APPROXIMATION OF CONTINUOUSLY DIFFERENTIABLE FUNCTIONS
NORTH-HOLLAND MATHEMATICS STUDIES Notas de Matematica (112)
Editor: Leopoldo Nachbin Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro and University of Rochester
NORTH-HOLLAND -AMSTERDAM
NEW YORK 'OXFORD 'TOKYO
130
APPROXIMATION OF CONTINUOUSLY DIFFERENTlABLE FUNCTIONS Jose G. LMVONA Facultad de Matematicas UniversidadComplutensede Madrid Madrid, Spain
YHc 1986
NORTH-HOLLAND -AMSTERDAM
NEW YORK OXFORD *TOKYO
(cl
Elsevier Science Publishers B.V., 1986
All rights reserved. No part of this publication may be reproduced, storedin a retrievalsystem, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.
ISBN: 0 444 70128 1
Publishedby: ELSEVIER SCIENCE PUBLISHERS B.V. P.O. BOX 1991 1000 BZ AMSTERDAM THE NETHERLANDS Sole distributors for the U.S.A. and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 52 VAN D ER B ILT AVE N U E NEW YORK, N.Y. 10017 U.S.A.
Library of Congress Catalogingin-F'ubliertion Data
Llavona, Joe6 G. Approximation of continuously differentiable functions. (Notas de matem6tica ; 112) (North-Holland mathematics studies ; 130) Includes index. 1. Differentiable functions. 2. Approximation theory. 3. Banach spaces. I. Title. 11. Series: Notas de matedtica (Rio de Janeiro, Brazil) ; no. 112. 111. Series: North-Holland mathematics studies ; v. 130. QU.N86 no.ll2 CQ4331.53 510 s C515.83 86-19924 ISBN 0-444-70128-1
PRINTED IN THE NETHERLANDS
To Ana, A ida and Bea
This Page Intentionally Left Blank
vii
The purpose o f t h i s book i s t o expose t h e b a s i c r e s u l t s about a p p r o x i m a t i n g c o n t i n u o u s l y d i f f e r e n t i a b l e r e a l f u n c t i o n s . The f i r s t chapt e r r e f e r s t o f u n c t i o n s d e f i n e d on m a n i f o l d s l o c a l l y o f f i n i t e dimension, and i n c l u d e s , among o t h e r t h i n g s , N a c h b i n ' s theorem about d e s c r i p t i o n o f dense subalgebras i n t h e a l g e b r a o f c o n t i n u o u s 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 i n t h e s p i r i t o f Weierstrass-Stone theorem f o r continuous f u n c t i o n s
,
p u b l i s h e d i n 1949; and a l s o d e n s i t y theorems f o r t o p o l o g i c a l and polynomial a l g e b r a s o f c o n t i n u o u s 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 . The r e s t o f t h e book i s devoted t o t h e a p p r o x i m a t i o n o f c o n t i n u o u s l y d i f f e r e n t i a b l e funct i o n s on a Banach space. There has been c o n s i d e r a b l e i n t e r e s t d u r i n g t h e l a s t few y e a r s i n f u n c t i o n t h e o r y i n i n f i n i t e dimensional spaces, and i n p a r t i c u l a r t o a p p r o x i m a t i o n o f " c o m p l i c a t e d " f u n c t i o n s d e f i n e d on a Banach space by " s i m p l e r " o " n i c e r " f u n c t i o n s . For example, i n t h e complex case, t h e r e has been work done on polynomial a p p r o x i m a t i o n o f a n a l y t i c f u n c t i o n s
,
d e f i n e d on Runge o r p o l y n o m i a l l y convex s e t s i n i n f i n i t e dimensional spaces.
I n t h e r e a l case, t h e r e has been i n t e r e s t i n t h e general problem
o f a p p r o x i m a t i n g i n one o f s e v e r a l t o p o l o g i e s , c e r t a i n c l a s s e s o f d i f f e r e n t i a b l e f u n c t i o n s by smoother ones, such as p o l y n o m i a l s o r r e a l a n a l y t i c f u n c t i o n s . I n t h i s book we make a s y s t e m a t i c s t u d y o f t h i s problem w i t h r e s p e c t t o f i v e t o p o l o g i e s o f normal use, and a l s o o f t h e c l a s s e s o f c o n t i n u o u s 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 a s s o c i a t e d w i t h them. We p r e s e n t t h e v e r s i o n s o f Whitney and Nachbin theorems f o r i n f i n i t e dimensional spaces.
F i n a l l y we show i m p o r t a n t r e s u l t s about homomorphisms i n a l g e b r a s
o f c o n t i n u o u s 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 and a v e r s i o n o f t h e Paley-Wiener -Schwartz theorem i n i n f i n i t e dimensions. To summarize, we can say t h a t t h e main o b j e t i v e o f t h i s book i s t o present, t a k i n g t h e c l a s s i c r e s u l t s o f t h e t h e o r y as a s t a r t i n g p o i n t ,
viii
Foreword
t h e d i f f e r e n t contributions in t h e l a s t few years of mathematicians such as Abuabara, Aron, Bombal, Ferrera, Gomez, Guerreiro, Lesmes, Nachbin, P r o l l a , Restrepo, Sundaresan, Valdivia, Wells, Wulbert, Zapata and myself among o t h e r s . The main f e a t u r e s of t h i s book a r e : 1.- For the f i r s t time the work knits together some important
and very recent r e s u l t s in approximation of continuously d i f f e r e n t i a b l e functions such a s : extension of Wells' theorem a n d Aron's theorem f o r t h e f i n e topology of order m ; extension of B e r n s t e i n ' s and Weierstrass' theorems f o r i n f i n i t e dimensional Banach spaces ; extension of Nachbin's and Whitney's theorem f o r i n f i n i t e dimensional Banach spaces ; automatic continuity o f homomorphisms in algebras of continuously d i f f e r e n t i a b l e functions ...e t c .
2.- The book describes some of t h e most important moderin features of a very rapidly expanding a r e a , which abounds in q u i t e i n t e r e s t i n g and challenging oper: problems.
3 .- Very a c c e s s i b l e . Sel f-cont.ained. A more d e t a i l e d d e s c r i p t i o n of the book:
Chapter I shows the most important general r e s u l t s about approximation of continuously d i f f e r e n t i a b l e functions on real manifolds locally of f i n i t e dimension. I t s t a r t s with Weierstrass' theorem about polyng mial approximation o f continuously d i f f e r e n t i a b l e functions and shows Nachbin's theorem about dense subalgebras i n the algebra of Cm functions endowed with the compact open topology. I n order t o study the problem of describing dense subalgebras in topological algebras of continuously diff e r e n t i a b l e functions , we introduce m-admissible algebras a n d c h a r a c t e r i z e m-admissible algebras among t h e i r closed subalgebras. Finally we study modules on strongly separating a l g e b r a s , obtaining a description of dense polynomial algebras r e l a t e d t o Stone a n d Nachbin conditions.
T h e r e s t o f t h e book i s devoted t o t h e approximation of continu ously d i f f e r e n t i a b l e functions on a Banach space E . Chapter I1 i s dedicated t o approximation f o r the f i n e topology of order m. Wells' a n d Aron's theorems a r e extended a n d we present a nonl i n e a r c h a r a c t e r i z a t i o n of superreflexive Banach spaces.
Foreword
ix
Chapter I 1 1 b r i n g s o u t s e v e r a l r e s u l t s on a p p r o x i m a t i o n f o r t h e compact-compact t o p o l o g y o f o r d e r m, and f u r t h e r m o r e a c h a r a c t e r i z a t i o n of f i n i t e t y p e continuous p o l y n o m i a l s space c o m p l e t i o n f o r t h i s t o p o l o g y . Chapter I V i s an e x h a u s t i v e s t u d y concerning t h e p r i n c i p a l spaces of weakly continuous f u n c t i o n s on Banach spaces. The bw-topology and t h e c o m p l e t i o n o f these spaces a r e s t u d i e d . S p e c i f i c a l l y t r e a t e d i s t h e p o l y nomial case. Chapter V shows t h e u n i f o r m l y weakly d i f f e r e n t i a b l e f u n c t i o n s c l a s s and p r e s e n t s an e x t e n s i o n o f B e r n s t e i n ' s theorem. Chapter V I d e a l s w i t h a p p r o x i m a t i o n f o r t h e compact open topology
o f o r d e r m.
An e x t e n s i o n o f W e i e r s t r a s s ' theorem f o r i n f i n i t e
dimensional Banach spaces i s g i v e n . Chapter V I I goes i n t o t h e weakly d i f f e r e n t i a b l e f u n c t i o n s c l a s s . We i n t r o d u c e t h e bounded weak a p p r o x i m a t i o n p r o p e r t y and o f f e r some r e s u l t s on polynomial a p p r o x i m a t i o n o f weakly d i f f e r e n t i a b l e f u n c t i o n s . Chapter V I I I i s d e d i c a t e d t o 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 n cont i n u o u s l y d i f f e r e n t i a b l e f u n c t i o n spaces. Many o f t h e d e n s i t y r e s u l t s o b t a i n e d i n t h e p r e v i o u s c h a p t e r s and
€-products o f c o n t i n u o u s l y d i f f e r -
e n t i a b l e f u n c t i o n spaces a r e used. Chapter I X d e a l s w i t h polynomial a l g e b r a s o f c o n t i n u o u s 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 . An e x t e n s i o n o f N a c h b i n ' s theorem i s found. Chapter X d e l v e s i n t o t h e c l o s u r e o f c o n t i n u o u s l y d i f f e r e n t i a b l e f u n c t i o n modules. An e x t e n s i o n o f W h i t n e y ' s theorem i s g i v e n . Chapter X I develops a s t u d y o f homomorphisms between a l g e b r a s o f u n i f o r m l y weakly d i f f e r e n t i a b l e f u n c t i o n s . The a u t o m a t i c c o n t i n u i t y problem o f these homomorphisms i s t r e a t e d and t h e f u n c t i o n s i n d u c i n g these homomorphisms a r e c h a r a c t e r i z e d . Chapter XI1 f i n a l l y shows a v e r s i o n o f t h e Paley-Wiener-Schwartz theorem i n i n f i n i t e dimensions. The book i s f i n i s h e d up w i t h an appendix d e d i c a t e d t o W h i t n e y ' s S p e c t r a l Theorem. T h i s book can be used by graduate s t u d e n t s t h a t have t a k e n courses
i n D i f f e r e n t i a l Calculus
, Topology
and F u n c t i o n a l A n a l y s i s and a r e
i n t e r e s t e d i n t h e Approximation Theory and I n f i n i t e Dimensional A n a l y s i s .
Foreword
X
I hope t o have served a l s o t h e a p p l i e d mathematician, t h e p h y s i c i s t and t h e engineer. The book i s reasonably s e l f - c o n t a i n e d and i t s r e a d i n g w i l l g i v e them a good o p p o r t u n i t y t o a p p l y t h e b a s i c p r i n c i p l e s o f D i f f e r e n t i a l Calculus and F u n c t i o n a l A n a l y s i s . On t h e o t h e r hand, we t h i n k t h a t i t can be u s e f u l as a r e f e r e n c e book f o r p r o f e s s o r s i n t e r e s t e d i n t h e s u b j e c t . Except f o r chapter 11, t h e t r e a t m e n t o f t h e s u b j e c t has n o t appeared i n book form p r e v i o u s l y . The area described, i s r a p i d l v expanding, and abounds i n q u i t e i n t e r e s t i n g and c h a l l e n g i n g open problems, many o f which a r e discussed i n t h e book. F i n a l l y I would l i k e t o express my g r a t i t u d e t o P r o f e s s o r Leopoldo Nachbin f o r b r i n g i n g up t h e i d e a f o r t h i s book. l i k e t o extend my h e a r t f e l t thanks t o Richard
M.
I
would a l s o
Aron and J a v i e r G6mez
G i l f o r t h e i r c o l l a b o r a t i o n and a d v i c e .
I s i n c e r e l y thank Anna S t e e l e f o r h e r h e l p i n p r e p a r i n g t h e E n g l i s h m a n u s c r i p t and P i l a r A p a r i c i o f o r h e r e x c e l l e n t e f f o r t s i n t y p i n g it.
Jos6 G. Llavona Madrid, June 20, 1986.
xi
CONTENTS
..................................................... Chapter 3 . PRELIMINARY RESULTS .............................. 0 . 1 F u n c t i o n s on l o c a l l y compact spaces ............... 0.2 W h i t n e y ' s theorems ................................ 0.3 M u l t i l i n e a r mappings and p o l y n o m i a l s .............. 0.4 Polynomials a l g e b r a s .............................. Foreword
.......... ...................................
0.5
€ - p r o d u c t 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
0.6
A n g e l i c spaces
...................... ................................ 0.9 Holomorphic f u n c t i o n s ............................. 0.10 Weakly compactly generated spaces ................. 0.11 I n j e c t i v e spaces .................................. 0.12 Some a d d i t i o n a l theorems .......................... Chapter 1. APPROXIMATION OF SMOOTH FUNCTIONS ON MANIFOLDS .... 1.1 W e i e r s t r a s s l theorem .............................. Nachbin's theorem ................................. 1.2 1 . 3 m-admissible a l g e b r a s ............................. Nachbin m-algebras ................................ 1.4 1 . 5 Modules on s t r o n g l y s e p a r a t i n g a l g e b r a s ........... Dense polynomial a l g e b r a s ......................... 1.6 1.7 P o i n t w i s e d e s c r i p t i o n o f c l o s u r e s ................. 0.7
A b s o l u t e l y summing o p e r a t o r s
0.8
Realcompact spaces
1.8
Notes. remarks and r e f e r e n c e s
.....................
vii
1 1
5 8
11 12 14 15 16 17 17 18 18 23
23 26 29 36
38 42 44 48
xii
Contents
Chapter 2
.
SIMULTANEOUS APPROXIMATION OF SMOOTH FUNCTIONS .....
53
.....
53
Approximation f o r t h e f i n e t o p o l o g y o f o r d e r m
2.1 2.2
A nonlinear characterization o f superreflexive
...................................... remarks ..................................
57 62
Banach spaces
2.3 Chapter 3
Notes and
.
3.0 3.1
POLYNOMIAL APPROXIMATION OF DIFFERENTIABLE FUNCTIONS
......................................
Introduction order
m . B a s i c d e n s i t y p r o p e r t i e s ..................
Q u a s i - d i f f e r e n t i a b l e f u n c t i o n s on Banach spaces
....................... m On c o m p l e t i o n of (Pf(E;F); T c ) .................... Notes and r e f e r e n c e s ................................
3.3 3.4
.
WEAKLY CONTINUOUS FUNCTIONS ON BANACH SPACES
.
.......
. P r o p e r t i e s ..............................
4.1
Introduction
4.2 4.3
The bw and bw* t o p o l o g i e s
4.4
On c o m p l e t i o n o f spaces o f weakly continuous func-
Elementary
..........................
69 76 77 79
79 82
Weakly continuous and weakly u n i f o r m l y continuous
..........................
.............................................. Polynomial case .................................... tions
4.5 4.6 4.7
Composition o f weakly u n i f o r m l y continuous f u n c t i o n s Notes and r e f e r e n c e s
.
...............................
86 93 97 105 112
APPROXIMATION OF WEAKLY UNIFORMLY DIFFERENTIABLE FUNCTIONS ...........................
5.4
66
Preliminary Definitions
f u n c t i o n s on bounded s e t s
5.1 5.2 5.3
65
.
Basic topological properties
Chapter 5
65
Approximation f o r t h e compact-compact t o p o l o g y o f
3.2
Chapter 4
.
Introduction
115
.......................................
U n i f o r m 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 bounded s e t s
.
115 116
Extension o f B e r n s t e i n ' s theorem t o i n f i n i t e dime! s i o n a l Banach spaces
...............................
Notes and r e f e r e n c e s
..............................
120 125
Contents
.
Chapter 6
xiii
APPROXIMATION FOR THE COMPACT-OPEN TOPOLOGY
....
127
6.1
E x t e n s i o n o f W e i e r s t r a s s ' theorem f o r i n f i n i t e dimensional Banach spaces References
............. .....................................
127
6.2
132
APPROXIMATION OF WEAKLY DIFFERENTIABLE FUNCTIONS
133
.
Chapter 7
7.1
Weakly d i f f e r e n t i a b l e f u n c t i o n s
.
.
Some r e s u l t s on
L o c a l l y convex s t r u c t u r e ......
133
7.2
The bounded weak a p p r o x i m a t i o n p r o p e r t y .........
141
7.3
Polynomial a p p r o x i m a t i o n o f weakly d i f f e r e n t i a b l e
7.4
Notes. remarks and r e f e r e n c e s
weak compactness
functions
Chapter 8
.
....................................... ...................
SPACES OF DIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY ..........................
8.1
8.3 Chapter 9
On 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 n :paces
9.2 Chapter 10
of
........... ...........................
155
Notes and r e f e r e n c e s
159
POLYNOMIAL ALGEBRAS OF CONTINUOUSLY DIFFERENTIABLE
.......................................
Polynomial a l g e b r a s
. Extension o f
10.2
161
N a c h b i n ' s theorem
.......... r e f e r e n c e s ....................
t o i n f i n i t e dimensioneal Banach spaces
162
Notes, remarks and
166
ON THE CLOSURE OF MODULES OF CONTINUOUSLY DIFFERENTIABLE FUNCTIONS.........................
10.1
151
continuously d i f f e r e n t i a b l e functions
FUNCTIONS
9.1
151
E-products o f c o n t i n u o u s l y d i f f e r e n t i a b l e funct i o n spaces . A p p l i c a t i o n s .......................
8.2
144 146
169
E x t e n s i o n o f Whitney's i d e a l theorem t o i n f i n i t e dimensional Banach spaces ........................
169
r e f e r e n c e s .............................
176
Notes and
Chapter 11 HOMOMORPHISMS BETWEEN ALGEBRAS OF UNIFORMLY WEAKLY DIFFERENTIABLE FUNCTIONS 11.1
..................
177
R e p r e s e n t a t i o n s o f un f o r m l y weakly d i f f e r e n t i a b l e functions
............ ...........................
178
xiv
Contents
11.2
Homomorphisms between a l g e b r a s o f u n i f o r m l y
................ ....................................... remarks and r e f e r e n c e s ..................
weakly d i f f e r e n t i a b l e f u n c t i o n s
182
11.3
Examples
188
11.4
Notes.
193
Chapter 12
THE PALEY-WIENER -SCHWARTZ THEOREM I N INFINITE DIMENSION
12.1
......................................
The F o u r i e r t r a n s f o r m o f d i s t r i b u t i o n s w i t h bounded s u p p o r t i n i n f i n i t e dimensions
12.2
195
.........
196
Characterization o f the Fourier transform o f d i s t r i b u t i o n s w i t h bounded s u p p o r t i n i n f i n i t e
12.3
..................................... remarks and r e f e r e n c e s .................
dimensions
203
Notes
206
.
Appendix I. W H I T N E Y ' S SPECTRAL THEOREM REFERENCES
INDEX
....................
................................................
......................................................
INDEX OF SYMBOLS
...........................................
209 221 235 239
1
Chapter 0
PRELIMINARY RESULTS
0.1
Functions on l o c a l l y compact spaces.
N
R denotes t h e s e t o f r e a l numbers,
AT = U
numbers and
U
. If
Cml
we p u t
X
i s an i n t e g e r such t h a t
1
N,,,
m'
, R[Gl
-
G, a nonempty s e t o f R
> 1 and =
a1
t..
m
R , Ni
8
.+ ctn <- m ;
v a l u e d f u n c t i o n s d e f i n e d on a G ; and i f
denotes t h e a l g e b r a generated by
,
a s e t o f R-valued f u n c t i o n s on Rn as a s e t o f f u n c t i o n s on
0.1.1
10.
*
Given set
n
a E Mn
i s the set o f multi-indexes
denotes t h e s e t o f n a t u r a l
Definition.
G
6
O(Gn)
is
has t h e n a t u r e 1 meaning
X.
a t any p o i n t , if f o r every
,
g (x)
such t h a t x
E
separates p o i n t s if given
G
We say t h a t : i l
x # y, thcre esic!-s g
G i s strongZy
then
O(Rn)
, there
X
# g(y)
exists
g
;G 6
G
x,y B X,
does not vanish such t h a t
g ( x ) # 0;
separating i f it separates p o i n t s and does not vanish a t
any point. Let
X
be a l o c a l l y compact H a u s d o r f f space, and l e t
r e a l l o c a l l y convex H a u s d o r f f space. a l l nonempty compact subsets o f seminorms on
X
c(X) and
F
be a
w i l l denote t h e c o l l e c t i o n o f
cs(F)
t h e s e t o f a l l continuous
F.
Let
Co(X;F) = C(X;F)
0.1.2
X
to
Definition.
dimension.
If m
2
F.
Let
1 and
On
denote t h e v e c t o r space o f a l l c o n t i n u o u s
o f comP p a c t and p o i n t w i s e convergence r e s p e c t i v e l y a r e d e f i n e d i n t h e usual way ; we p u t T O = T~ and T O = T I f m > 1 and n c Rn i s an open P P ' h subset, C (R;F) w i l l denote t h e space o f Cm f u n c t i o n s f r o m R t o F. (See Treves [ l l , 540). f u n c t i o n s from
C(X;F)
X be a r e a l
the topologies
T~
and
T
Cm manifold which l o c a l l y has f i n i t e
A(X) denotes the maxima2 atZas on
X,
Zet Cm(X;F)
2
Chapter 0
f from X t o F such t h a t
denote the vector space of a22 functions
F =R
When
¶
we w r i t e
When each space
Cm(X)
Cm($(V);F)
for
Cm(X;R). Consider t h e l i n e a r mappings
i s endowed w i t h t h e topology o f compact
convergence ( r e s p . p o i n t w i s e ) o f o r d e r m y t h e corresponding p r o j e c t i v e topology on
Cm(X;F)
T o = c(X) x cs(F)
Let
If
m
1
i s denoted by
let
¶
the charts o f
Ac(X) A(X)
rm = Iml
l < m < m ,
T
m ~ resp. (
y = (K,a) 8
be t h e a t l a s
T;).
r a yf
E Co(X,F)
and d e f i n e
o b t a ined by r e s t r ic t ing
(Vi,$i)ieI
t o t h e i r r e l a t i v e l y compact open subsets.
x I x cs(F)
r,,
y = ( m Y i y a )e
f
Let
Cm(X;F)
6
and
define
N,'i o f a l l
where t h e sum i s taken o v e r t h e s e t ni = dim
$i(Vi)
such t h a t
denote t h e u n i o n o f a l l
rm
I k l = klt for
1
5
...t
kni
m <
k = ( k l,...ykn
) e i
5 m.
