,
28, 32,
80, 95, 1 2 7 , 130, 1 3 2 , i 65
l o c a l i z a t i o n of, 149 A r c h i m e d e a n v a l u a t i o n , 154 , A r e n s , R.F.,
11, 2 7 ,
Arens-Kelley
Theorem,
Aron, R.M.,
C
32 Csc, N.P.,
32
98
C a r n e i r o , J.P.Q.,
60, 148
204
Cay l e y - D i c k s o n a l g e b r a , i 6 C e n t r a l algebra, 65
B
Centroid, 65 Bachman, G . ,
159
Chalice, D.R.,
Beckenstein, E.
,
Bernstein, S . ,
107, 108
159
Bernstein-Nachbin
t i o n , 173
approxima-
t i o n p r o b l e m , 80 Bierstedt, K.D.,
18
C h a r a c t e r i s t i c cross-secC h e r n o f f , P.R.
159,162
C l i f f o r d a l g e b r a , 66
17, 59, 146,
149
Closed c o n v e x h u l l ,
27
C l o p e n s e t , 156
B i e q u i c o n t i n u o u s Convergence,
Cohen, P . ,
T o p o l o g y o f , 140
Collins, H.S.,
Bimodule,
Compact-open t o p o l o g y , 1,8 0 ,
186
Bishop, E . ,
1 9 , 2 2 , 23
66, 67,
o f , 1, 79 Convex h u l l , 27
135, 188 Boas, R . P . , J r . , 109 J., 57,
Compact o p e r a t o r s , 74 C o n t i n u o u s f u n c t i o n s , spaces
22
B i s h o p ' s Theorem, 1 9 , 2 3 , 7 2 ,
Blatter,
133
1 93
B i s h o p a n t i s y me t r i c de c omposition,
50
Cross-section, 70
25
Cunningham, F . , J r . , 28,
30
216
INDEX
C(X)-modules,
1 3 , 1 5 , 1 6 , 26,
72
Fremlin, D.H.,
133
Full algebra, 173
Cb(X),modules,
130
Functionals
,
extreme, 123
Fundamental w e i g h t s ,
107,
108, 112
D D e L a F u e n t e Antunez, A.,
52,
G
53, 57, 6 6 , 71 D e La madrid, J . G . ,
133
Garling, D.J.H.,
133
D e B r a n g e s , L., 1 8
G e n e r a t o r s , s e t o f , 82
Deutsch, F.,
G l i c k s b e r g , I . , 1 8 , 20, 23,
28, 123, 1 2 4
Dieudonng, J . , 4 6 , 9 8 , 1 5 9 , 1 8 2 ,
90, 135 Grothendieck, A.,
19 3 Dieudonnh , Theorem o f
,
46
Directed s e t o f w e i g h t s ,
148 79
133
Dorroh, J . R . ,
96, 138,
H
D u g u n d j i , J . , 55 Hadamard's p r o b l e m , Haydon, R . G . ,
E
Hewitt Ec,
property, 53
E i f f e r , L., Ellis,
R.L.,
50,
E.,
Hull, Stone-Weierstrass, 119, el28
147
Enflo, P.,
,
107
129, 133
193 144
I
Epsilon product,
138 Ideals, 13, 65, 169, 179,
E s s e n t i a l , 29 E x t e n s i o n Theorems
,
52
203
E x t r e m e f u n c t i o n a l s , 27
I n g l e t o n , A.W.,
Extreme p o i n t s ,
In] ective Tensor p r o d u c t ,
27
183
141
F J
F-convex,
194
F i b r a t i o n , v e c t o r , 25 F i e l d w i t h v a l u a t i o n , 153 F i g i e l , T.,
74
Fontenot, R.A.,
133
Jacobi i d e n t i t y , 66 J a c o b s o n , .N. , 66
J e w e t t , R . I . , 4,6 J o h n s o n , W.B., 74
67,
21 7
INDEX
K
N
Kaplansky, I., 13, 156, 159,
Nachbin, L.,
168, 169
3 , 6 , 35, 6 0 , 7 9 ,
80, 81, 88, 89, 108,
K a p l a n s k y ' s Lemma, 1 5 9
111, 1 1 3 , 1 1 5 , 1 6 5
K a p l a n s k y ' s Theorem, 1 5 6 , 1 7 7 , 178
,
185 N a c h b i n space, 79
Kelley, J.L., Kleinstuck,
27
N a r i c i , L.,
90
G.,
Krull, W.,
159
Nonarchime'dean, a p p r o x i m a t i o n
159,
theory, 153
K r u l l v a l u a t i o n , 159
l o c a l l y F-convex
k m -space,
space, 1 9 5
141
norm, 1 6 3 L
normed a l g e b r a ,
173
normed s p a c e , 1 6 3 Laursen, K.B., Lie algebra,
17
s e m i n o r m , 194
66
topological algebra,
local, 173
196
l o c a l i z a b i l i t y , 3, 1 1 5
v a l u a t i o n , 153
l o c a l i z a b i l i t y , s h a r p , 82 l o c a l i z a b l e under A, l o c a l l y F-COnVex,