Further, l e t
. Then
the family o f
N
n
,
r, semi-
d e f i n e s t h e t o p o l o g y !T on t h e space Cm(X;F) and P y E rm Y { f e Cm(X;F) Py(f) 5 1 y e rm i s a sub-basis o f c l o s e d neighbourhoods o f zero. norms
¶
¶
When
F =R
and dim(X) = n
i s d e f i n e d by t h e f a m i l y o f seminorms
The
PiYl
seminorm on
Cm(X)
t h e !T PiYl
satisfies
t o p o l o g y on t h e space ¶
i
6
I
1 e
1,
Cm(X)
, where
Preliminary results
The f a m i l y o f seminorms X
i s an open subspace o f Rn Definition.
f E C"
(X;F), When
F =
C:(X,K;F)
,
, such
CF(X;F)
rm ,
is also directed.
CF(X;R ) .
for
Given
K
f a CF(X;F)
.
T:
c(X)
8
,
such t h a t The f i n e s t
t h a t a l l the inclusions
m
and i s denoted by
, m=O
F =R
and
X
i s an open subset o f Rn
,
,
E
a r e continuous, i s c a l l e d l i n e a r i n d u c t i v e l i m i t
c CF(X;F)
When X
i s d i r e c t e d . F u r t h e r , when
endowed w i t h t h e t o p o l o g y induced by
topology o f order
tively
,y
Py
R , we w r i t e C:(X)
l i n e a r t o p o l o g y on
and
, then
ro,
E
denote t h e v e c t o r space o f a l l
supp(f) c K CF(X,K;F)
Y
C F (X;F) w i l l denote the vector space of a l l such t h a t s u p p ( f ) i s a compact s e t .
0.1.3
let
,y
P
3
T:.
is
m
a-compact o r
t h e spaces Cc(X)
i s arbitrary
and
CF(X)
respec-
w i t h t h e i r s t a n d a r d l o c a l l y convex i n d u c t i v e l i m i t t o p o l o g i e s
a r e o b t a i n e d i n t h e above way.
0.1.4
Definition.
A topological algebra i s an aZgebra endowed w i t h a
In
linear topology such t h a t m u l t i p l i c a t i o n i s separately continuous. p a r t i c u l a r , i f A i s a topological algebra and then 0.1.5
, the
0
closure of
,i s
Remark. The b i l i n e a r mapping
, C:(X)
A c CF(X) 0.1.6
i s a topological algebra.
Lemma.
r'
c
M c C F (X;F)
Let
_ I
rm
K
6
c ( X ) which contains
, f o r aZZ y
Proof.
W
Let
g
E
r'.
Then
f
E E
be a neighbourhood o f z e r o i n
W 1 = W n C:(X,K;F) exists a f i n i t e set
then
c
rm
such t h a t i f
&fl
and
.
c
CF(X;F) heC:(X,K;F)
Assume
and such t h a t f o r
that
i s a neighbourhood o f z e r o i n rl
As a conse-
f a C F (X;F).
supp(f)
M such M .
.
T:
M c C:(X;F)
AM c M
be nonempty and
f i n i t e , there e x i s t s
Py(f - 9 ) < 1
and
T:
Also, i f
a r e nonempty subsets such t h a t
t h a t there e x i s t s every
i s a subalgebra,
a subalgebra too.
i s s e p a r a t e l y continuous f o r t h e t o p o l o g i e s quence
B cA
supp(g) c K
.
and
Since
CF(X,K;F) and
,
there
P (h)
Y
5 1,
4
Chapter 0
for all g E
r',
e
y
implies
supp(g) c K
such t h a t
Letting Since
h = g
W
-f
h B W1.
that
P ( Y
and
f
B
5 1 , for all y
f -9)
g-f E W
it follows t h a t
i s a r b i t r a r y we g e t
By h y p o t h e s i ' s t h e r e e x i s t s hence
E
I".
.
r l (W t f )
g B
8.
0.1.7. Definition. When X is an open subset of Rn , the aZgebra of all f c Crn(X) such t h a t f and a l l i t s p a r t i a l d e r i v a t i v e s o f order 5 in are bounded i s denoted by CF(X) , An anaZogous d e f i n i t i o n f o r Cy(X) is,where
"bounded" i s replaced by "vmish.ed a t i n f i n i t y " .
denote t h e aZgebra of a21 f o r all x
6
f E CY(R) such t h a t
I f ( x ) I <_
AZso
, we
constant.
IxI
,
R , by C;,,(R).
The seminorms
m
T~
o f u n i f o r m c o n v e r g ence o f o r d e r
rn.
m n S (R )
denotes the vector space of a12
f E Cm(Rn)
generate t h e topology 0.1.8
Definition.
which s a t i s f y the foZZowing condition: f o r any i n t e g e r k , p e INn with /pi
5
m and
We sa y t h a t We e q u i p
E
f
>
0,
there e x i s t s
p >
0 such t h a t
5 m a r e r a p i d l y decreasing
a nd i t s d e r i v a t i v e s o f o r d e r
w i t h t h e f a m i l y o f seminorms
Sm(Rn) qk,p ( f ) =
1 ( 1-+ 1 x 1 2 1 ~a P f ( x )
max
I
(q
kYP
)
d e f i n e d by
,
x€Rn
0.1.9
Definition.
X be an open subset of Rn and
Let
V = (Va)J cx
6
n Mm,
be a family o f s e t s of weights on p o s i t i v e functions on
X
X t h a t i s upper-semicontinuous and We w i l l assume t h a t V has the foZZowing
.
properties:
i) For every x E X and V,(X) 0 ; ii) For every there e x i s t Cm Vm(X)
v
a
E V
a
and
c1 E
N i , there
a,
v
e x i s t s va E Va n E urn such that 6 5 cx and
~ E - Va-B~
be the algebra of a12
f
B
such t h a t Cm(X)
Va
such t h a t
5
Va* V,
Va
such t h a t E Va ,
Let a-B' aaf vanishes V
5
Preliminary results
at infinity for all
Mmn and va E Vcl
a E
.
and va define
Every such a
a seminorm f
+
m on C V,(X).
SUP
t va(x);aa f ( x ) l
,x
E
x 1
Under t h e t o p o l o g y generated by t h o s e seminorms, CmVm(X)
becomes a t o p o l o g i c a l a l g e b r a a l s o c a l l e d a weighted a l g e b r a . 0.2 W h i t n e y ' s theorems.
I n t h i s s e c t i o n we s t a t e a b a s j c theorem o f Whitney on t h e extens i o n o f mappings d e f i n e d on a c l o s e d subset, and t h e Whitney I d e a l Theorem ( W h i t n e y ' s s p e c t r a l theorem) concerning t h e d e s c r i p t i o n o f c l o s e d i d e a l s o f differentiable functions. L e t E,F be two Banach spaces; L(E;F) denote t h e Banach space o f k continuous l i n e a r maps from E t o F ; L ( E;F) denotes t h e Banach spaces o f = continuous k - m u l t i l i n e a r maps from E t o F ( i . e . , L o ( E ; F ) = F; L(k+lE;F) k k = L(E;L( E;F)) ; Ls( E;F) denotes t h e Banach space o f s y m e t r i c k - m u l t i -
l i n e a r maps from E t o F.
U
If
C E i s an open s e t and
m m E &, C (U;F)
denotes t h e space o f
a l l m-times c o n t i n u o u s l y F r 6 c h e t d i f f e r e n t i a b l e mappings f r o m (See Cartan
U t o F.
[l] and Dieudonn6 [l] ) ,
W h i t n e y ' s E x t e n s i o n Theorem can be viewed as a g e n e r a l i z a t i o n o f t h e f o l l o w i n g obvious converse o f T a y l o r ' s Theorem,
+
0.2.1 Pru o s i t i o n . Let
E,F
...
f o r k = 0,1, ,r , w i t h k ; Rk : U x U * Ls( E;F) by
f k : U * L s ( E;F),
k = 0,l
,... ,r
be Banach spaces, UC E open, f : U
for. x,y E U. Then f i s c l a s s Cr and d kf
f o = f . Define
= fk
, for
+
F
and
, for
k = 0,1,,
. . ,r
,
provided t h a t the following condition on the remainders is s a t i s f i e d :
For
xo E U
and
k = O,l,,..,r
ll R&XO II Y -
,
YY)
II +
0
as
y +xo.
x o r k
W h i t n e y ' s E x t e n s i o n Theorem i s a g e n e r a l i z a t i o n o f (0.2.1).
6
Chapter 0
0 . 2 . 2'
Theorem (hrhitney Extension Theorem) Let F be a Banach space, A c Rn a closed subset, and cr(r
( i ) If Rk : A x A + L s ( kRn ;F)
, then
II
f o r every
Rk(xl,x2)1/ <
E E
+
0
i s defined by
x06 A
f o r each
11 R k ( x l I x 2 ) 1 1 I1 x1- x 2 l lr - k i.e.,
F.
...,fr
..
x,y 6 A
+
0 ) f unct i on g : R~ + F provided t hat k n with f o = f , f k : A + L s ( R ;F) ( k = O , l ,... ,r), the condition f i l belot, i s s a t i s f i e d :
(11 f extends t o a there e s i s t s f o , f l , and f o r k = 0,1,. ,r
for
f : A
as
~ 1 ~ -+x 2X O
A
in
> 0 there e x i s t s 8 > 0 such t hat f o r a12 ~ 1 ~ 6x A, 2 r-k whenever II XI-x o l l ,]I ~ 2 xo I < 6. 11 XI-x2II
(11) f extends t o a C" function g : Rn+ F , provided that there e x i s t functions f o , f I , . , . , such t hat ( i ) holds for each k = 0,1,2,. . .
.
(I)
(111) I n k d glA =
fk
or ( I I ) , t he extension g
f o r a22 appropiate
of
f may be chosen so t hat
k.
For t h e p r o o f , see Abraham-Robbin[ll
and W h i t n e y [ l ] .
See a l s o
Margalef-Outerelo 113 f o r t h e i n f i n i t e dimensional case. The i d e a l subset of H.
which
theory i n the algebra
Whitney [ 2 ]
proved i n 1948
U
i s a open
r e s o l v i n g a c o n j e c t u r e by
Cm(U)
proved t h a t each c l o s e d i d e a l i n
the primary ideals t h a t contain Later orem
, where
Rn , i s based on W h i t n e y ' s i d e a l theorem. I n t h i s theorem,
Schwartz, t h e c l o s u r e o f an i d e a l i n Whitney
Cm(U)
L.
i s characterized. Specifically Cm(U)
i s the intersection o f
it.
B. Malgrange [l] p r o v i d e d a s i m p l i f i e d p r o o f o f t h i s the-
, following
Whitney's o r i g i n a l i d e a s and a l s o i n c o r p o r a t i n g t h e more
general module language d e r i v e d from
G.Glaeser [l].
Regarding t h i s theorem J.C.Tougeron [l], V.Poenaru [ l ] , L . S c h w a r t z [21 and
L.Nachbin 181 a l s o stand o u t . Let
F
be a f i n i t e dimensional v e c t o r space over R.
We c o n s i d e r
7
Prel iminary r e s u l t s
t h e space
Cm(U;F)
(resp. C"(U))
Cm
functions o f
F-valued ( r e s p . r e a l v a l u e d )
U, endowed w i t h t h e compact-open t o p o l o g y
c l a s s on
m y i.e.,
o f order
of all
t h e t o p o l o y generated by a l l seminorms o f t h e f o r m
II where
K
u.
i s a compact subset o f
I n a s i m i l a r way, we d e f i n e space o f a l l
C"(U;F)
( r e s p . C"(U))
U w i t h values i n
Cm-functions on
F
(resp. R )
m
and
endowed
, where now U and t h e
I
w i t h t h e t o p o l o g y generated by t h e f a m i l y o f seminorms K
as t h e
a r e a l l o w e d t o range o v e r t h e compact subsets o f
n a t u r a l numbers r e s p e c t i v e l y . N
If
a 6
, for
each
U we d e f i n e t h e map : Cm(U;F) *
T:
Also, i f in
{ k E Nn : Ikl 5 ml
i s the cardinal o f the set
M
Cm(U;F)
0.2.3
i s a submodule of
, we
denote by
Theorem. If
FN
0
M
Cm(U;F)
, i.e.,
a
Cm(U)-module c o n t a i n e d
the intersection
(Whitney ' s i d e a l theorem)
m
M i s a submodule o f
C (U;F),
-
the closure M of
Cm(U;F)
M in
A
M.
coincides w i t h
-
0.2.4
in
m
Theorem. Cm(U;F)
m
T x f E Tx(M)
If
f o r every
m
Ta f
p r o d u c t t o p o l o g y on sion
i s a vector
,
T:(M)
C"(U;F),
i s t h e module of a22 functions
The map T;(M)
M i s a submodule of x E U
and every
f in
Hence
Cm(U;F)
M
of
M
such t h a t
m 2 0.
i s a c o n t i n u o u s l i n e a r map
FN .
the cZosure
, when
considering the
, i f M i s a submodule o f Cm(U;F) ,
subspace o f
FN
, and
since
FN has f i n i t e dimen-
m -1(Ta(M)) m i s c l o s e d i n FN and so (T,)
i s c l o s e d . However,
Chapter 0
8
as we w i l l see i n chapter 10, i n i n f i n i t e d i m e n s i o n s t h i s does n o t g e n e r a l l y T h e r e f o r e , i t i s u s e f u l t h a t another more adequate f o r m u l a t i o n
occur.
o f Whitney's
i d e a l theorem t o be extended
to
i n f i n i t e dimensions be
given. M
If
Cm(U;F), we w i l l denote by
i s a submodule o f
M"
intersection
n
=
{ f c Cm(U;F)
,:
f o r each
the
> 0, t h e r e e x i s t s g
E
E
M
aeU
11
such t h a t
- a kg ( a ) l l 5
akf( )
, for
E
every
,
k
I k I 2 m) V
M
I n a s i m i l a r way, i f f o r each f o r every 0.2.5
> 0,
E
-
M of
M in
Cuo(U;F)
11
such t h a t
2
I f E Cuo(U;F): a U mciu akf(a) - akg(a)jl 5 E M =
.
6
Cm(U;F)
If M is a subrnoduze of coincides w i t h *M ,
For t h e p r o o f of theorems
0.2.3,
0.2.4,
Cm(U;F),
and 0.2.5
The f o l l o w i n g i s another way o f s t a t i n g Whitney's
M
The c l o s u r e o f a submodule t o the c l o s l r e
of
M
of
C"(U;F)
f o r the topology
f a m i l y ( w i t h parameters
,
c1
and
la1 5 m , where
f o r t h e T: T~
P
a E Wn
0.3
M u l t i l i n e a r mappings and p o l y n o m i a l s .
,
l.( E;F)
If
E
n E
W , denotes
mappings from
and
En
F
Cm(U;F)
-
the
t o p o l o g y i s equal d e f i n e d by t h e
f E Cm(U;F)-I\ a a f ( r ) l l
x E U.
a r e r e a l o r complex l o c a l l y convex spaces
=
F.
t h e l i n e a r space f o r a l l continuous n - l i n e a r We denote
of
F.
F o r any n - l i n e a r mapping
A
we d e f i n e i t s symmetrization
As
bY
where set
0
= ( o ~ , . . . ~ o ~and )
{ 1 , . ..,n
1.
on i s t h e s e t o f n!
I.
see appendix i d e a l theorem.
t h e l i n e a r subspace o f L('E;F) n a l l symmetric n - l i n e a r mappings by L s ( E;F). I f n = O , we s e t L(OE;F) = L,(OE;F)
to
on
x ) of seminorms
for
n
,
1.
m
c
Let m
Theorem.
cZosure
g E M
here e x i s t s
, (k
k
i s a submodule o f
.
permutations o f t h e
ER
Preliminary results
9
An n - l i n e a r mapping i s c o n t i n u o u s i f i t i s c o n t i n u o u s a t t h e o r i gin. ._ A continuous
n-homogeneous polynomial
composition o f t h e form diagonal o p e r a t o r o f
A0
E
, where A
An
into
E x
I n o r d e r t o denote t h a t
w i l l write
^A.
p =
... p
We w i l l denote
f
E L('E;F)
E
and
&
F
An
i s the
is a
x E.
corresponds t o P('E;F)
c o n t i n u o u s n-homogeneous p o l y n o m i a l s from A
from
p
A
i n t h i s way, we
t h e v e c t o r space o f a l l E
to
F; P(OE;F) = F
and
i
A = (AS). 0.3.1
(Nachbin [ 10 1,
The mapping
53).
i s a vector space isomorphism and ue have t h e " p o l a r i z a t i o n formula" A( XI ,.
. . 'xn)
=
-~
1
1
€1
...
n
A ( E ~ x+ ~ ... +
E,
E,
xn).
1112'E ] = * l , . . . ,En=fl We w i l l be i n t e r e s t e d i n t h e subspace generated by t h e c o l l e c t i o n o f f u n c t i o n s where
($n B y ) ( x ) =
on(x).y
$n
f o r each
x
6
Pf(nE;F)
Iy =
of
P('E;F)
@'.y (n6U , $ € E l , YEF)
E.
m
Let
P(E;F)
1
=
P('E;F)
be t h e space o f a l l continuous
n=O p o l y n o m i a l s from
E
F
and
m
Pf(E;F)
=
1
n=O _ continuous _ _ _ p o l y n o m i a l s o f l i n i t e t y p e from E
0.3.2
and
i.Vachbin [ I O I
531
P E P("EE;F , we s e t
II und
,
All
Pf(nE;F)
t h e space o f a l l
F.
~f E and F are normed spacesand i f A G L ( " E ; F )
10
Chapter 0
Then we have
Also if E and F are Banach spaces, then
0.3.3
Let
E and F be Banach spaces.
with respect t o the norm induced by
The completion o f
X
c l o s e d subset o f
X
Let
If
X
C(X)
n
E
111
P('E)
=
t h e space of
i s dispersed
c o n t a i n s an i s o l a t e d p o i n t ) and
sup norm , t h e n f o r every
0.3.4.
X.
,
PC('E;F)
.
P('E;F)
be a compact H a u s d o r f f space and
a l l s c a l a r continuous f u n c t i o n s on
Pf(nE;F)
is denoted by
P('E;F)
and in genera2 i s s t r i c t l y contained in Let
is a Banach space.
P('E;F)
.
PC('E)
E and F be t w o Banach spaces. For each
(every
E = C(X)
w i t h the
(See Aron 1 2 1 ) .
111
n E
let
PN( 'E ;F) be t h e Banach space o f a12 n-homogeneous nuclear continuous po2ynomiaZs from E t o F , endowed w i t h the nuclear norm 1) - 1 ) , PN('E ;F)
is characterized by t h e foZlouing conditions: n P( E;F).
(1)
P ~ ( " E ; F ) is a vector subspace of
(2)
PN(nE;F)
is a Banach space with the nucZear norm.
(3)
Pf(nE;F)
is a dense subspace of
sion mapping
P~("E:F)
-+
P ( n E;F)
and the inclu-
i s continuous.
p E Pf(nE;F)
( 4 1 For every
n PN( E;F)
11
the nuclear norm
is de-
PI1
f i w d by
where the infimwn i s taken over a l l representations
k 1
p =
j=1 The e x i s t e n c e o f t h e Banach space
J
fl
b.. J
i s assured i f
PN(nE;F)
E'
( 0 . 5 . 2 ) . (See Gupta t l ] ) .
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 Whenever t h e space
4 k.
PN(nE;F) i s considered
we w i l l assume t h a t
t h i s hypothesis i s s a t i s f i e d . Let
E
be a Cm-function
and
.
F
be Banach spaces
For each
j E
U
j
2 rn
U cE and
be open and
x E U
f :U
-+
the d e r i v a t i v e
F
11
Preliminary results
dJf(x)
6
Ls(JE;F).
dJf(,x) can be considered as a t o F.
and (0.3.2)
Taking (0.3.1)
i n t o account each d e r i v a t i v e
j-homogeneous continuous p o l y n o m i a l from E
Unless t h e c o n t r a r y i s e x p r e s s l y i n d i c a t e d , t h r o u g h o u t t h i s book we
w i l l take the d e r i v a t i v e s f : U
a l l mappings
derivative dJf(x)
E
+
F
djf(x)
E
P(’E;F).
That i s , Cm(U;F) c o n s i s t s o f
such t h a t f o r each
P(JE;F)
j
N
E
e x i s t s and t h e mapping
,j 2 m , and x E U t h e d J f : U + P(JE;F) is
continuous . 0.4
Polynomials a l g e b r a s Let
space.
be a t o p o l o g i c a l H a u s d o r f f space and F a l o c a l l y convex
X
The v e c t o r space o f a l l c o n t i n u o u s f u n c t i o n s f r o m
ed w i t h t h e compact-open t o p o l o g y 0.4.1
[Prolla [ 3 ] , 41. Let
X
to
F, endoq?
be a v e c t o r subspace.W
i s caZled
, i s denoted by C(X;F).
W c C(X;F)
a po2ynominl a l , g e b r a , i f it has any of t h e foZlowing equivalent properties. n _Z 1, given g
( 1 ) For each
(21 A = 16
that
A
0
f : $
F’
E
E
W and p E Pf(nF;F) ,p
o
g belongs t o W
, f e W l is a suhaZaebra of
such
C(X)
F c W.
0.4.2 ( P r o l a [ 3 ] ,4) (Weierstrass-Stone theorem f o r p o l y n o m i a l a l g e b r a s ) Let of W
W
c
C(X;F) be a v e c t o r subspace. The Stone-Weierstrass h u l l denoted by A(W), i s t h e s e t o f a l l f u n c t i o n s
i n C(X;F),
f EC(X;F)
such t h a t g(x)
such t h a t f ( x ) # 0, t h e r e i s g E W such t h a t
1)
f o r any
x EX
2)
f o r any
x,y e X
# 0. such t h a t
f(x) # f(y)
, there i s g e W
z
such t h a t
g(x) S(Y). We say t h a t W
i s a Stone-Weierstrass subspace i f
A(W)
c
w.
Suppose F is a Hausdorff space. Every s e l f - a d j o i n t polynomial is a Stone-Weiarstrass subspace. In p a r t i c u l a r , if W c C(X;F)
aZgebra E and F arc tm r e a l locally convex Hausdorff spaces, then is dense in c(E;F). 0.4.3
Let
A c C”(E;F)
E and
Pf(E;F)
F be two rcal Banach spnces. A polynomial, algebra
, (m 2 1) , is called
u A!achbin polynomial aZgebra i f the
foZloz&zg three conditions hold: (a) For every
x
E
E
, there is g
E
A
such t h a t
g(x) # 0.