v a l u e d f i e l d , 153
3
Non-associative
195
Non-locally M
Norm,
6 , 21,
Machado S . ,
5 7 , 74,
22,
8 1 , 1 6 7 , 172
nonarchimedean, 163
t r i v i a l , 163
,
Normed space, 162
Mandelbrojt, S.,
107 0
144,
M e r g e l y a n , S.N.,
5 9 , 109
P d e r g e l y a n ' s Theorem, 5 9 , 1 0 7
,
0 s t r o w s k i Is Theorem,
1 1 0 , 146
Module,
c o n v e x s p a c e s , 43
162
23, 24,
188
Meise, R . ,
a l g e b r a , 64 ,
172
154
P
3,7,
Monna, A.F.,
Morris, P . D . ,
28, 1 2 3 , 1 2 4
Multiplication algebra, Murphy, G . J . ,
p- a d i c ,
1 5 4 , 195
174
65
n u m b e r s , 193 v a l u a t i o n , 153 p a r t i t i o n of u n i t y , 8
21 8
INDEX
17 Pe&czyiiski, A . , 57 P - l o c a l i z a b l e , 80 p o l y n o m i a l , 61 Pedersen, N.W.,
Polynomial a l g e b r a ,
-sections,
173
81 S h a r p l y P - l o c a l i z a b l e , 81 S h i l o v , G . , 24 Shuchat, A.H., 43, 44 s i m p l e a l g e b r a , 14 S i n g e r , I . , 28, 98 S i n g e r , Theorem of, 31 S p a c e C,,(X ; E l , 80, 127 S p a c e Co(X : E l , 79, 118 S t e g a l l , C., 74 Stone, M.H., 7, 9, 54, 67, 69 Stone-Weiers t r a s s , Sharp l o c a l i z a b i l i t y ,
57, 182
lSt k i n d , 63 2rd k i n d , 6 3
57 21, 24, 57, 69, 74, 81, 90, 106, 167, 172 P r o p e r t y E c , 53 P r e n t e r , P.M.,
P r o l l a , J.B.,
Q Q u a s i - a n a l y t i c c l a s s , 107, 112 Q u a s i - a n a l y t i c c r i t e r i o n , 89 R
rapid1.y d e c r e a s i n g a t i n f i n i t y ,
81, 107 159, 162 R e s t r e p o , G., 76 Roy, N . M . , 28, 30, 32 Rudin, W., 8, 17, 18, 146 Rasala, R.A.,
h u l l , 67, 119, 128 s u b s p a c e , 67, 120,
129 , Theorem, 7, 9, 69,
119, 122; 127, 128, 156, 165 S t r i c t t o p o l o g y , 80, 196 Strobele, W.J., 28, S t r o n g s e t of g e n e r a t o r s , 82 S m e r s , W,H., 90, 98 T
T e n s o r p r o d u c t , 140 S
144, 145 S c h a f e r , R.D., 65 S c h o t t e n l o h e r , M . , 60, 148 S c h w a r t z , L., 139 , 140 , 144, 145 S e l f - a d j o i n t , 10 Seminorm, 193 Separating, 2 Schaefer, H.H.,
S e p a r a t i n g a l g e b r a of cross-
Theorem o f
32 B i s h o p , 23, 72, 137, 188 Dieudonng, 46, 193 Arens-Kelley,
Dieudonn&Kaplansky,
156 Mergelyan, 1 1 0 , 146 Rudin, 17 S i n g e r , 31
59, 107,
INDEX
Stone-Weierstrass
,
7, 9 ,
69, 119, 1 2 2 , 127, 156,
16 5
21 9
W a t e r h o u s e , !'J.C.,
159, 162,
169 Weakly c o n t i n u o u s
,
76
T i e t z e , 53, 1 2 1 , 189
Wells, J . , 9 8 , 1 2 7
Weierstrass , 69 , 1 8 5
Weiers t r a s s a p p r o x i m a t i o n Theorem, 6 9 , 1 8 5
Todd, C .
,
127
Weight,
Topology ,
25, 79
W e i g h t e d a p p r o x i m a t i o n , 79 1, 80 ,
compact-open,
hlulbert, D.,
57
195,
s t r i c t , 80,
127,
uniform,
7 9 , 118
2,
w e i g h t e d , 79
U l t r a m e t r i c i n e q u a l i t y , 1 5 4, 155 Ultranormal, 190 Uniform conve r g e n c e , t o p o l o g y 79, 118
Uniform Topology,
2,
79, 1 1 8
upper semicontinuous vector s p a c e of cross-sections,
26, 1 7 1 V
Valuation , archimedean, 154 n o r n a r c h i m e d e a n , 1 53
trivial, 153 Valued f i e l d s , 153 V a n i s h e s a t i n f i n i t y , 79 Vector f i b r a t i o n s ,
w Warner, S . ,
142
Zapata, G . I . ,
107, 110
zero algebra,
65
z e r o dimensional space, 155
U
o f , 2,
n
25, 1 7 1