12
Chapter 0
For every pair
ih)
X,Y
g ( x ) # sI(Y). ( , c ) For every x E E and
thut
E, x #
E
v
8
E
y ,tl-zere is g
,v #
6
A
such t h a t g e A
0 ,there is
sueh
dg(x)(v) # 0. Note t h a t
,
Pf(E;F)
P(E;F)
, Cm(E;F)
and
Cm(E;F)
a r e Nachbin
a l g e b r a s . Another i n t e r e s t i n g example o f a Nachbin polynomial
polynomial
E
a l g e b r a occurs when
has an m-times c o n t i n u o u s l y d i f f e r e n t i a b l e norm.
I n t h i s case
If E Cm(E;F) : f
has bounded s u p p o r t
i s a Nachbin polynomial a l g e b r a
0.5.
.
1
(See, Wulbert [11 ) .
E-product 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 . Let
dual E ' .
E
be a l o c a l l y convex H a u s d o r f f space, w i t h t o p o l o g i c a l
E;
We denote by
t h e space
El
endowed w i t h t h e t o p o l o g y o f
u n i f o r m convergence on a l l a b s o l u t e l y convex compact subsets of f o l l o w s from Mackey's
theorem t h a t t h e dual ( E i ) '
of
E;
is
E. E
It
(as
a v e c t o r space). Now l e t LE(Er;F) T : E;
-f
E
and
F
be two l o c a l l y convex Hausdorff spaces.
w i l l denote t h e v e c t o r space o f a l l continuous l i n e a r mappings F
endowed w i t h t h e t o p o l o g y o f u n i f o r m convergence on t h e e q u l
E'
continuous subsets o f
.
The space
LE(E;;F)
i s t h e n a l o c a l l y convex
Hausdorff space, whose t o p o l o g y i s generated by t h e f a m i l y o f seminorms T where and
p V
+
sup { p ( T ( u ) ) : u E V " )
ranges along a system o f seminomis d e f i n i n g t h e topology o f runs through a 0-neighbourhood base i n We d e f i n e t h e E E F = LE(F;
(Schwartz [4 E
E-product o f
E
and
E. F
by s e t t i n g
; E),
I). E
F
and
F
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 isomorphic.
F,
Preliminary results
0.5.1
(Schwartz [ 5 ] ,
Hausdorff space and
Th.2 m
, expos6
.
E
13
.
no 10)
Let F
be a locally convex
Then
0.5.2 (Grothendieck [I 1 1. A ZocaZly convex space E has Grothendieck's approximation property, if t h e i d e n t i z y map e can be approximated , uniformZy on every precompact s e t in
E
, by
continuous l i n e a r maps of f i n i t e
rank. I n E n f l o 111 i t i s shown t h a t t h e r e i s a Banach space which f a i 1s t h e a p p r o x i m a t i o n p r o p e r t y .
E
If
,
erty
i s quasi-complete
,
then
i f and o n l y i f t h e i d e n t i t y map
l y on every compact s e t i n
E
If
space
L ( E;F)
0.5.3
Let
E
and
F
E e
has t h e a p p r o x i m a t i o n propcan be approximated
, uniform-
E, b y 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 .
a r e l o c a l l y convex spaces, Lc(E;F)
denotes t h e
endowed w i t h t h e t o p o l o g y o f precompact convergence.
be any l o c a l l y convex space with dual E '
. The folZowing
properties of E are equivalent E' s E in
Lc(E;E)
(11
The cZosure of
12)
E' s E
(3)
For every locally convex space
contains the i d e n t i t y
map e .
E'
F
is dense i n 14)
i s dense i n
Lc(E;E).
F
,
F
, F ' s E i s danse i n
Lc(E;F).
For every localZy convex space
Lc( F;E).
(Schaefer 0.5.4
L(E;F)
Let
[I]
E
, 111,§9). Pc a Banach space. The fcZZowing are equivalent
(1)
E ' ha,- the approximation property.
(2)
For every Banach space
F
,
the closure of
i s i d e n t i c a l t o the space of compact maps i n
(Schaefer [ l ]
, III,9.5).
E'
L(E;F).
F in
14
Chapter
0.5.5
Let: E
0
be a quasi-complete Zocally convex Hausdorff space. Then
the folZowing are equivalent E
(1)
F
(21
E
P
F i s dense i n
(3)
E
IF
,
53)
.
E
E
F
, for
a l l Zocally o m v e x spaces
E
E
F
, for
a t 1 Banach spaces
.
( P r o l l a [31, 8
0.5.6
i s dense i n
E
F
.
(Bierstedt [ l l ) .
E be a quasi-complete ZocalZy convex Hausdorff space. I f
Let
i s a l s o quasi-complete and has t h e approximation property
El
has the approximation property
.
[I 3
(Kzthe
, then
, § 43 I .
E i s said t o have t h e bounded approximation property, there i s a constant C , 1 5 C < m , such t h a t f o r every E > 0 and
0.5.7 i f
has t h e approximation property
every
A Banach space
K
compact
in
E
, there
11 T ( x )
erator T on E such t h a t
0.5.8
Let
E
be a Banach space.
i s a f i n i t e rank continuous l i n e a r op-
5
E'
has the bounded approximation property
i f and only i f there i s a constant
compact s e t s
K c E and
f o r every
E,
T
x E K
TI1
C > 0 such t h a t f o r every pair
L c Eland every
rank continuous Zinear operator
, and 1 1
-xII
: E
E
+
> 0
, there
E such t h a t
5 C.
of
exists a finite
1) T I )
5 C
and
(See A r o n - P r o l l a [ l l ) . A systematic
study 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 i e s i s g i v e n i n Kb'the 111
[11
and L i n d e n s t r a u s s - T z a f r i r i
0.6.
.
A a e l i c spaces.
A t o p o l o g i c a l H a u s d o r f f space every r e l a t i v e l y c o u n t a b l y compact s e t (a)
A
(b)
For each
X
i s c a l l e d angelic
A c X
if for
the following hold:
i s r e l a t i v e l y compact.
x
E
A t h e r e i s a sequence i n A which converges
P r e l irninary r e s u l t s
to
15
x.
0.6.1
AZZ metrizabZe ZocaZZy convex spaces
,i n
p a r t i c u l a r a l l normed
spaces, are angeZic in t h e i r weak topoZogy.
[ l l , 3.3).
(Floret
0.7.
A b s o l u t e l y summing o p e r a t o r . Let
T E L(E;F)
E
and
i s called
F
be Banach spaces and
p - a b s o l u t e l y summing,
so t h a t , f o r e v e r y c h o i c e o f an i n t e g e r , we have
n
p
2
1
.
An o p e r a t o r
i f there i s a constant
K
...,x n l
in
and v e c t o r s i x l ,
E
-
TI ( T ) . The c l a s s o f a l l P i s denoted by TI (E;F) . P The 1 - a b s o l u t e l y summing o p e r a t o r s w i l l be s i m p l y c a l l e d a b s o l u t e l y sum-
The s m a l l e s t p o s s i b l e c o n s t a n t p-absolutely
and
operators. For every TI
plete.
P
0.7.1
If
.
L(E;F)
, n (E;F)
is
P
nP (E;F)
a l i n e a r subspace o f
L(E;F)
i n which t h i s space i s even com-
S and T are bounded Zinear operators whose composition i s
(ST) 5 ) I SIl r p ( T ) . P Every bounded Zinear operator T from 11 i n t o 12 is absolutely T ~ ( S T )5
defined then
sumning
p
d e f i n e s a norm on
(T)
K i s denoted by
summing o p e r a t o r s i n
T I ~ ( S )1). Tlland
TI
Every absoZuteZy summing operator i s 2-absoluteZy summing. (Lindenstrauss-Tzafriri
0.7.2
[ 1 1, 2.b).
Grothendieck-Pietsch Domination Theorem. Let E and F be Banack spaces and suppose that
T
: E
+
F is
a p-absolutely s d n g operator. Then there e x i s t s a regular Bore2 proba b i Z i t y measure
1-1
, defined
on
B(E')
, the
cZosed u n i t baZl in
E',
(in i t s w*-topoZogy) suck t h a t , if E denote t h e cZosure of E i n L P ( u ) P T = P G , where G : E -L E i s t h e natural mapping of E i n i t s
then
0
P
16
Chapter 0
P : E -+ F L (p)-compZetionof E , and P P is the unique continuous Zinear extension of T t o alZ of E G is a P'
Ep
originaZ norm i n t o
, the
continuous linear operator too. G
There a r e two t h i n g s about
t h a t must be mentioned. F i r s t
,G
G i s a weakly compact o p e r a t o r ; t h a t i s
unit ball in
E, i n t o a weakly compact s e t i n
f o l l o w s from t h e r e f l e x i v i t y o f
G
that
into
E
E
i s the r e s t r i c t i o n t o
C(B(E');w*)
takes
If
EP
,
.
p
If
, the closed z 1 , then t h i s
t h e n one need o n l y n o t i c e
P' o f the i n c l u s i o n operator taking
L 1 ( p ) ; on i t s way from
t h e i n c l u s i o n o p e r a t o r passes through compact. Next
p = l
B(E)
C(B(E');w*)
into
L p ( j l ) making i t , and
, G i s c o m p l e t e l y continuous ; t h a t i s
%
M > 0
an
f o r each
I / xnll
such t h a t x' E E'
as
<
E M
x o = weak l i m n xn
and
, weakly
if
(x,)
, then
is
there i s
x'xo = l i m x'xn n as a c t i n g on B ( E ' ) , we g e t
for all
w e l l . Viewing
,
Ll(p) G
takes weakly
G
convergent sequences t o norm convergent sequences. I n f a c t , a weakly convergent sequence i n
,
E
n
and
l i m x n ( x ' l = x o ( x ' ) f o r each x ' E B ( E ' ) and l x n ( x ' ) l 2 M h o l d i n g n f o r each x ' 6 B ( E ' ) By Lebesgue's bounded convergence theorem, t h i s
.
g i v e s us
Since t h e operator
P : Ep
-+
F i s weakly continuous as w e l l G
as continuous, t h e above p r o p e r t i e s o f
weakly compact and c o m p l e t e l y continuous.
-R e a l c m a c t
0.8.
Let
ous
,
X
(Diestel [ 2
T
.
T is
1, p. 60-61).
spaces. be a t o p o l o g i c a l space and
r e a l - v a l u e d f u n c t i o n s on
C(X)
Z(X)
If
X.
% i n X , a nonempty s u b f a m i l y X
a r e passed along t o
0 of
t h e s e t o f a l l contin!
denotes t h e s e t o f a l l
Z(X)
zero-
i s c a l l e d a z - f i l t e r on
provided t h a t (i) @
(ii)
if
( i i i ) If
+0 Z1,Z2 E Z 6 0
Q
, Z'
then E
Z(X)
Z1 fl Z 2 E Q, and and
Z'
2
Z
,
then
Z' 6 0
.
17
Preliminary results
By a z - u l t r a f i l t e r
on
X
i s meant a maximal z - f i l t e r ,
one n o t c o n t a i n e d i n any o t h e r z - f i l t e r . We c a l l a z - f i l t e r c o r d i n g t o t h e i n t e r s e c t i o n o f a l l i t s members i s nonempty.
A c o m p l e t e l y r e g u l a r space
X
i s realcompact
fixed
i.e., ac-
i f every z - u l t r a -
t i l t e r w i t h tne countable i n t e r s e c t i o n property i s f i x e d .
0.8.1
Properties of realcompact spaces.
.
(1)
Every LindeZZf space i s realcompact
(2)
Every closed subspace of a realcompact space is realcompact.
13)
If X
i s realcompact , and each point of X i s a
’
then every subspace of X i s realcompact.
Let T be a continuous mapping from a realcompact space
(4)
X
i n t o rz space
of
Y is realcompact.
( G i l l m a n - J e r i s o n [11
0.9
,
8).
Holomorphic f u n c t i o n s .
U be an open subset o f a complex Banach space E .
Let tion
Then t h e t o t a l preimage of each realcompact subset
Y.
f : U
C
+
A func-
i s s a i d t o be G-hoiomorphic (Gateaux-holomorphic),
if
t h e f u n c t i o n d e f i n e d by
where D = { A e IC : x t Ay E U). A funcx,y a E i s s a i d t o be holomorphic i f i t i s G-holomorphic and
i s a n a l y t i c f o r every tion
f :
U
+
t
continuous. (Nachbin 1131j
.
0 10 Weakly compactly generated spaces.
when i t (W C G) Both separable and r e f l e x i v e spaces
A Banach space i s weakly compactly generated has a weakly compact t o t a i subset a r e p a r t i c u l a r cases o f (Diestel [ l l ) .
W C G
.
spaces.
18
Chapter 0
0.10.1
W C G space
Every
i s weakly Lindelgf.
(Talagrand [ l l ) .
0.11
I n j e c t i v e spaces. A Banach space
space
F containing E F
t i o n from
onto
E
i s s a i d t o be i n j e c t i v e
,
i f f o r e v e r y Banach
as a subspace, t h e r e i s a bounded l i n e a r p r o j e c -
E.
I n j e c t i v e spaces can be c h a r a c t e r i z e d by e x t e n s i o n p r o p e r t i e s f o r operators.
The following three assertions concerning a Banach space E are
0.11.1
equivalent. (1) E zs i n j e c t i v e .
T B L(E;Z) there i s a
and every
F
For every Banach space
(2)
f
3
E
6 L(F;Z)
, every
which extends
For every p a i r o f Banach spaces Z
13)
Banach space
3
T
F and every
z
. T 6 L(F;E)
h
T
there i s a
-
(Lindenstrauss
U.12.
L(Z;E)
6
T.
uhich extends
Tzafriri [ll, 2fj.
Some a d d i t i o n a l theorems.
I n t h i s s e c t i o n a s e r i e s o f r e s u l t s w i l l appear t h a t w i l l be used i n c e r t a i n places o f t h e book.
Taking i n t o account t h a t these re-
s u l t s belong t o d i v e r s e areas ( s u c h as t o p o l o g y , f u n c t i o n a l a n a l y s i s ,
. ..
etc)
we recommend t h a t t h e r e a d e r s k i p t h i s s e c t i o n and r e t u r n t o
i t when so i n d i c a t e d .
0.12.1
Let
subspace from
5
S
to
rf
X of
be a t o p o l o g i c a l space. We w i l l say t h a t a t o p o l o g i c a l
X
i s C-embedded
in
X , i f e v e r y continuous f u n c t i o n
R can be extended t o a continuous f u n c t i o n from X t o R . X
i s a normal space
, every
closed s e t i s
C-embedded.
(Gillrnan-Ge;el-iscn [ l ] , 3 0 ) .
0.12.2
Let X be a topologicaZ space. The following are equivalent
Preliminary r e s u l t s
19
X is c o i l e c t i o m i s e normal.
(1)
For any tcanach space Y , any closed A of X and every continuous map : A + Y , there e x i s t s a continuous extension i2)
-
g : X + Y
g.
of
(Dowker [ l l ) . Every Hausdorff compact space is collectiontlise normal.
0.12.3 Let X be a Hausdorff completely regular space ana E be a Hausdorff l o c a l l y convex space . Let C ( X ) ( r e s p . C ( X ; E ) ) be t h e l o c a l l y convex space of the continuous real-valued ( r e s p . €-valued) f u n c t i o n s on x , w i t h t h e compact-open topoiogy. The space subset
Y of X
is bounded on
C[X)
is barrelled, zf and onZy if any given closed
, such t h a t every continuous r e a l valued f u n c t i o n on Y (Y c X
a pseudocompact s e t )
C(X)
The space
, then
Y
X
i s compact.
is bornological space, if and onZy if
X
is
realcompact .
(iuacnbin [121- Shirota [11 ) . If
C ( X ) is barreZZed and E is a Frgchet space, then
C(X;E)
is barre 2led.
If X is a realcompact space and E is metrizabZe is borno Logica L . (Mendoza [ i l - Schmets [ l l ) .
0.12.4
(See a l s o Schmets [2])-
be t h e space of a l l continuous mappings from
C(X;Z)
Let
, then C ( X ; E )
X
co
to
Z , where
U K n , K n is compact, Z is metric, and A c C ( X ; Z ) n=l be zhe topology of pointiJisc convergence i n X on C ( X ; Z ) . Then
Let
T
P
7
f o r every witii
X =
f
f
E
ti 'p
A
i in
C ( X ; Z ) ) there is a countable subset
D cA
.
(Floret [ l l , 3.8). Let X be a Hausdorff topnlogical space and E a l o c a l l y convex space over K (K = R o (c) . Let C ( X ; E ) be t h e space of a l l continuous
0.12.5
20
Chapter 0
functions takirlg
A
w W
C(X;X)
i s localizable in
C(X;E)
E, enduwea w i t h t n e compact-open t o p o l o g y . I f
intu
i s a subalgebra o f
tnat of
X
and
under
Stone
Y
=X
C
C( X;R)
L(X;E)
8
in
C(Y;t)
, such
that
flY
belongs
f o r each equivalence
.
f o r modules
C(X;E)
.
.
W c C ( X;E)
Every A-module
(Prollat31,
§
is
4 ,1.5).
E be an i n f i n i t e dimensional Banach space. Then there
Let
e x i s t s a sequence c$j(,
$,
f
w/Y
be a subazgebra
localizable uxder A i n 0.12.6
i f t h e compact-open c l o s u r e
C(X;E)
(m0d.A).
- Wgierstrass theorem A
i s an A-module, we w i l l say
in
i s the set o f a l l
t o t h e compact-open c l o s u r e o f class
W c C(X;E)
A
E ' such t h a t
// $11
=
1 for all
n e
N
0 i n t h e cr(E';E)-topology. (Josetson [ 11- Nissenzweig 11 1).
and
0.12.7
-+
Let
E be a separable Banach space.
The following are equivalent: E
(1)
contains
no subspace isomorphic t o 1
(21 Each bounded subset o f dense zn i t s o(E";E') -closure.
'.
E " i s ci(E";E')-sequentially
(Rosenthai I1 I).
0.12.8
E
Let
be a Banach space. The following are equivalent
(1)
E
(2)
There e x i s t s
i s non-reflexive. f E E ' which does not achieve i t s norm.
( F l o r e t 111, 5.31.
0.12.9
Let
E
be a Banach space
.
E
i s reflexive
,i
f and only i f i t
i s f a l s e t h a t f o r each number 8 < 1 it i s possibEe t o embed o f bounded functions defined on a s e t
A i n such a way t h a t A contains
the p o s i t i v e integers and, for each p o s i t i v e
E
with
E i n a space
n
, there
i s a member zn
Of
Prel iminary r e s u l t s
a E A
.
0.12.10 of
e
n components of zn a22 are
where t h e f i r s t
21
I tll 5
and
1 f o r a22
(James [23).
Every separable Banach space E
i s isometric t o a quotient space
( L i n d e n s t r a u s s - T z a f r i r i [l], p.iO8 ).
11.
0.12.11
if E
, we
i s a H a u s d o r f f l o c a l l y convex space
t o p o l o g i c a l and a l g e b r a i c duai spaces by a dual p a i r
, we
>
E'
w i l i assume t h a t
and
E*
w i l l denote t h e
r e s p e c t i v e l y . Given
i s a subspace o f
F
G*
,
using
the canonical i n j e c t i o n .
Let E be a Hausdorff ZocaLZy convex space such t h a t ( E ' ; a ( E ' ; E ) )
.
i s separable
Let x0 be a point o f the algebraic duaL of
E U
the Zinear huZZ o f
( E ' ;a(E';F))
then
{XO}
F
E'. I f
is
.
i s separable
(Valdivia [ z l j .
Let E be a Banach space and F a
of
E ' . Let z
H
be the linear h u l l o f
of
F,
I / Unll
Proof.
.
Hence there e z i s t s a subset {un:n
Let
: E'*
be t h e subspace o f
G
E'*/G
-+
[(E'"/G)'
;a((E'*/Gj'
follows t n a t
t
111 , t h e n
f o r each
n
$ = 0 E
N
s a t i s f i e s (J(un)
6
NIl
=
0
u(F; .ir(H))
. then
E
and
(F; u ( t ' ; t ) ) 10
n(z)
IT(E).
and
t h e above r e s u l t
,
it
separable. T h e r e f o r e , t h e r e e x i s t s n(H)
I n o t h e r words (I = 0
F
By h y p o t h e s i s
$(un) = 0
satisfies
,
(J E
if
. Ubviously
n
satisfies
we can cnoose each
f o r each (J(un) = 0 un
with
. Let
A be a subset
i n the o(E";E'j
of
,
.
1 and a c c o r d i n g
; E'*/G)
is
{un : n E M } c F such t h a t if $
norm 1
@ 6 H
ortogonal t o
El*
the canonical p r o j e c t i o n
From t h e c a n o n i c a l i d e n t i f i c a t i o n between
n e
subspace
n E IU ,then (I = 0 .
for m y
T
E U {z}
1 f o r each n E IN ,such t h a t i f
=
separable
z i s not continuous in ( F ; u ( E ' ; E ) ) and Zet
that
E E"such
u(E' J E )
of a 8anach space
E
.
Assume t h a t every
( E ' ; o ( E ' ; E ) ) .Then A zs weakLy r e l a t i v e l y compact.
Let A be a subset of a quasi-compLete space
E
.
ous reaZ valued f u n c t i o n on E i s bounded on
A
, then
tiveZy compact.
Z
-closure of A i s continuous on every separable subspace
(valaivia [ l l j .
(Valdivia
111 ) .
I f every weakly continu-
A i s weakZy reLa
Chapter 0
22
il)
E be a Hausdorff l o c a l l y convex space, K be a compact topological space and 1-r be a p o s i t i v e measure defined on the Borel subs e t s of K such t h a t p ( K) = 1 if f : K + E i s a contznuous mapping 0.12.12
Let
.
and t h e convex h u l l
c o ( f ( K ) J of
, i f- E is complete) c o ( f ( K ) ) such t h a t
cular
, then
f(K)
i s r e l a t i v e l y compact ( i n p a r t i
there e x z s t s a unique vector
I _ _
y
E
f o r every
X'
e E ' , I t i s defined b y
f dp = y . JK
(2)
If v i s any p o s i t i v e measure defined on t h e Borel subsets of
some scalar multiple
w1
with v i n the place of co(f(K))
.
Therefore
of p
, the
w i s such t h a t
. In t h i s
. Thus
K
,
( i ) hoZds
case y i s not necessariZy
resuZt (11
valued Bore1 measures. (Rudin
vl(K) = 1
in
can be extended t o compZex [I 1 .3.26 ; 3 . 2 7 ) .
23
Chapter 1
APPROXIMATION OF SMOOTH FUNCTIONS ON r1ANIFOLDS
T h i s c h a p t e r begins w i t h a s e c t i o n d e d i c a t e d t o W e i e r s t r a s s ' theorem on t h e a p p r o x i m a t i o n o f d i f f e r e n t i a b l e f u n c t i o n s by Dolynomials. S e c t i o n s 1.2, 1 . 3 and 1.4
a r e c e n t e r e d around t h e problem o f d e s c r i b i n g
denses subalgebras i n t o p o l o g i c a l a l g e b r a s o f d i f f e r e n t i a b l e f u n c t i o n s .
,
N a c h b i n ' s theorem (Cm(X), :T
c h a r a c t e r i z i n g denses subalgebras i n t h e a l g e b r a
) , i s given.
The concept o f
m-admissible a l g e b r a s i s i n -
troduced, a l l o w i n g us t o o b t a i n d e n s i t y r e s u l t s i n t h e s p i r i t o f Nachbin's m m n theorem, f o r t h e a l g e b r a s (CT(X), T~ ) , (C!(X), and S (R ) .
TI)
Sectionsl.5,
.
i n (CF(X,F), ):T
1.6 and 1.7 a r e focused on a p p r o x i m a t i o n r e s u l t s
The problem o f c h a r a c t e r i z i n g A-module
A c CF(X)
c l o s u r e i s s t u d i e d , where
M
c CF(X,F)
i s a n o n t r i v i a l a l g e b r a which i s
s e p a r a t i n g and does n o t v a n i s h a t any p o i n t on
X . T h i s assumption a l l o w s
us t o reduce t h e a p p r o x i m a t i o n problem t o t h e compact open case, and t h e a p p r o x i m a t i o n on m a n i f o l d s i s reduced t o t h e a p p r o x i m a t i o n on open sub-
Rn.
sets o f
The above r e d u c t i o n s a l l o w u s t o o b t a i n a d e s c r i p t i o n o f
dense polynomial a l g e b r a s r e l a t e d t o Stone and Nachbin c o n d i t i o n s . F u r thermore space
F
we o b t a i n , under some r e s t r i c t i o n s on t h e module
, a pointwise description o f
M
and t h e
r e l a t e d t o Whitney c o n d i t i o n s .
1.1 W e i e r s t r a s s ' t h e o r - . L e t (p,)
denote t h e sequence o f p o l y n o m i a l s d e f i n e d by: 1
-+ 2 (x' - p i ( x ) ) Then, i t i s c l e a r (i)
-1 5 x 5 1
y
.
by i n d u c t i o n t h a t :
Ospn(x)sIxl,
From ( i ) i t f o l l o w s t h a t
n s N (p,)
,
- 1 s x s l
.
i s a bounded and i n c r e a s i n g sequence.
24
Chapter 1
From D i n i ' s theorem i t f o l l o w s t h a t
-
lim n+
g(x) = belongs t o t h e
-(ii) 1x1 e P(R) Let
Since fa ,m
f,,,(x)
-
T~
fa
,
p n ( x ) = 1x1
c l o s u r e of polynomials i n
-
T~
,,, : R
-f
closure o f
P(R)
in
C( [ - l , l ] ) .
Hence
C(lR).
R be t h e mapping g i v e n by:
[Ix-al t (x-a)]
=
1 5 x 5 1
y
, from ( i i )
i t follows that
e K R T i n C(R). Let
ga,,(x)
Since
9
a,b
=
: R +R
be t h e mapping g i v e n by:
i'
,
x 5 a
m(x-a)
,
a < x < b
m(b-a)
,
x >_ b
g
- f b Y m ) we have t h a t a,b - (fa,, l a r y the trapezoidal function:
belongs t o t h e c l o s u r e o f Let
F
P(R)
in
gaYb e
p0
in
c(R).
Simj
C(R).
be a r e a l l o c a l l y convex H a u s d o r f f space and
m
E
<.
1.1.1 Lemma. P(R;F) Proof.
is
m
Tu
- dense in
.
m E N , only t r i v i a l modifications being f e C(R:F) , I = [ a,b] and a e c s ( F ) a ( f ( x ) - f ( y ) ) < 1/2 i f x,y e I L e t 6 > 0 be such t h a t
We prove t h e r e s u l t f o r
necessary f o r t h e case be given.
c"(R;F)
m=-
.
Let
Approximation o f smooth f u n c t i o n s
I
and I x - y
< &.Choosing
a trapezoidal p a r t i t i o n o f the u n i t y { $ l , . . . , $k}
~ u p p ( @5~ 6) , and l e t t i n g
w i t h l e n g t h of
25
,
xi E s ~ p p ( $ ~ )i,= I ,...,k
we have t h a t
Each
k
k
T: $ i ( X ) f ( X i ) ) ' i 1= l $ i ( X ) ( d f ( X ) i=l
-
a(f(x)
-
6 P(R)
@i
in
,
C(R)
5 1/2
-f(xi)))
p1 ,.
t h e r e f o r e we can f i n d
x
Y
. . ,pk
6
I
E P(R)
such t h a t :
So we conclude t h a t :
Assume t h a t Let
f 8 Cmtl(R;F)
P(R;F)
,r
i s dense i n
1 and a e c s ( F )
q E P(R;F)
be such t h a t
, let
-
x > -r 1
8
E'
u =f(x)
such t h a t
q(x)
l(u) =
-
h(-r) =
a(u)
is
.
. ,
such t h a t :
= f(-r)
.
x E [-r,r],
For
ci
.
If
h = l ( f -4)
,the mean
t E (-r,x)
,
[-r,r]
P(R;F)
i
is
=
T:
.
O,,..,m+l
-
dense i n
Cm(R;F)
m EN.#
1.1.2 Theorem
m -ru
111 <-
and
on
Then by i n d u c t i o n we have t h a t
Proof.
N
g=f'
, a ( f ( x ) - q(x)) = h(x) ( x + r ) l ( f l ( t ) - q ' ( t ) ) 5 2 r c i ( g ( t ) - p ( t ) ) 5 1 . Hence < 1
for all
m E
Letting
From t h e Hahn-Banach theorem t h e r e e x i s t s
v a l u e theorem a p p l i e s and f o r some = h(x)
.
, q(-r)
q' = p
.
f o r some
be g i v e n
p e P(R;F)
from t h e assumption t h e r e e x i s t s
Let
Cm(R;F)
The s e t
P(R";F)
is
Assume t h a t t h e r e e x i s t s n E - dense i n Cm(R n ;F) f o r a l l
TmU
N
Under a n a t u r a l i d e n t i f i c a t i o n we have
, m
in
dense n
and
2
1 F.
,
ci
n n (R ;F).
such t h a t
P(Rn;F)
F i x one such p a i r .
=
Chapter I
26
; F) = Cm(R ; Cm(Rn ; F ) )
Cm(Rntl
as t o p o l o g i c a l v e c t o r spaces g
6
T m n C (R ; F) -+ p
.
Given
Ig E
p E P(R)
t h e mapping
Cm(R ; Cm(Rn ; F ) )
i s continuous, hence from t h e assumption
Since
i s arbitrary i t follows that
p
m n P(R;C (R ;F)) = P(R)
; F ) = P(Rntl
Cm(Rntl
m n (R ;F) c P(Rntl
; F)
.
, i t follows that
Also, from (1.1.1)
Hence
IC
;F)
and we o b t a i n theorem 1.1.2 by i n d u c t i o n . #
1.2. Nachbin's theorem. Let n, m
E N,m
X
Cm
H a u s d o r f f m a n i f o l d o f f i n i t e dimension
.
2 1 , and G c C m ( X )
Theorem.
1.2.1
be a r e a l
RIG1 i s
?!-dense
in
m C (X)
, if and
only i f t h e fo2-
lowing conditions holds : ( i ) G i s strongly s e p a r a t i n g . ( i i ) For every x 6 X and g E G
there e x i s t s Proof.
Assume t h a t
f E Cm(X)
+
, such
dg(x)(v) # 0
that
RIG1 i s :T
f(x) E R
v E Tx(X) = t a n g e n t space a t
-
dense i n
Cm(X)
i s continuous f o r every
{ f E Cm(X) : f ( x ) = 01
and
x
.
v e T,(X)
,v #
0
,
f 0,
x E X
Since t h e mapping
, it follows that , x # y , are
I f E Cm(X) : f ( x ) = f ( y ) 1
proper c l o s e d subalgebras; hence c o n d i t i o n (i)h o l d s . Also, g i v e n and
,v
.
the set o f a l l
g
6
Cm(X)
such t h a t
x E X
dg(x)(v)=O
i s a p r o p e r c l o s e d subalgebra; hence c o n d i t i o n ( i i ) h o l d s . Conversely, l e t K c X be compact. L e t W c X be open, w i t h
Approximation o f smooth f u n c t i o n s
Kc W
w
and
compact.
1 . 2 . 2 . Lema. i s t h e mapping
gl
There e x i s t s
. . YgN)
@= (91 ,.
,.. . , gN 6
Cm
For every
(iii)
If b i s the o r i g i n i n RN , then 0 B
x
W
G
0
@(W) + R
i t follows that
$ : RN + R.
-
Since
0
f
0
@(w). @ ( W ) C RN
n.
By a p p l y i n g t h e i n v e r s e f u n c t i o n
Cm
diffeomorphism f r o m
i s c l a s s Cm.
i s class
n.
From ( i i ) i t f o l l o w s t h a t
@ i s a local
0-l : @(W) + W
0-l :
(0.2.2)
d @ ( x ) has rank
submanifold o f dimension
theoremywe have t h a t
RN
C m - homeomorphism.
(ii)
@(W)
+
is a
@/W : W
+
@ : X
the following conditions hoZd:
(i)
Therefore
f
such t h a t i f
G
, then
Assume t h a t lemma 1 . 2 . 2 i s t r u e . is a
27
Cm.
If
f
W
to
@(W).
Cm(X), t h e mapping
6
From W h i t n e y ' s e x t e n s i o n theorem
@-I : @ ( K )
-+
R extends t o a
Cm - f u n c t i o n
Therefore,
+(D) =
d @(W) we can assume t h a t
From W e i e r s t r a s s '
0.
theorem 1 . 1 . 2 i t f o l l o w s t h a t t h e r e e x i s t s a sequence o f p o l y n o m i a l s pr(xl,.
.. ,xN) k apr
(*) Since
such t h a t :
r+m
pr(0)
+
s t a n t term. I f
k
3 fr
(r
0
a kiil
u n i f o r m l y on I01 m)
+
f r = pr
-
k
3 f
o
, we
u
@(w), f o r
can assume t h a t each
0 = pr(gl
,...,gN) , f r o m
a l l k e Nny I k l pr
5
i s without con
(*) i t follows that:
u n i f o r m l y on K, f o r a l l k e Nn
, Ik
r +m
Since each
fr e R[Gl t h e p r o o f o f theorem 1 . 2 . 1
i s finished
#
P r o o f o f lemna 1 . 2 . 2 . functions ( i = 1,2,3) , N = Ni w i l l now be d e f i n e d . The f i r s t N1 w i l l a s s u r e t h a t b
d
t h e second
W ; and t h e l a s t
NP t h a t
d @ ( x ) has r a n k
n
f o r each
x
6
N1
+
N2
+ NB
@(w) , b B RN;
N 3 t h a t @ w i l l be i n j e c t i v e . For each
x e
there exists
gx
6
G
such t h a t
gx(x) # 0
.
m.
Chapter 1
28
U,
Let
c Ux
that
1
w
Since
u . .. u
YgN, e G
91 Y
.
t e U,
f o r every
U
x e X
# 0
dfl(x)(ul)
.
# (o,...,O)
, ul fly
,1
G
6
.
.
5
,..., A n
,
= ei
L(hnun) =
+ clY2e2
# 0 , 15 i 5 n
Cm
e
2
... t
clYnen
+
...
‘2,nen
+
. . . . . . . . . . . . en
diffeomorphism a t
t h e r e e x i s t { V l ,VP
Let
u 2 e Tx(X) ,
,...,f n )
u ~, . . . , u ~e T ~~ and and c o n s i d e r
.
For some constants
L
*
i s o n t o and t h e r e f o r e i t i s an
By a p p l y i n g t h e i n v e r s e f u n c t i o n theorem we have t h a t
isomorphism.
,... ,b
fA(t))
,
dfn(x)(u)en
+
From t h i s i t f o l l o w s t h a t t h e mapping
,...,
i = l,.*.,k
we have
L(hzU2) =
i = 1,2
Otherwise we remark be such t h a t no
i s t h e canonical b a s i s i n Rn
L(hlul)
i s a local
such t h a t
I n t h i s way we o b t a i n
... t
L(u) = dfl(x)(u)el + el,...,en
8.
-1
.
XI
UN1 such
n-1
dfi(x)(u.) = 0 , 1 5 i < j < n L e t f, = ( f l J t h e l i n e a r mapping L :. Tx(X) + R n d e f i n e d by :
where
....,
Using t h e h y p o t h e s i s we o b t a i n
f l y..., f n 6 G such t h a t d f i ( x ) ( u i )
and
XI
I n particular, there exist = 0
dfz(x)(up) # 0
such t h a t
f x = fl.
5 k
x e
fle G
Say
Hi = Li ( 0 )
If
vanishes i d e n t i c a l l y .
n ...n Hk) 2 n - k . # 0 , such t h a t d f l ( x ) ( u 2 )
f2E G
0.
n = l we t a k e
..., f k
t h e dim(H1 u2
U
exist functiork
f o r every
, u1 #
e Tx(X)
I n case
the following : l e t Li = d f i ( x )
, there
gx(t) # 0
such t h a t
such t h a t
(g1(x),...,gNl(x)) Let
X ,
c
i s compact, t h e r e e x i s t
Therefore
Nl*
, U,
of x
be an open neighbourhood
,,. .,Vb 1
there e x i s t is a
Cm
x
. By compactness we conclude
such t h a t f o r each
open c o v e r i n g o f
f: ,...,f,!, e G
diffeomorphism from
fx that
such t h a t
Vi
Hi(t)
= (fl(t)
,
t o an open s e t o f Rn.
Approximation o f smooth f u n c t i o n s
i 9Nlt(i-l)n+j = fj
i = l , ...,b
’
29
, N 2 = nb .
j = l,...,n
F i n a l l y c o n s i d e r t h e compact s e t
w-
Q = w x
(x,y) E R
If
then
CVlX
Vl
x # y
,
open neighbourhood u(x,Y) f o r e v e r y ( ~ , n )E U (XYY) such t h a t
u
...u
1
Vb x Vb
.
t h e r e f o r e f o r each
, and
o f (x,y)
.
(x,y) E Q there e x i s t s
f E G
such t h a t f ( S ) f f ( u )
By compactness t h e r e e x i s t
hl
,. .. ,hC
E G
Let
gN1+N2+ i
,
= hi
i
=
1, ...,c
,
, @(x) =
Now i t i s c l e a r t h a t t h e f u n c t i o n @ : X + R N N =
N1 t
1.2.2. 1.3.
N2
N3 = c ( g l ( x ) , ....gN( x ) ) ,
, s a t i s f i e s c o n d i t i o n s ( i ) , ( i i ) and ( i i i ) i n lemma
t N3
# m-admissible algebras. I n t h i s s e c t i o n t h e concept o f m-admissible a l g e b r a i s i n t r o -
duced.
The most i m p o r t a n t spaces a p p e a r i n g i n
a r e examples o f m-admissible a l g e b r a s .
distribution
theory
We c h a r a c t e r i z e m-admissi b l e a 1
gebras among t h e i r c l o s e d subalgebras. 1.3.1
n
.
Definition.
Let X be a r e a l Cw
Hausdorff manifold of dimension
m
Topa(X) w i l l denote the c l a s s of a l l topological algebras
A such
that (i)
A
(ii)
m n ) c A. C ~ , ~ ( A C,(A )-
(iii)
A contains
(iv)
For any
i s a subalgebra o f
CF(X)
x E X
and
m C (X)
.
as a dense s u b s e t . v E T x ( X ) = tangent s p a c e a t x
,
30
Chapter 1
f 8 A
the mappings
* f ( x ) e R and A
An element
on X.
The topology
E
6
(XI w i l l
Top!
A
on
f
A * d f ( x ) ( v ) s R are continuous. b e c a l l e d an m-admissible a l g e b r a
w i l l be denoted b y
T ~ .
Exampl es : 1)
Cm(X)
endowed w i t h t h e t o p o l o g y T :
.
2)
C:(X)
endowed w i t h t h e t o p o l o g y
,
endowed w i t h t h e coarser o f t h e t o p o l o g i e s f o r which
3 ) C:(X) t h e i n c l u s i o n s C:(X) = CC(X)
4) Crn(X) in
Let
a r e continuous f o r a l l
m 6
CmVm(X) be a weighted a l g e b r a .
N
where
For a p p r o p r i a t e
we g e t as p a r t i c u l a r cases t h e t o p o l o g i c a l a l g e b r a s r n n and S (R ) Denoted by EmVm(X) t h e c l o s u r e o f C:(X)
.
C!(X)
CmVm(X).
CmVm(X) = EmVm(X)
When
f dmVm(X)
CmV,(X)
,
as i n t h e p a r t i c u l a r cases
, i t f o l l o w s e a s i l y t h a t CmVm(X)
mentioned above when
c C!(X)
.
V
choices o f
T:
,
E Top!(X).
Even
.
EmVm(X) E Top!(X)
i t can be shown t h a t
I n f a c t t h i s i s a consequence o f t h e c o n t i n u i t y o f t h e mappings g E A where 1.3.2
, Q e CiI1(R)
E A
rp.9 A =
+
Definition
. '9,
f
TOP:,^
.
* gh e A , h e CF(An)
w i z z denote the class of ~ Z Z A
E
TO~!(X)
TA
C c ( g l y . . . y g n ) c R[g,,...ygn]
f o r each
m CC(X).
(X) i s c a l l e d an m - a d m i s s i b l e a l g e b r a a ,c o f compact t y p e . Further,as a consequence o f t h e n e x t p r o p o s i t i o n
A e Topm
An element
on X
(XI
m
Ciyl(g,) 3 . .
g 6 A
n = dim(X) ,then
n m h that i f
g,
and
CmVm(X).
i t f o l l o w s t h a t a l l t h e known examples o f rn-admissible algebras a r e o f
compact t y p e . However
t h e q u e s t i o n remains whether
or not. 1.3.3
Proposition.
Then A E TOP!
YC
(X).
Let
A B Top!
(X)
be such t h a t
Topm ( X ) = Topr(X) a ,c
rn T ~ ( C : (X) <_ -ri
.
Approximation o f smooth f u n c t i o n s
m I t i s enough t o prove t h e p r o p o s i t i o n i n t h e case .rAICC(X) =
Proof.
L e t I $ ~ e C;,l(R) bourhood o f 0
I$ E C F (Rn) m
in
C:
in
X
q
and
e CF (K)
f
and
go,
... ,gn
.
Then
WK
pjyl(f)
<
m
E Cc ( X I
E
and
u
K = supp(go)
n C: (K)
W
Jo c I
( K ) . Hence t h e r e e x i s t s
such t h a t Let
be g i v e n
T~
i s a compact subset o f 0
31
W
T
m
i.
a neigh-
... u
supp(gn)
i s a neighbourhood o f
f i n i t e , 1 E R1,
and
imply t h a t
j e Jo
> 0
E
f 8 WK
be a polynomial on Rn+1 w i t h o u t c o n s t a n t term and d e f i n e
.
(x1 Y . .
YXn+l
) e Rn+'
.
Then i t i s c l e a r t h a t J, e Cm (Rntl)
.
-1 0 $j )..., hn = gn 0 $ -1 j . Since m n h o y . . . , h n e Cb ( $ . ( V . ) ) , f o r e v e r y C( e 1, there e x i s t s a constant J J ccL > 0 such t h a t on @ . ( V . ) we have J J For
j
e J o fixed, l e t ho= go
for all
m ntl C (R )
.
,
where t h e sum i s t a k e n on t h e s e t Nn+' lal > 0 such t h a t From t h i s i t follows t h a t t h e r e e x i s t s a c o n s t a n t c j E
From ( 1 . 1 . 2 )
i t follows t h a t the set o f a l l m s t a n t term i s T ~ dense i n the set o f a l l
q(0) = 0 . s e t i n Rn+' , t h e n g i v e n
H
Since
laY$l 5
E'
on
H
= { ( g o ( x ) ,...,gn(
for a l l
there e x i s t s a constant
Ifwe choose
E'
E'
> 0 y
c'
> 0
such t h a t
cjyl-
j ,1
and t h e p r o o f i s f i n i s h e d . #
x)) : x
e P(Rn+')
q E
C (R
e X}
, we can choose e Nr+l,
m
ntl
without con
)
such t h a t
i s a compact subq such t h a t
From t h i s i t f o l l o w s t h a t
such t h a t
E'< E
, j e Jo
i t follows that
.
32
Chapter 1
1.3.4
B
Definition.
x
Let
be a l o c a l l y compact Hausdorff space.
o f continuous functions on X has p a r t i t i o n s of u n i t y on compact K of X and any f i n i t e
subsets of X i f f o r any nonempty compact subset open covering
Vl
,...,Vr
el
=
1 on
t
... t
8,
1.3.5. Lemma.
,i
K
there e x i s t 01,...,0,.
, 0 2 ei 2
. Then
1 and
X,
,x #
y 6 X
g = 1 on a neighbourhood of
y and
0
B
such t h a t
,
c Vi
supp(Bi)
y
,
i = 1 ,..., r . B
g = 0
X,
g
there e x i s t s
Cc(X)
c
B
8
on a neighbourhood
5 g 5 1. The s u f f i c i e n c y w i l l
C l e a r l y t h e c o n d i t i o n on B i s necessary.
Proof.
E
B has p a r t i t i o n s of u n i t y on compact subsets of
f and only i f f o r any
such t h a t of
K
of
Let X be a locally compact Hausdorff space and
be a subalgebra X
A set
x
be a consequence o f t h e f o l l o w i n g r e s u l t : g i v e n neighbourhood o f
g,
x, t h e r e e x i s t s
E
B
E
X
such t h a t
and
V
, an
open
gx = 1 on a n e i g h
, 0 5 g, 5 1 and supp(gx) c V . I n f a c t , f r o m t h e hyp o t h e s i s t h e r e e x i s t s g e B such t h a t g = 1 on a neighbourhood o f x and 0 <- g <- 1. I n case supp(g) c V , we t a k e g, = g . Otherwise , l e t H = ( X -, V ) n supp(g) . F o r every y 6 H t h e r e e x i s t s h E B bourhood o f x
Y h = 1 on a neighbourhood o f x , h = 0 on a neighbourY Y hood V o f y and 0 5 h < 1 Since H i s compact , t h e r e e x i s t Y Y y, ,...,yn E H such t h a t H c V U U V Then i t i s enough Yl Yn t o t a k e g, = g hy h 1 Yn Now l e t K be a nonempty compact subset o f X and V1,. ,Vr such t h a t
.
-
...
.
...
.
..
be an open c o v e r i n g o f t h a t f o r every compact and that every
.
c Vi
.
.
Using t h e r e s u l t a l r e a d y proved i t f o l l o w s
there exists
neighbourhood o f
supp(g,) K
x E K
K
x
i E {l,
such t h a t
g,
...,r l
= 1
, g,
on
and d e f i n e
B
and
Wx
a
, 0 5 gx 5 1
F c K f i n i t e such
By compactness t h e r e e x i s t s
.
u W, L e t Fi = I x e F : W, c Vil XEF Fi i s nonempty. For i e { l , . . . , r 1 f i x e d c
W,
E
We can assume t h a t
,
let
Fi = { x i
,..., x l l
Approximation o f smooth f u n c t i o n s
= 9
fl
4 1 ~=
If 0
5
5
$i
x1
... +
fl +
1 and
33
, t h e n $i E B and sup^($^) c Vi . A l s o 1 on u W, s i n c e $i = 1 - ( 1 - gxl) . ,
fl
$i =
xsFi
... ( 1 -
Then
.
)
gx
1
ei e
,el + ... +
B
,
i = l,...,r
Now we d e f i n e
1.3.6. Lemma.
Or
= 1 on
r ,S
Let
that
g(x) = s go e Bo
bourhood o f and hence
and
, go and
y
B
,
v
, be
.
given
, $( -m,r]
. Then
B cB
$ 0
Let
g(y) =
r'
.
Let
go= $
= 1 on a neighbourhood o f
0 5 go 5 1
.
c V,
supp(ei)
= 0
B
x
E
E
Tx(X)
Let X be a r e a l
X be given
.
x
$ E C(R)
has p a r t i t i o n s
g E €3
0 < r'< r such
, B o = B n Cc(X) . , g o = 0 on a neigh-
i t follows that
compact subsets o f
Bo
¶
X .#
Cm Hausdorff manifold of dimension
Assume t h a t
, v # 0 , there
g
o
B c Co(X)
, + = 1 on a
r ' be such t h a t
From (1.3.5)
has p a r t i t i o n s o f u n i t y on
Lemma.
and l e t
x # y
x,y E X
s
i s a s t r o n g l y separating algebra, there e x i s t s
B
Then
every
and
Assume t h a t there e x i s t s
0 < r
such t h a t
neighbourhood of S , 0 5 $ 5 1 and of u n i t y on compact subsets of X.
1.3.7.
5 ei 5 1
Let X be a l o c a l l y compact Hausdorff space and
and r e a l numbers
Proof.
0
as we wanted .#
be a strongly separating subalgebra.
Since
,
K
exists
G
c
Cm(X)
n
i s such t h a t f o r
g E G f o r uhich
dg(x)(v) # 0 .
Chapter 1
34
m of x such t h a t C ( X ) I V,
Then there e x i s t s a neighbourhood V,
I n t h e p r o o f o f lemma 1.2.2
Proof. a
g,
6
Let
V,
.
W,
f E Cm(X)
+(Y) =
$ E C:
1.3.8.
io
(Rn)
Theorem.
subalgebra.
and
A
g E B
B
B=A
g-l
g = gxIWx is
is a
Cm d i f f e r e n t i a b l e .
such t h a t
y E gWx)
,
Y
E
Rn
.
g(Wx)
, as we wanted t o prove.#
(X) and assume t h a t B = A i s a cZosed a ,c i f and onZy if t h e foZlowing conditions hold:
x
6
X and
.
v E Tx(X)
,v
X
.
f 0 , there e x i s t s
.
dg(x)(v) f 0
B = A
v E Tx(X)
and
Conversely go,...,gn
E
m ,c
,n (X) ,
B,
,
Since
C:
, the
v # 0
set o f a l l
g B A
such t h a t
i s a p r o p e r c l o s e d subalgebra, hence c o n d i t on ( i i ) h o l d s .
dg(x)(v) = 0
Top,
Y
( X ) c B , from ( 1 3.5) i t f o l l o w s Also, g i v e n has p a r t i t i o n s o f u n i t y on compact subsets o f X
B
x E X
S-l)(Y)
0
has p a r t i t i o n s of u n i t y on compact subsets of
P r o o f . Assume t h a t
hence
such t h a t and
6 Toprn
( i i ) For every
E
x
of
gx(Wx) c R n
f ( V x = I$ .g( V,
Let
Then
(i)
A
we observed t h a t t h e r e e x i s t s
let 0(Y) ( f
that
I
be a compact neighbourhood o f x c o n t a i n e d i n t h e i n t e r i o r of m n L e t 8 B Cc(R ) be such t h a t e = 1 on g(Vx) , supp(8) c g ( N x ) .
Given
Then
Wx
Gn and a neighbourhood
b i j e c t i o n o n t o t h e open s e t
m n
c Cc(G ) V.,
B
C:
(B!)
, assume
t h a t c o n d i t i o n s ( i ) and ( i i h o l d . L e t m m = B fl C, (X) and E Cc (R") . Since
+
, Bc
= dim(X)
it follows that
c B,
.
Since
C:
(X)
i s dense i n
A
, it
i s enough
C! (X) c Bc . L e t f E C F (X) and K = s u p p ( f ) . C o n d i t i o n (ii)i m p l i e s (lemma 1.3.7) t h a t f o r e v e r y x E K t h e r e e x i s t s a corn-
t o show t h a t
Approximation o f smooth f u n c t i o n s
V,
p a c t neighbourhood condition ( i )
of
xl,..
on
K
Since
and
(i)
e K
r
supp(ei)
,
Vx,
c
I V,
e Cc (B:)I
such t h a t 8
€il,...,
r
erf
t
( B n ) l V,
.
V,
Since
u
K
and
ei
-
e Bc
eif
and by is
K
... u ix. r
c B such t h a t 81t...t8r
i n particular
1
...
C;
Illx
X1
there e x i s t
elf +
f =
.,x
f
e
flV
such t h a t
we can assume t h a t
compact, t h e r e e x i s t By c o n d i t i o n
x
35
E
, i = 1, .. . ,r.
Bc
(B:)
CF
= 1
,
i = 1,
...,r ,
f e Bc.#
i t follows that
Theorem. Let
be a topobgy on C F ( X ) such that m A = (CF (X) , T ) E Topaac (X Let B C A be a closed subalgebra. 1.3.9
T
.
Then B = A if and only if t e fo2lor;ing conditions hold: B
(i)
is strongly separating.
( i i ) For eoery x E X , and v ezists g E B such that d g ( x ) ( v ) # 0
.
Proof.
Assume t h a t
continuous
.
B =A
Tx
, it
{f e A : f(x) = f ( y ) }
,
condition ( i ) holds.
Condition
,
(X)
S i n c e t h e mapping
x e X
f o r every
E
v # 0 , there
f E A
-t
f(x)
e R is 1 and
follows that { f e A : f ( x ) = 0
x f y , a r e p r o p e r c l o s e d subalgebras. Hence (ii)
f o l l o w s as i n t h e p r o o f o f theorem
1.3.8. Conversely, assume t h a t c o n d i t i o n s ( i ) and ( i i ) h o l d .
4
E
C y (Rn)
l,C:
be such t h a t
YB) = C i y l
A E Top:
ayL
such t h a t
(X).
(6). $(O,
...,O ) C
Remark.
Bi
B2 o f
C F (Eln)=
B
Let Then
because
X
.
i t follows that
From (1.3.8)
B
has p a r t i t i o n s
i t follows that
B=A.#
x E X
e A : dg(x)(v)
A
A e
i n t h e above theorems a r e inm Top, (X) , t h e r e e x i s t c l o s e d YC
such t h a t l e t t i n g
{ i , j } = {1,2}
(i) b u t does n o t s a t i s f y c o n d i t i o n ( j )
s a t i s f i e s conditior:
6, = { g
n = dim ( X ) .
C o n d i t i o n s ( i ) and ( i i )
B 1 and
instance given
1 and
i s a s t r o n g l y s e p a r a t i n g subalgebra
dependent i n t h e sense t h a t g i v e n subalgebras
=
C;,l(B)*
, f r o m (1.3.6)
o f u n i t y on compact subsets o f 1.3.10
, ,0)
B c C,(X)
Since
CiY1(B) c B
+(O,..
e Tx(X) , v # 0 , we can t a k e 01 and BP = {g 6 A : g(x) = 0 1 .
and =
v
, then
.
For
Chapter 1
36
1.4.
Nachbin m-algebras. The Nachbin m-algebra concept i s i n t r o d u c e d i n t h i s s e c t i o n ,
and Nachbin's theorem i s extended t o t h i s new c o n t e x t Nachbin t y p e theorems f o r t h e a l g e b r a s (CF (X), Sm(Rn)
. More
, (Cy
T;)
specifically,
(X)
, T!
) and
a r e obtained.
X
I n t h i s section 1.4.1.
Definition.
C" m a n i f o l d o f dimension n.
denotes a r e a l
G c Cm(X) s a t i s f i e s conditions
A subset
(N)
i f
( i ) G i s strongly separating. (ii) g
6
.
G such t h a t d g ( x ) ( v ) # 0
E
1.4.2.
Remark.
A.
dense i n and
g
E
A
-+
A
Let
From
1.4.3.
dg(x)(v)
Definition.
i b l e algebra on
E
R ,x , that
E
Tx(X)
E
G cA
A
X
,
,v # 0 ,
E
Tx(X)
(X) .A
there e x i s t s
R[Gl i s g(x) E R
be such t h a t g
6
A
-+
, i t f o l l o w s , as i n t h e
satisfies conditions
Top!
E
v
(N).
i s caZZeii a Nachbin m-admiss-
o r ,just a Nachbin m-algebya i f dense mbalgebras of (N) , EquivalentZy
, given
a nonempty
then R[G1 i s dense i n A i f and onZy i f G s a t i s f i e s
consequence o f theorem 1.3.9
Corollary. Let T be a topology on
1.4.4
)
and
G
As a s t r a i g h t f o r w a r d
T
(X)
E
are described by conditions
G of A , conditions ( N ) ,
v
Top!
6
Let
X
subset
X and
t h e c o n t i n u i t y o f t h e mappings
p r o o f o f theorem 1.3.9
A
x
For every
C:
(X) such t h a t
Toprn (X) . Then (C: ( X ) , T ) i s a Nachbin m a,c From ( 1 . 3 . 3 ) and ( 1 . 4 . 4 ) we have
we have
(C:
(X)
- aZgebra.
1.4.5. CorolZary. (c:
(x)
(b)
Let A
E
,T
y )
is a Nachbin m-algebra
Toprn [X) a ,c
.
.
Then CF(X) is dense i n A .
CoroZZary. Let X and Y be Cm manifoZds of f i n i t e dimension.
1.4.6.
If A
(a)
8
Toprn ( X x Y), a ,c
then
Cy(X) @ Cy(Y)
i s dense i n
A.
,
Approximation o f smooth f u n c t i o n s
Now presented i s t h e concept o f s u f f i c i e n t c o n d i t i o n s so t h a t an
-
be a Nachbin m 1.4.7.
-
supporting family, providing
admissible topological algebra w i l l
algebra.
Let A be a topological space of functions defined
Definition.
.
on another topological space
X A family (X.) of nonempty closed J jEJ X i s called a supporting f a m i l y f o r A i f given B c A closed
subsets of
we have f E B
f E A
a d
m
37
when
f
I Xj
E
,j
BIXj
E
.
J
Examples. (1)
The f a m i l y which c o n s i s t s o n l y o f
(2)
.
c"'(x)
xj
Let
(Xj)jEJ
such t h a t f o r e v e r y
aj '
c
For every
'
1.4.8.
j E J n
a E N ,
X. J
cmvm ( ~ " 1 ) .
Theorem (i)
.
, there
exists
, l e t Va
,j
A E Top:
Let
E
.
J
f o r which
j' 6 J
denote t h e s e t o f c h a r a c t e r i s Then
is a supporting
(Xj)jEJ
and assume t h a t the following holds:
(X)
for
There e x i s t s a supporting family (Xj)jEJ
A I X j c Cc ( X j )
that
i s a supporting family
be a d i r e c t e d f a m i l y o f c l o s e d subsets o f
t i c functions o f the sets family f o r
, for
j E J
a21
E
TA
j E J
,
B
C
Wx =
K
, for
a21
- algebra.
A
be a subalgebra which s a t i s f i e s c o n d i t i o n s (N).For f i x e d
I
B X J. has p a r t i t i o n s o f u n i t y on corn be g i v e n . I n f a c t , assuming t h i s , l e t f E C:(X)
i t i s enough t o show t h a t
X
j' implies t h a t f o r every
a neighbourhood of
such
A.
p a c t subsets o f Lemma 1.3.7
R [go,. ..,gn1
( g o ) * Cc (gI,...,g,,)c
Then A i s a Nachbin rn Proof. L e t
A
.
m
C;,l
(ii) go,...ygn
A.
(See 0.1.2).
(3)
R"
, t h e n (Ti)
Ac = (Vi,~i)iEI
If
i s supporting f o r
A.
I t i s c a l l e d the t r i v i a l supporting family f o r
for
X
Vx
there e x i s t s
Vxn X j *
Let
of
x
x1 ,. .
such t h a t
. , xr
e l ,..., 6,
E E
x
E
, there exists
K = supp(f)n X
flVx
E
K such t h a t
C:
j (Bn))Vx
K c Wx
'
.
By compactness * - .
" wxr
BIX.J be a p a r t i t i o n o f u n i t y on K
where
38
Chapter 1
subordinated t o t h e c o v e r i n g h =
Blh t
... t
implies t h a t
orh
(B
Bih E
and (En) c
:C
).. .,Wx
Wxl
B
r
.
If h = f
C F ( E n ) ) \ Xj.
eih E
hence
I Xj
then ( i i)
Condition
X
i = 1 , ...,r
f o r every
j Consequently f e B since j E J i s a r b i t r a r y . j ' L e t x,y E X x # y be g i v e n Since B i s a strongly j ' s e p a r a t i n g algebra, t h e r e e x i s t s g E B such t h a t g ( x ) = 1 and g ( y ) = O .
and
I X J.
f
E
BIX
.
@ E
Let
C i y l (R) be such t h a t
$I = 1
on a neighbourhood o f
0
d i t i o n s ( i ) and
imply t h a t
bourhood o f
4
g
E
,f
x
R[gl
from ( 1 . 3 . 5 )
(ii) = 0
5
BI
Xj
5 1.
$I
If
f = @ og(Xj
.
Also
y
and
f E Cc ( X . )
J
on a neighbourhood o f
from c o n d i t i o n ( i i )
,@
= 0
then con
f = 1 on a n e i g h
0 5 f 5 1.
f E
i t follows that
,
1
1 Xj
Since
' has p a r t i t i o n s o f u n i t y on compact subsets o f
as we wanted t o prove.
1.4.9.
0
and
on a neighbourhood o f
Hence X
j y
#
I n o r d e r t o prove t h a t c o n d i t i o n ( i i ) i n theorem 1 . 4 . 8 m (Cm (X) T u ) , (C! (X) )!T and Sm (Rn) i t
Remark.
holds f o r t h e algebras
i s enough t o a p p l y t h e theorem 1 . 1 . 2 . t h e proof of p r o p o s i t i o n
1.3.3
i n theorem 1 . 4 . 8
conditions ( i i )
Also w i t h n a t u r a l m o d i f i c a t i o n s i n
i t follows t h a t
. That
(CF (X)
,):T
satisfies
c o n d i t i o n ( i ) h o l d s f o r every
case i s c l e a r .
1.4.10.
G
Corollary , Let
be a nonempty subset o f any o f t h e t o p o l g
m
g i c a l a l g e b r a s (Cm(X) ) T u ) , (C: ( X ) Then R[Gl i s dense i f and o n l y i f G
1.5.
,Tm ~
) (CT (X) ,T!) and Sm (R'). s a t i s f i e s conditions (N).
Modules on s t r o n g l y s e p a r a t i n g algebras. I n t h i s s e c t i o n we w i l l show t h a t f o r modules
m A c Cc (X)
over s t r o n g l y s e p a r a t i n g algebras
the : problem reduces t o t h e compact open case and t h e T
T:
M c C F (X,F)
approximation a p p r o x i m a t i o n on
m a n i f o l d s reduces t o t h e a p p r o x i m a t i o n on open subsets of Rn. Let
X
be a l o c a l l y compact H a u s d o r f f space and l e t
a r e a l l o c a l l y convex H a u s d o r f f space. i n addition that
X
When
m
1
i s endowed w i t h t h e s t r u c t u r e o f a r e a l
The topology t o be considered on
C:
(X;F)
i s T:
.
F
be
we w i l l assume
C" m a n i f o l d .
Approximation o f smooth f u n c t i o n s
1.5.1.
Lerrnna.
be a strongly separating algebra.
Then
..
Kc X
given
a compact subset of X and a f i n i t e open covering U1,. , there e x i s t s 6i '...,en which i s a p a r t i t i o n of u n i t y on E A
Un ofK,
K
m
A c Cc(X)
Let
39
subordinate
Proof.
t o the given covering. f E C T (X)
I t i s enough t o prove t h a t g i v e n
such t h a t R[fl
= 0
$(O)
, then
$
f belongs t o t h e c l o s u r e o f t h e a l y e b r a I n f a c t , assuming t h i s , l e t @ E Cm (R)
0
R generated by f.
over
be such t h a t
@ = 0
on
,
(-m,1/21
.
@ E Cm (R)
and
@ = 1 on a neighbourhood o f
1 dnd
Then A i s a s t r o n g l y s e p a r a t i n g subalgebra o f CF(X) such 0 5 @ 5 1 that $ o f E A f o r a l l f E K Hence t h e c o n d i t i o n s o f lemma 1.3.6
.
a r e f u l f i l l e d and lemma Then l e t can assume t h a t
.
m=O
Case 1:
f
1.5.1
follows.
f E C% ( X ) , @ E Cm(R) , G(0) = 0 be g i v e n and @ a r e n o t 0. Put K = s u p p ( f ) .
Given E > 0
-
0
f E
R
Case 2; m
2
~ 1
and
on
hi
all
, hi
> 0
$ 6 Cm(R).
a constant
all
Let
(Vi,@i) = f
o
$;
Hence f o r e v e r y ciYk
supp(q
o
f)
i t follows that
E Ac(X)
1
.
be g i v e n
Assuming t h a t
, see m
(0.1.2),
k E Nn
such t h a t
such t h a t
1 3 k ($ Ohi)l
5 c
S i n c e t h e s e t o f a l l such
c;,~> 0
such t h a t
letting
k
and
put
i s f i n i t e , we have
and a l l i t s p a r t i a l d e r i v a t i v e s up t o t h e o r d e r
Si(Vi).
constant
, f r o m (0.1.6)
f E R[fl
f(K).
.
.
n = dim @i(Vi) that
q
K, and
on
E
E
are contained i n
We
by W e i e r s t r a s s ' theorem t h e r e e x i s t s a
polynomial q, w i t h o u t c o n s t a n t term such t h a t I @ q l 5 1 Hence - I $ 0 f - q o f l 2 1 on X. S i n c e supD(@ o f ) @
.
m
a r e bounded
I k l < m , there exists a m~ , 1~ $ ( J ) ( h i ) I f o r j=O i s f i n i t e , there exists
y = (rn,i,l
1
1)
we h a v e .
~i E c"'(R).
r'
c Tm be f i n i t e and E > 0 be g i v e n . We can assume t h a t m i s f i n i t e . L e t I . denote t h e second p r o j e c t i o n o f r ' i n t o I . S i n c e f ( V i ) i s a bounded s u b s e t o f R f o r e v e r y i E I o , from ( 1 . 1 . 2 ) t h e r e e x i s t s a polynomial q on R , w i t h o u t c o n s t a n t Now l e t
term
, such t h a t
Chapter 1
40
Further , given y e r ' , t h e r e e x i s t s i e I,, and r > 0 such t h a t y = ( m , i , rl I ) . Hence, l e t t i n g IJJ = Q, - q ' , from ( 1 ) and ( 2 ) i t follows t h a t
Since c ! r(m+l) i s a constant and 1 ,m enough i t follows t h a t Py(Q,
o
f
-
q
o
r'
i s f i n i t e , taking
f ) 5 1 , for all
y
6
small
E
r'.
Now i t i s s u f f i c i e n t t o observe t h a t q o f e R [ f l a n d hence from (0.1.6) we conclude t h a t $I 0 f 6 R[f7.#
supp(q
0
f)
=
K,
1.5.2.
Theorem (reduction t o the compact open case). m Let M c Cc ( X ; F ) be a module over a strongly separating m algebra A c C F ( X ) . Then M = T - closure of M in !C ( X ; F ) .
.
m
Proof. I t i s c l e a r t h a t M ~ T ! - closure of M Let f - closure of M and assume f # 0 . From ( 1 . 5 . 1 ) t h e r e e x i s t s e e A such t h a t 0 = 1 on s u p p ( f ) . F i x one such e . Given a f i n i t e r ' c rm , we can assume m < m . From ( 0 . 1 . 5 ) i t follows t h a t t h e r e e x i s t s g E M such t h a t Py{O(f - 9 ) ) 2 1, f o r every y e r ' . Since ef = f and s u p p ( f ) , supp(O9) a r e contained i n supp(O), from ( 0 . 1 . 6 ) we conclude t h a t f 6 M , since eg e KM c M.# 1.5.3.
Theorem. -
(reduction t o the &elidean c a s e )
Assume t h a t m strongly separating algebra
f
R
1 and l e t A c C:(X)
+-'
if and only i f f P ( $ ( V ) ; F ) , f o r a21 charts
E
in
.
Given
belongs t o the (v,$)
be a module over a
M c C F (X;F)
6
f T!
8
C:
(X;F) , then
-closure o f
M
o
-1
Q,
A,(x).
Proof. The "only i f " p a r t i s c l e a r . For t h e converse , assume t h a t f f 0. Given a f i n i t e subset r ' of rm , we can assume t h a t m i s f i n i t e and t h e r e e x i s t f i n i t e s e t s I. c I and S c cs(F) such t h a t
r'
=
that
Im} x I . x S I Also, by enlarging I,, i f necessary, we can assume V i , i 6 I . , i s an open covering of s u p p ( f ) . F i x a compact
Approximation o f smooth f u n c t i o n s
neighbourhood
a E S.
Let
of
K Bi
6
supp(f)
A ,i
B
and
E To,
41
5 B for all
such t h a t a
E cs(F)
be a p a r t i t i o n of u n i t y on
ordinated t o the covering
n
f i x a compact neighbourhood
Hi
,
Vi
of
.
i E I.
Further
supp(f)
supp( ei)and
a chart
$i IVi
.
sub
i E Io,
f o r any
( V i , q i ) E Ac(X)
such t h a t -
Hi c Vi c Vi c Ui
(1) Let
E,E'
M
of
o
> 0 -1
Jii
be g i v e n .
Since
Cm($i(Ui)
in
and f
; F)
B ( a k [if -gi)
(2) Hence i f
hi = e i ( f - g i ) , k
B ( a (hi
(3)
Let since
, there
o$il))
i, j
6
I.
5
0$i11)
on
E'
we can choose
5
E
on
be g i v e n .
T~
M
gi E
$i(Hi)
-
closure
such t h a t l e t
qi(Ui),
such t h a t t h e r e e x i s t s a
.
Let
n
Hi = 6
compact, t h e r e ex s t s a c o n s t a n t n Cm(T;F) , k 6 Nm
S =
qj(Uj
n
Ui)
T = qi(U. fl Ui)
Hence t h e r e e x i s t s a c o n s t a n t
J cilj
> 0
CH
,i
E I.
j
=
ni
and f o r
be nonempty open s e t s i n R n
I ) : S
Cm d i f f e o m o r p h i s m
H c T
.
7
, H
T . Hence g i v e n 0 , such t h a t f o r a l l
=
+
v.J n
.
i t f o l l o w s hio$T1=O J
# 6 hence n
S, T
k E E,l 'i
all
n k E Wmi
for a l l
vj
If
, for
small enough such t h a t
E'
supD(hi)= Otherwise , V . n Vi i' J n k E 111 i t f o l l o w s from (1) t h a t
The f o l l o w i n g can be observed
Let
exists
1
all
6
m
belongs t o t h e
o
ni = dim Jli(Ui)
ting
h
= $i
Hi
,$
=
qi
o
-1 qj
.
such t h a t from t h e above remark
42
Chapter 1
and ( 3 ) i t f o l l o w s t h a t
(5)
5
'i,j,k Let
1
g =
Bi
k e
f o r every
'i,j,E
.
gi
(0.1.5)
Then
Nm'i . implies that
g 8
ieI,
Also
1
If j
e , ( f -gi). iaI, ( 4 ) and ( 5 ) we g e t f -9 =
n Since
Nmi
I, and
p ( f -9) < 1
a way t h a t mark t h a t
are f i n i t e
Y
f
a p p l y (0.1.6)
and
g
6
, we
I.
,a
6
S
and y = (m,j,a)
can choose
, f o r every
c
y
E
r' .
f s
, from
small enough i n such
Now i t i s enough t o
have t h e i r supports c o n t a i n e d i n
and conclude t h a t
w.
re
K , i n order t o
M.#
1.6. Dense polynomial algebras. I n t h i s s e c t i o n we o b t a i n dense polynomial a l g e b r a s i n
(C:
,in
a s i m p l e way, a d e s c r i p t i o n of
(X;F) i T; )
r e l a t e d t o Stone and Nachbin
conditions. Let
X
be a l o c a l l y compact H a u s d o r f f space and
l o c a l l y convex H a u s d o r f f space.
When
m 2 1
,X
Hausdorff m a n i f o l d l o c a l l y o f f i n i t e dimension. s i d e r e d on every
x
6
C T (X;F)
is
7 7
F
denotes a r e a l
a real
Cm
The t o p o l o g y t o be c o n
.
A s e t A c C m ( X ) , m 2 1 , s a t i s f i e s c o n d i t i o n (N,) , i f f o r X and v e T,(X) , v # 0 , t h e r e e x i s t s h E A such t h a t
d h ( x ) ( v ) # 0.
M c Ccm ( X i F ) be a poZynomia2 aZgebra. Then M i s 1 the dense i f and only i f M i s strongly separating and i n the case m s e t F ' 0 M s a t i s f i e s condition (No) 1.6.1
Theorem,
Let
.
43
Approximation o f smooth f u n c t i o n s
Proof.
Assume t h a t
M
i s dense.
M has a l s o t h i s p r o p e r t y
hence
f e C:(X;F)
The mapping
, and
x e X
i s continuous f o r a l l
C:
(X;F)
. When
m
-f
e F
f(x)
i s strongly separating 1
,
,
the c o n t i n u i t y o f the
mapp ing s
f
C;
8
Ft0 M
implies that h
(X;F)
C:
E
F', M
1
m (X).
dX,V
Since
dx
(0.12.5)
i s dense i n
,
A = F'o M
Conversely, l e t From
(X)) # 0
(C:
x
when
, v e
X
E
,
v # 0
Tx(X)
i t follows that
(No).
satisfies condition
A.
,
dh(x)(v) B R
,v
1 eF'
A l s o t h e mappings
-
Hausdorff, t h e a l g e b r a over
,
(X)
f E C:
o
C,
=
(X)
a r e continuous.
+
A
N
=
A I F c M.
1.5.2
and theorem
F
Since
i s s t r o n g l y s e p a r a t i n g and
N
is
i s a module
i t follows that
N, hence M,
C,(X;F).
rn 2 1
Now l e t
.
It i s clear that
i s a module over
A
,
i t i s enough t o prove t h a t K 0 +-I i s hence a c c o r d i n g t o theorem 1.5.3 m 'I -dense i n Cm($(V) ; F) f o r a l l (V,$) e Ac(X) . From ( 1 . 5 . 2 ) i t U
follows that dense i n
A
=T:
C:(X).
'c
closure o f
Hence
3
I F
0 E CT (+(V))
from (1.1.2) conclude t h a t 1.6.2 C,
$I-' i s dense i n
0
Let
dense i n
A c Cc(X)
C c (X;F)
f 8 C F ($(V);F)
, let
-
N
.
a
s u p p ( f ) . Hence
, if
n = d i m $(V),
closure o f
C F ( $ ( V ) ) BI F.
From t h i s we
Cm(+(V);F).,
be a strongly separating algebra. Then
In p a r t i c u l a r , Cc(X) I F
i s dense
in
(X;F).
1.6.3 C,
N
i s T:
belongs t o t h e
f
which i s a subset o f
Corollary.
A I F is
Also given
0 = 1 on
i t follows that
8 P(Rn;F)/Q(V)
A
i t follows that
C F ( + ( V ) ) I F.
be such t h a t
that
( X I I F.
C:
Given (V,+) e Ac(X) CF(X)
and f r o m (1.2.1)
N
Corollary.
Let
Cc (XI)
I
C,(X,)
X1
,Xz
be l o c a l l y compact Hausdorff spaces. Then
i s dense in C, (X, x X2;F) is dense in C, ( X i x Xz).
(XI) I Cc(Xz) I F
.
In particular
44
Chapter 1
m m 2 1 , be an algebra which v e r i f i e s 1.6.4. Coroltary. Let A c Cc ( X ) conditions ( N ) hen A B F i s dense i n C: ( x ; F ) . ~n particular ,
.
C:
(XI
I
Corollary. Let
1.6.5.
f i n i t e dimension. m E
for a l l
CT (x, 1.7.
IT
.
x2)¶ for
x
(x;F).
i s dense i n C;
F
be Cm
Xl,X2
(X,)
Then C:
IC :
~n particular , C;
m
all
manifolds, t h a t are ZocaZly of m (X,) I F i s dense i n Cc ( X I x X 2 ; F )
(x,)
(XI I):C
is
dense
in
IT.
E
P o i n t w i s e d e s c r i p t i o n o f closures.
M
I n t h i s s e c t i o n we o b t a i n a p o i n t w i s e d e s c r i p t i o n o f
M
under some r e s t r i c t i o n s on t h e module
T:
and t h e space
for
The
F.
corresponding d e s c r i p t i o n i s r e l a t e d t o Whitney c o n d i t i o n s . Let
be a r e a l
X
F a r e a l l o c a l l y convex H a u s d o r f f space.
dimension and
t o be considered on
1.7.1
Lemma.
the !T
0.f
h
M,
6
L
R~ ,m 2 1 and
A c Cm (Y) which v e r i f i e s conditions
5
m'
C"(Y)
i t follows t h a t the
from ( 0 . 2 . 3 )
cm
m' E
The topology
,
of L i s the s e t o f a l l
Assume t h a t
(1.2.1)
T~
Let Y be an open subspace
- cZosure , E > O and
Proof.
M
is
C F (X;F)
a moduZe ouer an aZgebra y E Y
Cm Hausdorff m a n i f o l d , l o c a l l y o f f i n i t e
i t comes t h a t
m
, there
h E Cm(Y) f o r which
satisfies T:
h
-
hoe L
exists
closure
c
c"'
(Y)
(N). Then
, given
such t h a t
t h e s t a t e d c o n d i t i o n s . From of
belongs t o t h e
L i s an i d e a l , hence
T!
-
closure o f
L
in
(Y). That t h e c o n d i t i o n s a r e necessary f o l l o w s from t h e d e f i n i t i o n
m of T U .# 1.7.2
Lemma
.
f 6
Fix
2
1 and F has the approximation property.
f 6 C F (X;F)
Then f o r a l l
Proof.
Assume t h a t m
F'o
f B F ,
f E C F (X;F)
and l e t a f i n i t e
r'
c
rm
be g i v e n . We can
45
Approximation o f smooth f u n c t i o n s
m
assume t h a t
i s f i n i t e and
.
a E cs ( F )
f i n i t e and
Let
r'
I ' x {a} , where
= {ml x
,
E Ac(X)
(Viy$i)
i E I'
I 'c I i s
,
be
t h e cor-
responding s e t o f c h a r t s . (X) , n k E N, ,
E A,
The f o l l o w i n g o b s e r v a t i o n can be made: g i v e n (V,$) then
ak(f
@ - l ) ( $ ( V ) ) i s r e l a t i v e l y compact i n
0
F, f o r a l l
n = dim $ ( V ) . I n f a c t , t h e r e e x i s t s a c h a r t ( U , $ ) E A, ( X ) V c U i s compact and $ = $ [ V . Hence a k ( f OC+-') ( $ ( V ) ) =
ak
(f
$ - l ) ( $ ( V ) ) c 8 k ( f ~ $ - ' ) ( $ ( v ) ) which i s compact. For i E I' l e t ni = dim $i(Vi) and p u t
H i s a r e l a t i v e l y compact subset o f
Then exists
h E E'
g = h
o f
E
81
a(1 If
-
F
.
Hence g i v e n E > 0
E
2
h(1))
E
.
1 B H
for all
, f r o m t h e above i n e q u a l i t y f o l ows
g E E'
o
~
f
B
_
5 E
1.7.3.
, for all
1
y E
r'
for all
'i k E N "'i
.
.
and supp(g) c s u p p ( f ) _
f 6 E l c f I E.#
that
,
small enough we o b t a i n PY(f - 9 )
Since
there
such t h a t
) Taking
such t h a t
, from (0.1.6)
we conclude
D e f i n i t i o n . Given 1 5 m < m , M c C m (X;F) , f E Cm(X;F) and has weak approximate contacts of order m w i t h M a t the point
x E X , f
x
if f o r every
exists
(V,$) E A,
such t h a t x E V and 1 E E '
g 6 M f o r which lak(l(f-g)o $
where
(X)
n = dim $ ( V ) .
3 $(x))[
In the ease
5
1
F =R
f o r a22
k E Ni
, we omit
"weak".
,
,
,
there
Chapter 1
46
1.7.4.
Mc
Let
Assume t h a t m
Theorem.
1 and F has t h e approximation property.
CE (X;F) be a module over an aZgebra A c CF (X) which s a t i s f i e s
(N) and assume t h a t
conditions
F'
o
M P F c
M.
If m i s f i n i t e , a given f 6 C: (X;F) beZongs t o R i f and o n l y i f f has weak approximate contacts of order m with M a t every point. Proof. Given ( V , $ ) l ak ( l o g o + - l ) ( + ( x ) ) \ f 8
M
then
x 6 V
and
=
11 [ a k ( g o + - ' ) ( r n ( x ) ) l ~ I l ( e ) 1s R
-f
we have
,
f o r a11 9 6 C: ( x ; F ) . i s a continuous seminorm, hence i f
satisfies the stated condition.
f
F o r t h e converse, assume t h a t o f order
1 6 F'
Ac(X)
1 E F
Also the function
,
6
m w th M
l o f e l o M
a t every p o i n t .
has weak approximate c o n t a c t s
f
1
Given
I n f a c t , according t o
6 F',
1
(1.6.4),
we c l a i m
i s an i d e a l
0
( V , $ ) E A, ( X I , 1 0 t h e g i v e n cond t i o n on
o
,
Given
s a module over a s t r o n g l y s e p a r a t i n g a l g e b r a .
i n particular
that
+-'
i s a module o v e r CF ( $ ( V ) ) , hence from 1 f, s t a t e d f o r 1 , E > 0 , and (1.7.1) , i t
follows t h a t
(1
0
f)
o
$
-1
-
belongs t o t h e !T
-
1
c l o s u r e of
o
M
o
+-I,
- Then from (1.6.1)
we o b t a i n
1
o
f
6
1
flc 1
M.
1
i s ar-
be a module as i n theorem
1.7.4,
o
o
Since
b itrary i t follows that F'
o
f P F
c
F'o
M a
To f i n i s h t h e p r o o f , i t i s enough t o a p p l y
1.7.5
f
B
CoroZlary.
M
Let
. Then
C y (X;F)
f E
t a c t s of every order w i t h 1.7.6
Corollary.
f
(X)
B
C:
every order
. Then 2
c Cm (X;F) C
R
i f and onZy i f
2
I
f has weak approximate
1 and l e t
i f and only i f , f
f 6
a t every point of
cH.
(1.7.2),#
M a t every point of
Assume t h a t m
m with
F
F cF'o M
con
supp(f).
I c CE ( X )
be an i d e a l ,
has approximate contacts of supp(f).
47
Approximation o f smooth f u n c t i o n s
Given a homomorphism T from
C F (X;F)
1.7.7
Lemma.
group
G , there e x i s t s a smallest closed subset o f X c a l l e d t h e support and denoted by supp(T) such t h a t f E CF (X;F) and
of T supp(f)
n
Proof.
Let
(XjijEJ
f
such t h a t
E
T ( f ) = 0.
=>
supp(T) =
and
supp(f)
.
X' = 6
X
(X
# 0
f
.
=
6
imply
Let
X'
T ( f ) = 0.
of
x
This
denote t h e i n t e r -
f E C y (X;F)
and t a k e
Assuming t h a t
u
n
j belongs t o i t .
X
s e c t i o n of t h e f a m i l y (XjljEJ
n
xj
denote t h e f a m i l y o f a l l c l o s e d subsets
Cc (X;F)
f a m i l y i s nonempty s i n c e supp(f)
i n t o an a d d i t i v e
such t h a t
there exists a f i n i t e
Jo c J
6. e C F ( X ) , j E J o y be a J p a r t i t i o n o f u n i t y o n - s u p p ( f ) s u b o r d i n a t e d t o t h e g i v e n c o v e r i n g . Then supp(f) c
such t h a t
Xj)
.ieJ n
1
f =
jEJ
j e J,
.
0. f J Hence
Let
T(6.f) = 0 since supp(ejf) n X j = 6 J T ( f ) = 0 and X ' i s t h e r e q u i r e d s e t . #
and
for all
Let G be a topoZogiea1 v e c t o r space and m 2 1. Assume t h a t X i s an open subset of R n and F has t h e approximation prop1.7.8
Proposition.
.
erty
Given
C F (X;F)
f
T : C F (X;F) * G k
i s such t h a t
a continuous l i n e a r mapping
3 f = 0 on
supp(T)
,for
, if
n k E Nm
all
T ( f ) = 0.
then Proof.
Let
M
denote t h e s e t o f a l l
g E C y (X;F)
such t h a t
.
We n o t i c e t h a t t h e c o n c l u s i o n i s c l e a r when supp(g) II supp(T) = d supp(T) = X. Otherwise, M i s n o t reduced t o 0; a l s o i t i s a polynomial a l g e b r a and a Since
T
CF (X) module.
vanishes on
From ( 1 . 7 . 4 )
M y we conclude t h a t
,
i t follows that
T(f) = 0
f E
w.
by c o n t i n u i t y .#
1.7.9
Remark. We n o t i c e t h a t under a n a t u r a l i d e n t i f i c a t i o n we have m Cm ( X ; F ) ' c Cc ( X ; F ) ' . A l s o i f X i s an open subset o f Rn , t h e r e e x i s t s a sequence
S c X
if
k
6
( 0 . ) i n C F (X) J and f E Cm (X;F)
such t h a t
a r e such t h a t
,1 n , t h e same h o l d s f o r e j f .
1.7.10
Corollary.
Let
ejf * f
akf = 0
on
Further
S, f o r a l l
Hence we have
X be an open subset o f
.
f o r a l l f.
R n and assume t h a t
has the approxinxrtion property If T 6 Cm (X;F)' and f E Cm (X;F) k suck t h a t a f = 0 on supp(T),for a l l k E Nmn then T ( f ) = 0 .
F
is
Chapter I
48
m
Given
f
C:
E
F
E
x
f(x) = 0
such t h a t
(X)
and
.
E
let
X
1;
denote t h e s e t o f a l l
C: ( X ) . I f T T 1.7.11 Proposition. Let T be a linear topology on P’ x E X ; then the maximal id eal s i n C: (X) are given by t he family m i n p m t i c u l a r they are T-closed. Conversely, i f Ix i s T - closed f o r all x E X , then T > T
It
P’
> T - P Z ( 1 ) = Cx E X : g ( x ) = 0 Proof. Assume t h a t
,
K C X
K
compact
, be
I be an i d e a l i n C F ( X ) . I f m I n fact I)= 6 , t h e n I = Cc ( X ) .
and l e t
T
,g
E
given.
For a l l
that
f ( x ) # 0 ; we t a k e
cf2, c > 0
that
g x ( x ) = 1 and
0
nite
sum
gx
o f these
i.
supp(0) c
H
Define
I and
Then
g o = hg
that
supp(f) c K
E
C:
follows that
of h
K.
C:
6
(X) = I
1;
n
Z(1) 1;
Let
(X) f = f
m
K
I c1:
.
Since
.
!I I
let such
I such
E
1/2 on a cornbe equal 1 on K and
CF (X)
g
h = e/g
I.
, I
there e x i s t s a fi-
K
on
. Hence f o r
go6
g,
8
every
Since
. Then
K
8
f
C:
6
X \H.
(X)
such
,
i s arbitrary
it
# 6 and i t i s c l e a r
Z(1) x,y
0 on
H and
X
,x
f y
CT
I f x 6 X i s such t h a t m c I x and I i s maximal
1;
is
the ideal Z(1) = { X I , we ob-
.
I = I,
Conversely, assume t h a t
P
6
I i s maximal
i s properly contained i n
Then t h e seminorm T
0
putting
c o n t a i n s o n l y one p o i n t , s i n c e f o r
i t follows that
tain
obtaining
such t h a t
.
Now assume t h a t that
by g
g o = 1 on
we have
thus
f
there exists
By compactness o f
, denoted
g,
p a c t neighbourhood
.
x 6 K
f
* If(x)l i s
T
-
closed f o r a l l
continuous f o r a l l
x 6 X
x
.
X.
E
Since
i s generated by t h e f a m i l y o f t h o s e seminorms, we conclude t h a t
1.7.12 Corollary. The maximal i d e a l s i n In particular , they are closed.
C:(X)
are given by
I;
2
T
y
x
T
8
1.7.13 Corollary. Let 6 : CT(X) + R be an algebra homomorphism. Then there e x i s t s one land only one) point x in X such t hat 6 ( f ) = f ( x ) , for all f E :C (XI.
1.8.
Notes
,
remarks and r e f e r e n c e s .
Chapter 1
i s based f u n d a m e n t a l l y on Nachbin [ l
I
and Zapata
P ‘I
x.
Approximation o f smooth f u n c t i o n s
t11, t21
49
.
, Nachbin went t o t h e U n i v e r s i t y o f Chicago f o r a two y e a r v i s i t from 1948-1950 , a t t h e i n v i t a t i o n o f Stone. W h i l e t h e r e , I n 1948
he had t h e o p p o r t u n i t y , i n 1949
,
t o p r e s e n t a t AndrG W e i l ' s Seminar t h e
t h e n r e c e n t a r t i c l e "On i d e a l s o f d i f f e r e n t i a b l e f u n c t i o n s " by H a s s l e r Whitney, j u s t p u b l i s h e d i n volume 70 11948) o f t h e American J o u r n a l o f Mathematics.
[21 and theorem 0.2.3.
See Whitney
I r v i n g Segal asked him:
After his lecture
,
how about a s i m i l a r r e s u l t f o r a l g e b r a s o f
continuously d i f f e r e n t i a b l e functions, along the l i n e s o f the Weierstrass-
.
Stone theorem?
I n o t h e r words
, t h e problem was t o d e s c r i b e t h e c l o s u r e
o f a subalgebra o f c o n t i n u o u s 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 ,
or equivalently,
t o d e s c r i b e t h e c l o s e d subalgebras o f c o n t i n u o u s l y d i f f e r e n t i a b l e functions, i n the s p i r i t
o f t h e Weierstrass-Stone theorem.
t o o u r knowledge, s t i l l unsolved.
See (1.8.1)
T h i s problem i s ,
for
Nachbin's conjec-
A l s o see Nachbin [ 2 3 .
ture.
Pressed by S e g a l ' s q u e s t i o n i n 1949 Nachbin Studied t h e n o t e worthy case o f dense subalgebras Coming back t o
,
Segal's question
mulate the f o l l o w i n g conjecture. orem (0.2.3)
A
d e f i n e d by
f(x) = f(y) 1.8.1
If
Cm(U)
for all
, U c Rn
, are
f
If f
If
x,y
E
U
and A i s a subalgebra of Cm(U) m belongs t o t h e closure of A i n Cm(U) f o r T " if land always
Conjecture.
then f
B
Cm(U)
only i f I , for every compact subset K of U contained i n some equivalence c l a s s modulo U/A and every E > 0 , there i s g e A such t h a t
-
1 a"g(x)
a"f(x)
1
< E for any x e K and any p a r t i a l d e r i v a t i v e
order a t most equal t o
aa of
m.
There i s a more n a i v e c o n j e c t u r e , which i s e a s i l y seen t o be f a l s e . One
m i g h t indeed c o n j e c t u r e t h a t e v e r y subalgebra
which i s c l o s e d f o r T : m=O
,
A
and
i s a l s o closed f o r the topology
of
Cm(U)
.
For P t h i s i s indeed t h e case ; as a m a t t e r o f f a c t , t h e statement
t h a t T\
T~
have t h e same c l o s e d subalgebras o f Co(U) = C(U) P i s e a s i l y seen t o be e q u i v a l e n t t o t h e Weierstrass-Stone theorem. TO
A
a r e e q u i v a l e n t when
.
A
E
subsumed by i t .
open, c o n s i d e r t h e e q u i v a l e n c e r e l a t i o n
U, a c c o r d i n g t o which
on
Nachbin was l e d t o f o r
i t i s t r u e , t h e Whitney i d e a l the-
and N a c h b i n ' s theorem (1.2.1)
i s a subalgebra o f
U/A
t o o b t a i n t h e theorem 1.2.1.
Chapter 1
50
1.8.2
ExampZe.
C'(R) of a22 f
Let A be the subazgebra of
B
f ( l / k ) = f ( 0 ) for a22 k = 1 , 2 , ..., and moreover 1 k= 1 is cZosed f o r T; but it is not d o s e d f o r -cl Then A P' such t h a t
C'(R) f'(l/n)/r?=Oc
Regarding Nachbin's theorem i t i s i n t e r e s t i n g t o p o i n t out papers by Khourguine , J-Tschetinine, N . [11 and Reid [11. I n t h e f i r s t one the authors gave a c h a r a c t e r i z a t i o n o f C m [ O , l l among i t s closed subalgebras, under the influence of S t o n e ' s r e s u l t s . See Stone 111. I n t h e second one Reid , motivated by the construction of a d i s t r i b u t i o n s theory f o r compact groups, provided several Nachbin type theorems in dif f e r e n t topological algebras. I t should be pointed out mentioned t h a t Reid was unaware of Nachbin's paper, Nachbin [ l l , since he obtained a l s o some p a r t i c u l a r cases of Nachbin's theorem, b u t using a d i f f e r e n t approach. 1.8.3 Remark. The theorem 1.1.2 can be obtained from de l a Vallie Poussin's extension, t o d i f f e r e n t i a b l e functions , of Weierstrass theorem on polynomial approximation, see Vallee Poussin [ l l , and the r e s u l t due t o L.Schwartz t h a t C y (Rn) 81 F i s dense in C: (Rn;F) (See p r o p . 10 of Schwartz [ l l and prop. 4.4.2 of Treves H I ) .
.
1.8.4 Remark. The question whether every A E Top! ( X ) i s a Nachbin m-algebra has a negative answer. I n f a c t , l e t I$ 6 C: (R) ,I$ f 0 and
Mk
= Sup
u(x)
=
{ l $ ~ ( ~ ) ( x:) Ix E R ) ,
inf
Mk C -
Idk
: k E
PI1 , x
k
8
6
N
.
R and
Also l e t
U
=
{u''~ : k
=
1,2 ,...I.
Then
-
i s a ( d i r e c t e d ) s e t of weights on R such t h a t U < U U . If V o = V1 = U then A = CIVm(R) E Top; ( R ) ( s e e 0.1.9 and 1.3.1 (example 4 ) ) . Let A D be t h e algebra of a l l polynomials on R . Then A 0 i s a subalgebra of A which s a t i s f i e s conditions ( N ) and as a consequence of Corollary 1 in Zapata [31 i t follows t h a t A D i s n o t dense. Hence A i s n o t a Nachb n m-algebra U
.
1.8.5. Remark Since (Cf ( X ) , 7:) and Sm (R n ) a r e weighted algebras i t i s enough t o apply remark 8 a n d lemma 1 in Zapata 141 t o conclude t h a t these algebras a l s o s a t i s f y ( i i ) of theorem 1.4.8 However , t o prove in general t h a t condition ( i i ) of theorem 1.4.8 holds f o r a weighted algebra, we need t o use s o l u t i o n s of the Bernstein approximation problem f o r d i f f e r e n
.
Approximation of smooth functions t i a b l e functions ; see Zapata [ 5 1 1.8.6
51
and 161.
Remark.
A l i s t of open problems r e l a t e d t o the ideas brought out in t h i s chapter can be found i n Zapata 111.
To conclude, I would l i k e t o make t h e observation t h a t weighted
spaces o f d i f f e r e n t i a b l e functions have been considered by various w r i t e r s , e , g t by Baumgarten [ l l . The weighted approximation problem f o r different i a b l e functions was investigated by Zapata t51.
This Page Intentionally Left Blank
53
Chapter 2
SIMULTANEOUS APPROXIMATION OF SMOOTH FUNCTIONS
T h i s c h a p t e r i s composed o f two s e c t i o n s .
The f i r s t p r e s e n t s
some a p p r o x i m a t i o n r e s u l t s f o r t h e f i n e t o p o l o g y o f o r d e r m. A l s o demon s t r a t e d i s t h a t on Banach spaces which a r e n o t
U 1 - smooth , c e r t a i n
smooth approximations a r e n o t p o s s i b l e i n t h e f i n e t o p o l o g y . The second s e c t i o n i s d e d i c a t e d t o a n o n - l i n e a r c h a r a c t e r i z a t i o n o f s u p e r r e f l e x i v e Banach spaces.
I t i s proved t h a t a Banach space
is
U'- smooth i f and o n l y i f i t i s s u p e r r e f l e x i v e . 2.1.
m.
Approximation f o r t h e f i n e t o p o l o g y o f o r d e r Let E
v e c t o r space of
, F be r e a l Banach spaces. As usual l e t us denote t h e - mappings a n E i n t o F by Cm(E,F). Here d i f -
Cm
Then t h e s e t s :
f e r e n t i a b i l i t y i s understood i n t h e FrGchet sense.
where on
f
E +R
B
Cm(E;F)
,
and
E(*)
? 0
i s an a r b i t r a r y continuous f u n c t i o n
c o n s t i t u t e a b a s i s f o r a t o p o l o g y on
t i o n extends i n a n a t u r a l way t o
Cm(M,N)
modelled on t h e Banach spaces
and
E
F
where
Cm(E;F) M,N
are
.
This d e f i n i -
Cm- m a n i f o l d s
respectively.
I n s p i r e d by c e r t a i n simultaneous theorems o f E e l l s and Mc A l p i n , Smale and Q u i n n , M o u l i s proved t h e f o l l o w i n g theorem. 2.1.1
Theorem. L,et F be an arbitrary Banach space ( a ) c ~ ( c ~ ; Fis ) dense in
C ' ( C ~ ; F ) equipped v i t h C'-fine
top0 zogy . ( b ) C"(1,;F) topoZogy
, (k
B
El).
is dense in
C2k-1(12;F)
equipped w i t h Ck -fine
54
Chapter 2
Here
and
co
12
a r e equipped w i t h an e q u i v a l e n t norm which
C" away from t h e o r i g i n .
is
The theorem 2 . 1 . 1
has been f u r t h e r extended t o m a n i f o l d s
modelled on H i l b e r t spaces as f o l l o w s : 2.1.2 Theorem.
M,N
Let
be separable paracompact
led on the Hilbert spaces E and Then t h e s e t of
Cm- mappings on
. Let
F
Cm- manifolds mode!
N i be a submanifold of
N
.
M i n t o N transversal t o N i is dense
C1(M,N) endowed w i t h C'-fine topology.
in
F o r t h e p r o o f s o f t h e theorems 2.1.1 and 2.1.2 see M o u l i s [11. F u r t h e r g e n e r a l i z a t i o n s o f t h e theorem 2.1.1 have been cons i d e r e d by Heble 111 ,who has proved t h e f o l l o w i n g theorem. 2.1.3
Theorem,
Given
f
B
Let H be a separable Hilbert space ,Q c H an open s e t .
Cm(R;F),
E(
* ) > 0 an arbitrary continuous function on H , and there e x i s t s 9 8 Cm (Q;F)
R,
+
there e x i s t s a dense open subset W c R
such t h a t g E Cm(W;F) and f o r j = 0,1, ...,m,II d j g ( x ) - d j f ( x ) I / f o r each X E 2 . This is a l s o true f o r H = 12' , P 2 1 i n t e g e r
H
E(X)
,
, and
= CO.
A Banach space
E
i s s a i d t o be
U'-srnaoth
a n o n t r i v i a l uniformly continuously d i f f e r e n t i a b l e on
E
w i t h bounded s u p p o r t .
f
f u n c t i o n means t h a t
E
i f there exists
real-valued function
Here u n i f o r m l y continuous d i f f e r e n t i a b l e
C'(E;R)
and
df : E
+
i s u n i f o r m l y contin!
E'
ous. 2.1.4
=a.
E be a Banach space and
Let
continuously d i f f e r e n t i a b l e e n t i a l of
f
.
(a)
f 1B
5
/f(x) (b)
,in
i s U.C.D.
and
R be a uniformly
function, and
If B i s a bounded subset o f
, i .e., there f ( y ) I 5 MI1 x - Y / I
-
If the
particular
Proof. (a) L e t
+
df
be t h e d i f f e r
Then
i s Lipschitzian
X,Y E B
zian
(U.C.0)
f: E
Br(0) Br(0)
supp(f)
"s
I
E
, then
the r e s t r i c t i o n
p o s i t i v e number
M
such t h a t f o r
.
i s bounded
, then
f i s globally Lipschit
f i s uniformly continuous.
be an open b a l l such t h a t i s bounded
,
sup xEBp(O)
11
Br(0) > B
df(x)ll
im.
.
Since
Thus i f
M
f
is
Simultaneous a p p r o x i m a t i o n o f smooth f u n c t i o n s
55
t h e preceeding supremum t h e mean v a l u e theorem i m p l i e s t h a t
-
If(x)
5 MI1 x
f(y)I (b)
-YII
for
Br(0). (a) .#
i s a consequence o f
Lemma. If E is a
2.1.5
XSY 8
U'-smooth Banach space and A is a positive
real number , then there is a uniformly continuously differentiable realvalued function f on E with f ( 0 ) = 1 and f ( x ) = O if 11 x I I 2 X . Proof.
Since
E
is
U1-smooth t h e r e i s an
real
U.C.D.
-
valued function
say B y i s a bounded subset o f E. g # 0 and supp g) w t h g ( a ) # 0 , and a be a p o s i t i v e number such t h a t a(B - a ) c B X ( O . D e f i n e f ( x ) = l / g ( a ) g ( x / a + a ) . I t i s v e r i f i e d t h a t U.C.D. r e a l - v a l u e d f u n c t i o n w i t h t h e s u p p ( f ) c B A ( 0 ) , and f i s a g
such t h a t
Let
a
E
B
f(0) = l.# The compositionsand p r o d u c t s o f u n i f o r m l y c o n t i n u o u s 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 a r e i n general n o t u n i f o r m l y c o n t i n u o u s l y d i f f e r e n t i a b l e . T h i s remark m o t i v a t e s t h e n e x t lemma.
Lemma. If f,g are two uniformly continuously differentiable realvalued functions on a Banach space E , and the s u p p ( f ) or supp(g) is bounded, then f g is uniformly continuously differ.entiable. 2.1.6
-
Proof.
For d e f i n i t e n e s s l e t t h e
any p o s i t i v e number and tance of
x
from
B.
s u p p ( f ) , say B , be bounded.
V = i x : d(x;B) Then
V
,
s i n c e df, dg
then I/df(x)
-
df(y)ll
If(x) - f(y)l <
E
,
IP(f.q)(x) - d ( f * g ) ( y ) l I
+/I
E,I/
<
/g(x)
5
such t h a t
dg(x)
-
M > 0
i s the d i s
such t h a t
and t h e r e s t r i c t i o n s
6 > 0
continuous, t h e r e i s a
be
i s a bounded open s e t and B cV.From (2.1.4)
i t f o l l o w s t h a t there i s a constant
Now if E > 0
A 1 where d(x;B)
<
Let A
-
are uniformly
, and i f 1 1 x-yII f o r x,y E E , and
0 < 6 <(1/2)A
dg(y)ll <
g(y)l<E if
fIV,g/V
E
x,y E V
IPf(x)Il Ig(x)-g(y)
I
+
.
If(Y)
Now n o t i n g t h a t
I I/
-dg(y)I/ + for d(f*g)(x) = 0 and a p p l y i n g ( * ) i t i s
d d x ) l / I f ( x ) - f ( ~ ) l+ I g ( Y ) l 1 1 d f ( x ) - d f ( Y ) l l 9 > 6 i f x 6 B and y 6 V , x 6 B , 1 1 x - yII 2 v e r i f i e d from t h e c h o i c e of 6 t h a t i f / / x - y I I < 6
< 6
56
Chapter 2
11 d ( f * g ) ( x ) -
then a
.
5 ME
d(f*g)(y)ll
f .g
Hence i t f o l l o w s t h a t
is
f u n c t i o n w i t h bounded support.#
U.C.D.
The f o l l o w i n g two p r o p o s i t i o n s a s s e r t t h a t c e r t a i n smooth app r o x i m a t i o n s a r e n o t p o s s i b l e i n t h e f i n e t o p o l o g y o f o r d e r 2.
If E i s a non U'- smooth Banach space and F i s a U' - smooth Banach space , and f : E -+ F is a uniformly continuously d i f f e r e n t i a b z e function, then f o r every bounded open s e t B c E, f( a B ) 2.1.7
Proposition.
i s dense in
. Furthermore
f(B)
d i f f e r e n t i a b l e maps on E Proof.
Let
x
such t h a t
BE
there i s a and
z
B
n G U.C.D.
$(f(x)) = 1 B y and
Since
is
F
g : E +R
Let
E\B
on t h e s e t
then f i = f e
f ( x ) $ f(aB) smooth
closure o f f(aB).
(2.1.5)
G
2f(a,
assures t h a t
F w i t h support i n
on
be d e f i n e d by
.
, the
B.
on
f ( x ) and an open s e t U'-
real-valued function $
.
g = 0
.
14
=
aB,
F coinciding on
BE w i t h c e n t r e a t
Then t h e r e i s a b a l l
are two uniformly continuously
f 2
I f possible l e t
B ,
E
+
if fi,
g(z) =
BE
if
$(f(z))
S i n c e $ vanishes i n a neighbourhood
_ I
of
V
it i s clear that
f(aB)
= { z : d(z;B)
g
i s a C'-
function.
, and
f o r some A > 0
< A}
E'
a r e r e s p e c t i v e l y t h e norms i n t h e spaces guing as i n t h e p r o o f o f (2.1.6) valued f u n c t i o n on is
Now c o n s i d e r i n g
the inequality
, F'
i t follows that
and g
L(E;F),
is a
U'-smooth c o n t r a d i c t i n g t h e h y p o t h e s i s on
E.
all
6.
-
real
U.C.D.
w i t h i t s s u p p o r t i n t h e bounded s e t
E
and Thus
E
The second p a r t o f t h e
p r o p o s i t i o n i s a d i r e c t consequence o f t h e f i r s t p a r t . #
Proposition.
2.1.8
a U'as
11
XI1
m
-+
, then
Proof. f(x)ll
Let
2
f : E
/I
p(x)lI
If
p :E
smooth Banach space and
U'-
-+
F i s a function such t h a t
there i s no nontrioiaZ
second d e r i v a t i v e on E
11
E be a non
Let
smooth Banach space.
-+
.
+
F
F such t h a t
11 f ( x ) l ( 5 I (
be a n o n t r i v i a l
Let
X ~ EE
C2-function
with
p(x)ll
F be
p(x) f w i t h bounded
+
0
.
C 2 - f u n c t i o n such t h a t f(xo) # 0
and
R
be a p o s i t i v e
Simultaneous a p p r o x i m a t i o n o f smooth f u n c t i o n s
11 x
number such t h a t i f
-xoll
2 R
then
B = {x
:I1
1)
p(x)
(2.1.7
i t follows t h a t i f
i n f(B
but t h i s contradicts the fact that
2.1.9 p
:E
, completing
x E a6
for
Corollary. +
E
on
function g
(a) (b)
E
+
+
F
11
/I .
1/211 f ( x o )
RI
f(x)ll
then f ( 8 )
5 (1
5
p(x)ll
From i s dense
1/211 f(xo)11
the proof.
E, F be as i n t h e preceeding proposition
F be a bounded f u n c t i o n w i t h
C 2 - function on
Proof.
Let
/I 5
x-xoII <
57
, and
p(x)
f i s not a
I/ xII
0 as
+
C3- function
, and
. Iff is a
+
, there
a C3-
F does not e x i s t such t h a t
11 f ( x ) - g ( x ) \ l 5 ( 1 P ( x ) l l and II d 2 f ( x ) - d ’ g ( x ) I I 5 II P ( x ) I I
I f possible
, l e t t h e r e be a f u n c t i o n g : E
satisfying the inequalities
( a ) and ( b )
.
Since
f
-f
F
o f class
i s not o f class
C3
C3,
f f g. Thus , i t may be assumed t h a t 11 f ( 0 ) - g(O)11 = a > 0. From ( a ) and ( b ) i t f o l l o w s t h a t ( f - 9 ) i s a n o n t r i v i a l C 2 - f u n c t i o n w i t h
bounded second d e r i v a t i v e and
11 f ( x ) -
g(x)ll
2 11
, contradicting
p(x)ll
t h e preceedicg p r o p o s i t i o n . #
2.2
A n o n l i n e a r c h a r a c t e r i z a t i o n o f s u p e r r e f l e x i v e Banach spaces. I n t h i s s e c t i o n i t i s proved t h a t a Banach space i s U’-smooth,
i f and o n l y i f i t i s s u p e r r e f l e x i v e . If
E,F
in
F , i n symbols
if
E << F
a r e Banach spaces, E
i s s a i d t o be f i n i t e l y represent@
, i f f o r each f i n i t e dimensional subspace X o f E , and p o s i t i v e number E , t h e r e i s a subspace Y o f F , depending on X and E , such t h a t t h e r e i s an isomorphism T on X o n t o Y w i t h 1 1 T I / 1 1 T - l I 1 5 1 + E . A Banach space F i s s a i d t o be s u p e r r e f l e x i v e , E << F
implies E i s reflexive. A u s e f u l concept i n t h e t h e o r y o f f i n i t e r e p r e s e n t a t i o n i s t h e
concept o f an u l t r a p o w e r s e t and
r
r e a l - v a l u e d f u n c t i o n on Now i f
o f a normed l i n e a r space.
(E,ll
11)
f u n c t i o n on S, l e t
l i m f(s) =
r
i s a normed l i n e a r space, and
limll f(s)l/
If\ =
norm on t h e v e c t o r space q u o t i e n t space o f
S, l e t
V
r
.
o f bounded
V modulo t h e k e r n e l o f
Let
S
be an i n f i n i t e
f i s a bounded sup E X :It E S , f ( t ) > A1 6 r l .
be a n o n t r i v i a l ( f r e e ) u l t r a f i l t e r on
S.
f
If
i s a bounded E-valued
It i s verified that
I I
E-valued f u n c t i o n s on
I I
i s a semiS , and t h e
equipped w i t h t h e q u o t i e n t
58
Chapter 2
norm i s known as t h e u l t r a p o w e r o f and i s denoted here by then
E(S,r)
.
E(S,r)
E associated w i t h the p a i r
E<
m e t r i c w i t h a subspace o f an u l t r a p o w e r
A Banach space
i f and o n l y i f
F(S,r)
i s s a i d t o be
E
E i s a Banach space
I t i s known t h a t i f
i s a Banach space, and
of
smooth
(S,r) , i s iso-
E
F.
, i f f o r a l l x # 0,
x s E
y E E
exists f o r a l l
. If
t h e n i t i s known t h a t
t h e l i m i t i n ( * ) e x i s t s a t an x, f o r a l l y
,\I
2.2.1
= \ I y J J .I t
xi1 = 1
.
E,
6
A smooth Banach space
E
i s c l e a r t h a t u n i f o r m l y smooth i m p l i e s
i s stiperrefZexive i f and onZy i f
A Banack space E
Theorem.
1
i f the l i m i t i n (*) i s attained uniformly
i s s a i d t o be u n i f o r m l y smooth f o r a l l x,y U -smooth.
11 G x [ I =
E E ' and
G,
E
i s isomorphic with a tiniformZy smooth Banack space. For a formal p r o o f , see Van D u l s t [ l l . See a l s o i n t h i s n e c t i o n James [11 and E n f l o
con-
[ll.
The n e x t two r e s u l t s show t h a t
U'-smoothness
i s finitely
inherited. 2.2.2 E
U'-
If E i s
Theorem.
smooth
, then
L e t t h e norms o f
Proof.
-
and
, there
E(S,r)
be r e s p e c t i v e l y
E +R
w i t h i t s support i n the u n i t b a l l o f
, and
smooth
E
E
E E(S,r)
is
U'
Since
x
{x(s)lses
f Now
l ' m f(x(s))
r e s e n t a t i v e of
.
E
> 0
Further since e E(S,r)
exists.
{x(s)lsEs
,
, there i s a J6 E r - f(y(s))/ < E for
f*(x) = l i m
r
Since t h e s u p p o r t o f
, see
(2.1.5)
Let
x.
{y(s)lSes
.
Since
, f*
is a
f
be another r e p
. Hence -
f(y)[
<E.
r e m e s e n t t h e same equivalence c l a s s
such t h a t all
Let
(1
x(s)
-
y(s)ll < 6
if
s E J6.
s 6 J 6 .Thus l i m f ( x ( s ) ) = l i m f ( y ( s ) ) .
r
r
i s a r e a l - v a l u e d f u n c t i o n on E ( S , r ) . i s i n the u n i t b a l l o f E , f*(x) # 0 implies
f(x(s))
f
E
11 11 and (I( ((1 .
f, f # 0 , on
such t h a t I I x - y [ 1 < 6 i m p l i e s ( f ( x )
{y(s)lsEs
Hence I f ( x ( s ) ) Hence i f
function
i s u n i f o r m l y continuous by (2.1.4)
f
6 > 0
there i s a
i s a U.C.D.
be a r e p r e s e n t a t i v e o f
bounded f u n c t i o n
if
of
every tiZtrapouer E ( S , r )
U'- smooth.
is
Simultaneous a p p r o x i m a t i o n o f smooth f u n c t i o n s
111i111 5
.
1
r
J E
that there i s a set
Hence t h e s u p p ( f * )
Since
df : E
11 x ( s ) l l 2
such t h a t
E'
-t
1 for all
s E J
. Thus
E(S,r).
i s i n the u n i t ball of
i s a u n i f o r m l y continuous mapping w i t h
bounded range, by proceeding as i n t h e
-x,y
59
preceding paragraph
,
i t i s ver-
E ( S , r ) , and I x ( s ) l s E s , { y ( s ) l s e s a r e r e p r e s e n t a t i v e s r e s p e c t i v e l y , t h e n l i m d f ( x ( s ) ) ( y ( s ) ) i s independent o f t h e r e p of i'. For E E(S,r) define resentatives {x(s)l,{y(s)l o f x and i f i e d that i f
z,;
I;()
E
l i n e a r f u n c t i o n a l on and
.
1;m d f ( x ( s 1 ) ( y ( s ) )
=
,
Ih(s)lseS 6
x,;
It i s verified that
, since
E(S,r)
Let
now
and f *
5 €11
yII
5 6 . Then t h e r e i s a s e t J
2 ~ 1 h1 ( s ) l l
/eX(,)(h(s))I
U.C.D.
is a
f
leX(y)I
6 > 0 such t h a t
lilhlll
i s bounded
df
i s a continuous
.
Now i f
6
8
E(S,r),
E
> 0,
then
where i t i s noted t h a t , s i n c e there i s a
1;
.
Hence
i s differentiable a t
lim
r
with
E
function, given if
11 yII 5
r
such t h a t f o r a l l s e J ,
1 ex(s)(h(s))l
6 for all
x E E.
~ ( ( ( h if ( ( ((((hlll:6
5
d f * ( x ) = li.
S i n c e d f i s a u n i f o r m l y c o n t i n u o u s map on
E
-t
E ' i t follows
( E ( S , r ) ) ' i s u n i f o r m l y c o n t i n u o u s once a g a i n working w i t h s u i t a b l e members o f r as has beendone i n t h e p r e c e d i n g p a r t s t h a t t h e map d f * : E ( S , r )
o f t h e p r o o f . Thus 2.2.3 Proof.
is
U'-
smooth.#
CoroZZary. If E is U' - smooth
, and
F<<E
, then
F is U'-
smooth.
The c o r o l l a r y f o l l o w s from t h e preceeding theorem t o g e t h e r w i t h
the f a c t t h a t some u l trapower
2.2.4
E(S,T)
+
F<<E
i f and o n l y i f
E(SJ) o f
F
i s i s o m e t r i c w i t h a subspace o f
E .#
Remark. S i n c e a s u p e r r e f l e x i v e Banach space i s isomorphic w i t h a
u n i f o r m l y smooth Banach space, and U'-
smoothness i s i n v a r i a n t under
isomorphisms i t f o l l o w s t h a t i f a Banach space then i t i s
U'-
smooth.
E
i s superreflexive
,
60
Chapter 2
2.2.5
Theorem
.
I f E is a
U'-
smooth Banach space
, then
it is refleg
ive. Proof.
0 <
Let
e
< 1.
real-valued function
if
1 1 xII
e
2
.
Since
Lemma 2.1.5
f
on
E
f
is
U.C.D.
M such t h a t i f
integer
see Theorem set
W
B(X)
such t h a t
E
if
, (1
h e E
If possible l e t
assures t h a t t h e r e i s a U.C.D.
h(l5
f(0) = 1
0 < 1
E
< 1
, and , there
f(x) = 0 i s a positive
, then
be n o n r e f l e x i v e . Then by a theorem o f James,
, i t follows t h a t there i s a set X containing the
0.12.9
o f positive integers o f bounded r e a l - v a l u e d
norm, i s o m e t r i c w i t h
Ey
, and
a subspace
f u n c t i o n s on
X
L
o f t h e Banach space
w i t h t h e supremum
, such t h a t f o r
a d m i t t i n g a sequence
n z l
,ie W
z (i) = 8, 1 5 i 5 n n z n ( i ) = 0, i > n
Let if
x n Y O= n >- 1
(n,k)
,ie W ,
1 2 zn , x O Y n = -
,
k > 1
f o r which
.
zn
Clearly11
for
n
2
1
, and
3
xnYk =
~ ~ , <~ 1l lf o r a l l p a i r s o f . Consider t h e polygonal
xnYk i s defined
4 zn -
4
-
n+k
integers path
Pc L
d e f ined by
2M -1
where
M
i s t h e p o s i t i v e i n t e g e r chosen t o s a t i s f y t h e i n e q u a l i t y ( a )
i n t h e preceeding paragraph.
Consider t h e d e r i v a t i v e
df(0) o f
f a t 0.
) = 0 i f and o n l y i f d f ( O ) ( x M) = O , 2 YO 0,2 and d f ( O ) ( x ) i s p o s i t i v e ( n e g a t i v e ) i f and o n l y i f d f ( O ) ( x ) is 2 YO 0,2M n e g a t i v e ( p o s i t i v e ) . Since P i s connected t h e r e i s a 5 8 P such t h a t By o u r c h o i c e o f
d f ( O ) ( E ) = 0.
If
x
n,k
, df(O)(x
61
Simultaneous a p p r o x i m a t i o n o f smooth f u n c t i o n s
then
= -21e
t;(j)
- --1 8, -1e I if
S(j)E[
Thus i f
2
M j = 2 -in j ,B W
2
e , 7e ) o r
S(jo) =
-
then
+
1 <- j -< ZM
.
Now i f
c1
=
-
51
thus chosen has t h e p r o p e r t i e s
The
m
5 l ( j ) >_
and
ZM - i o
-
z'-l
f o r at least
5'
8
P such t h a t d f ( E 1 ) ( 5 ' )
Q = {jl 1
I- e , e 1
5
j
5 2 M }c W ,
df(S1)
imply e i t h e r ( i ) (61 + a t l e a s t 2M-2
t
.r) ( j )
2
j E Q
.
Let
o r ( i i ) i s t h e case.
1
Si)(Sk) i=l
= 0
or
j E Q
.
( i i ) (51
-
that
{siliM
,
df(O)(S1) = O , j
5
2
M
.
noted i n
0 {2M , - 4e ~ o1 r
=
52
3-l K or -
These o b s e r v a t i o n s +')(j)
+'
2
20
for
a c c o r d i n g as ( i )
{ti}
in
L
k0 m
for
such t h a t
,
k
1
i=1
Si(j)
2
11 S i l l 5
1 r;i ,
1 <- k -< M. From o u r c h o i c e of k
f,M,
otherwise
, t h e r e s t r i c t i o n o f C1 t o t h e
i=l df(
W
Repeating t h i s procedure i n d u c t i v e l y i t f o l l o w s
t h a t t h e r e i s a sequence k-1
5
8
Then as b e f o r e t h e r e i s
has range e i t h e r i n t h e s e t
2e
integers
1
j
From t h e p r o p e r t i e s o f 6
0.
except p o s s i b l y f o r one v a l u e o f
5'
.
= 5
5 M1 ,
w ,
j e
for all j
0
51
,115111
values o f
, since
t h e preceeding paragraph set
=
1 - 4
, choose
1 >_ ZM-'
Next c o n s i d e r t h e d e r i v a t i v e a
,
S(j) =
j,
6.
,
W
E
1 8 f o r some j o s W , 1 5 j o 2 2M ( w h i c h i s t h e case
S(jo) <
F;(jo)E- C-
if
1 5 j < 2M- i o - 1 , j
if
, together w i t h the i n e q u a l i t y
11 1
i=l
Sill
2 k0 i t f o l l o w s
Chapter 2
62
M
M
k
, completing
a contradiction
k- 1
k-1
t h e p r o o f o f t h e theorem.#
The n e x t theorem p r o v i d e s t h e c h a r a c t e r i z a t i o n o f
U‘-
smooth
Banach spaces. 2.2.6
Theorem.
E is
A Banach space
U’-
smooth if and only if E is
superreflexive. Proof.
From (2.2.3)
F<< E
and
i t follows that i f
and (2.2.5)
, then F
. Thus
i s reflexive
E
E
is
U’- smooth
i s s u p e r r e f l e x i v e . The
converse f o l l o w s from t h e remark 2.2.4. 2.3
Notes and remarks. T h i s c h a p t e r i s m a i n l y based on SundaresanL 11 I n Wells [ 11 i t i s shown t h a t Banach space
U’-
.
c a f a i l s t o be
smooth, w h i l e i n Aron [ l l i t i s proved t h a t t h e Banach spaces
K an i n f i n i t e compact H a u s d o r f f space a r e n o t
U’-
C(K)
,
smooth. Now these
r e s u l t s a r e a consequence o f theorem 2.2.6. Let
E
be a non
Banach space. From (2.1.9) an a r b i t r a r y
U’-smooth Banach space and
F
be a
U’- smooth
i t f o l l o w s t h a t i t i s i m p o s s i b l e t o approximate
C 2 - f u n c t i o n on
E
-+
F
by a C 3 - f u n c t i o n i n t h e f i n e to-
pology o f o r d e r 2. S i m i l a r r e s u l t s a r e known, see f o r example Aron [1 1, f o r t h e Banach space
C(K) = E
and
F =R
,
K an i n f i n i t e compact Haus
d o r f f space. For fundamental work on s u p e r r e f l e x i v e space see James [ 11, and f o r a comprehensive account o f these spaces t h e i n t e r e s t e d r e a d e r i s ref e r r e d t o t h e monograph o f Van D u l s t [ lI o n t h e s u b j e c t . o f u l t r a p o w e r s , see S t e r n [ 1 ]
For an account
and H e i n r i c h [ll.
I t would be v e r y u s e f u l t o s t u d y g e n e r a l i z a t i o n s o f t h e theorems
2.1.1
and 2.1.3.
For i n s t a n c e t h e f o l l o w i n g c o n j e c t u r e has n o t y e t been
r e s o l v e d , even i n t h e separable case.
63
Simultaneous a p p r o x i m a t i o n o f smooth f u n c t i o n s
2.3.1
Conjecture.
Banach space.
Let
H be a H i l b e r t space , and F be an a r b i t r a r y
Then t h e s e t
topology o f order m
,m 2
Cm(H;F)
i s dense i n
Cm(H;F)
f o r the f i n e
0.
Although t h e above c o n j e c t u r e has been s t a t e d as a theorem i n v a r i o u s p l a c e s i n t h e l i t e r a t u r e (see f o r example Sundaresan-Swaminathan
[l]), we a r e n o t f a m i l i a r w i t h any c o r r e c t p r o o f o f t h i s r e s u l t .
This Page Intentionally Left Blank
65
Chapter 3
POLYNOMIAL APPROXIMATION
3.0
Introduction.
C"(E;F)
Let
E
and
F
OF
DIFFERENTIABLE FUNCTIONS
be r e a l Banach spaces
. We
endow
t h e space o f m-times c o n t i n u o u s l y F r g c h e t d i f f e r e n t i a b l e
K
EY (for m E
valued f u n c t i o n s on order m y i.e.,
) w i t h t h e compact-open t o p o l o g y o f
w i t h t h e t o p o l o g y o f u n i f o r m convergence o f t h e f u n c t i o n s
and t h e i r d e r i v a t i v e s o f o r d e r
5 m on t h e compact subsets o f E.
,
which we denote by T :
topology
F-
This
i s generated by a l l seminorms o f
t h e form:
* sup
f 6 Cm(E;F)
where
N
j e
l o g y :T
j < rn E IW
on
Cm(E;F)
Co(E;F) = C(E;F) E to F
and
K
: x E K }
i s a compact subset o f
i s d e f i n e d i n t h e obvious way. When
E . The t o p 9
m = 0
,
i s t h e v e c t o r space o f a l l continuous f u n c t i o n s from i s t h e t o p o l o g y o f compact convergence.
T\ = T
F = R
If
{I] dJf(x)I/
A
and
i s a subalgebra o f
Cm(E;R) = Cm(E)
we
may s t a t e t h e f o l l o w i n g problem : under which c o n d i t i o n s may t h e d e n s i t y of
A
in
C"(E)
be assured?.
I f dim(E) <
my
we have N a c h b i n ' s theorem.
(See theorem 1.2.1). I n 1975
by t r y i n g t o extend N a c h b i n ' s theorem when
dim(E) =
m,
I found a c o u n t e r - example t h a t a l l o w s u s t o prove t h e n o n d e n s i t y o f t h e polynomials o f f i n i t e t y p e
Pf(E)
, e v i d e n t l y a fundamental example o f
a Cm(E) subalgebra s a t i s f y i n g t h e Nachbin c o n d i t i o n s ( s e e ( 0 . 4 . 3 )
m
2
Cm(E)
when
3.0.1
Counterexample.
space.
Then
2
Pf(H)
and
E Let
in
i s a H i l b e r t space o f i n f i n i t e dimension. H
i s n o t T:
be a r e a l i n f i n i t e dimensional H i l b e r t
-
dense i n
C2(H).
66
Chapter 3
Proof.
We c o n s i d e r t h e f u n c t i o n
For each
x e H
morphism.
If
, d2p(x)
$ 8 Pf(H)
mensional subspace o f f o r each
= 2j
,
H'.
x e H, d 2 $ ( x )
p = H +R
, where
then
j = H
d$(H) c F,
d e f i n e d by +
x1I2= ( x l x )
i s the canonical iso-
H'
where
x e H
So, f o r each
1)
p(x) =
F
i s a f i n i t e di-
d 2 $ ( x ) e L(H;F)
, i.e.,
does n o t belong t o t h e s e t o f t h e continuous
H i n t o H ' . Since t h i s s e t i s open i n L(H;H'), we
isomorphisms on can conclude t h a t
p
can n o t be approximated i n
C2(H)
by continuous
reduced t o a p o i n t . # polynomials o f f i n i t e t y p e over any compact {XI Remark,
T h i s counterexample was o b t a i n e d s i m u l t a n e o u s l y by Lesmes.
See Lesmes [ l ] ,
A t this formulation t h e space
p o i n t , by extending Nachbin's r e s u l t w i t h t h e same
, researt:h Cm(E;F)
c o u l d be d i r e c t e d toward f i n d i n g t o p d l o g i e s i n
which would a l l o w t h e r e s u l t t o be extended.
I n t h a t sense, P r o l l a and m y s e l f found t h e compact-compact topology o f o r d e r
m, which p l a y s a fundamental r o l e i n t h e a p p r o x i m a t i o n
t h e o r y o f c o n t i n u o u s 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 , as we w i l l see i n t h e n e x t s e c t i o n s . I n p a r t i c u l a r , we prove a Nachbin t y p e theorem f o r i n f l n i t e dimensional Banach spaces. (See c h a p t e r 9). 3.1
Approximation f o r t h e compact-compact t o p o l o g y o f o r d e r
m. B a s i c
density properties. Let differentiable
Cm(E;F)
be t h e space of m-times c o n t i n u o u s l y F r i c h e t
F-valued f u n c t i o n s on
E.
3.1.1 De fin itio n . Ve endow Cm(E;F) with t he ZocaZZy convex compactcompact topology of order m . This topoZogy, which we denote T~m , i s generated by a l l seminorms of the form
where K i s allowed t o range over t h e compact subsets of
E
, and
dOf(x)(y) = f ( x ) . For t h e sake o f s i m p l i c i t y
, we
do n o t t a k e
m
=tm
, as
this
case r e q u i r e s minor v a r i a t i o n s o n l y . When
m=O
,
i t follows that
T:
=
T =~ T,,
T:
.
If
E
is
67
Polynomial a p p r o x i m a t i o n o f d i f f e r e n t i a b l e f u n c t i o n s
i n f i n i t e dimensional and 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 , t h e n t h e comm m pact-open t o p o l o g y o f o r d e r m 2 1 , T, , i s s t r i c t l y f i n e r t h a n in
rn
,
O f course
induces i n
T~
L(E;F)
t h e compact-open t o p o l o g y . I f
L(E;F)
is
E
T~
W
t h e norm t o p o l o g y and
.
induces
T~
f i n i t e dimensional b o t h
t o p o l o g i e s a r e t h e same. The f o l l o w i n g theorem p r o v i d e s a c h a r a c t e r i z a t i o n o f approxi-
-
mation o f d i f f e r e n t i a b l e functions f o r the : T 3.1.2
Let E be a Banach space and
Theorem.
topology.
2 1. The following
m
statements are equivaZent:
F
space
(1) Crn(E)
.
I
m
F i s -rc-dense i n Crn(E;F) f o r each Banach m
( 2 ) The i d e n t i t y i n E belongs to the ~ ~ - e l o s u rof e (3)
The i d e n t i t y in E belongs to t h e
(4)
E
P r o o f . ( 1 ) =>
H
has the approximation property
(2)
= Cdu(y) : u E Cm(E) IE
E
(4).
( 3 ) =>
,y
m c-cZosure of
E E l c E ' r E.
belongs t o t h e
Before proving t h a t
T:
E'I E
,
.
To v e r i f y t h a t ( 2 ) =>
i s obvious.
that the i d e n t i t y i n
T
Cm(E)IE.
,
3)
let
I t can e a s i y be v e r i f i e d
-
(4) =>
H. T r i v i a l l y
closure o f (1)
,
we w i 1 g i v e a
p r e l i m i n a r y lemma. 3.1.3
Lema.
Let
E and F be two r e a l Banach spaces , w i t h E having = {f 0 u : u 8 E ' ~ E I . ~f f 6 c ~ ( E ; F ) md
the approximation property then f beZongs t o t h e Proof. that
k
Let
11 x I I 2
M
c E
be compact and
f o r every
Af.
cZosure of
T:-
E
.
> 0
x F. K. S i n c e
Let
M > 0
be a c o n s t a n t such
i s u n i f o r m l y c o n t i n u o u s on
f
we have (1)
11
there exists
61 > 0, such t h a t i f
x - y I I < 6 1 , t h e n [I f ( x ) - f ( y ) I I < E L e t j be an i n t e g e r 1 5 j
sociated w i t h constant
C
djf
If
,y
6
E
o n l y on
x,y E
j
K and z
and 8
and
I
5
m
.
The j - l i n e a r mapping as-
being continuous, i t f o l l o w s t h a t there e x i s t s
, depending (2)
x E K
E
K ,
such t h a t :
with 11y-zll
5 1
then
a
K,
68
Chapter 3
11 dJf(x)(y) The mapping
djf : E
Cll Y -Zll
5
dJf(X)(Z)II
P ( j E ; F)
+.
-
i s u n i f o r m l y c o n t i n u o u s on
K,
thus :
(3)
t h e r e e x i s t 62 > 0
-
such t h a t i f
x 6 K
,y
E E
and
b y ( 2 ) and ( 3 ) . # Proof
that (2)
(1)
i n t h e o r e m 3.1.2.
then (1) i s t r u e (see (1.1.2)). be a compact s u b s e t o f to
E l i s denoted by
3.1.4 E
and
Remark.
Let
> 0. I f
El = u ( E )
p
0
u E P f (E )
E E' P E
,K f
there exists
dJP(x)(Y)ll <
f 6 Cm(E;F),u
p E Cm(El)
(0 5 j 5 m)
E
P
F
such t h a t
(X, Y E u ( l 0 )
, u E E ' I E , K b e a compact s u b s e t o f E , t h e r e s t r i c t i o n o f f t o E l i s d e n o t e d by g.
f 6 Cm(E)
From W e i e r s t r a s s ' t h e o r e m ( 1 . 1 . 2 )
Since
> 0. I f
dim(E) <
El = u ( E ) , t h e r e s t r i c t i o n o f
I n general, l e t E
g. By ( 1 . 1 . 2 )
I/ dJg(x)(Y) -
and
E
First of all, if
, the proof
the following proposition.
there exists
(4)
=>
(1)
p 6 P(E1)
i n t heorem
such t h a t
3.1.2,
shows
Polynomial a p p r o x i m a t i o n o f d i f f e r e n t i a b l e f u n c t i o n s 3.1.5
69
Let E be a Banach space with the approximation m Pf(E) i s T c - dense i n Cm(E )
Proposition. Then
property.
.
3.1.6 CoroZZary. Let E and F be two r e a l Banach spaces, w i t h having t h e approximation property. Then Pf(E;F) i s m in
E
dense
c~(E;F). Proof.
If
m=O
f r o m theorem 3.1.2 P r o p o s i t i o n 3.1.5
, i t f o l l o w s from W e i e r s t r a s s - S t o n e ' s theorem. I f m 2 1, i t f o l l o w s t h a t Cm(E) B F i s T: - dense i n Cm(E;F). proves t h a t is
Pf(E) B F = Pf(E;F)
3.2
-
T:
dense i n
Cm(E) B F.#
Q u a s i - d i f f e r e n t i a b l e f u n c t i o n s on Banach spaces. B a s i c t o p o l o g i c a l properties.
[ll and Kurzweil [l]show t h a t
The works by Bonic-Frampton
f o r c e r t a i n Banach spaces, t h e behaviour o f a d i f f e r e n t i a b l e f u n c t i o n i s quite restricted.
In fact
separable Banach space and
, Whitfield E'
[11
has shown t h a t i f
E
is a
i s n o t a s e p a r a b l e space i n t h e d u a l norm,
t h e n t h e r e a r e no nonzero d i f f e r e n t i a b l e f u n c t i o n s w i t h bounded s u p p o r t on
E.
Examples o f such spaces a r e
11 and t h e Banach space
a l l r e a l - v a l u e d continuous f u n c t i o n s on that
C[O,ll
I
As a r e s u l t
,
of
One may v e r i f y d i r e c t l y
[O,ll.
c o n t a i n s an isomorphic image
i s n o t separable.
C[O,ll
of
Lm(O,l).
Hence, C [ O , l I
I
t h e c l a s s o f d i f f e r e n t i a b l e f u n c t i o n s on
such spaces i s t o o small t o be u s e f u l . d i s j o i n t c l o s e d subset of t h e space.
F o r i n s t a n c e , i t does n o t separate Goodman
[11 shows t h a t t h i s separ-
a t i o n problem does n o t a r i s e i f F r i c h e t d i f f e r e n t i a b i l i t y i s r e p l a c e d by Furthermore, he shows
t h e weaker c o n d i t i o n o f q u a s i - d i f f e r e n t i a b i l i t y .
t h a t any bounded u n i f o r m l y continuous f u n c t i o n on a r e a l s e p a r a b l e Banach space i s t h e u n i f o r m l i m i t o f q u a s i - d i f f e r e n t i a b l e f u n c t i o n s . On t h e o t h e r hand, c o r o l l a r y T
m C
-
dense i n
Cm(E;F)
.
3.1.6
proves t h a t
U n f o r t u n a t e l y t h e space
g e n e r a l l y complete and t h e r e f o r e m c o m p l e t i o n o f ( P ~ ( E ; F ) ,- r C ) .
(Cm(E;F), T; )
(Cm(E;F),
Pf(E;F)
is
-rF )
i s not
does n o t r e p r e s e n t t h e
I n t h i s section a representation o f the completion o f (Pf(E;F),~F) i s o b t a i n e d , u t i l i z i n g t h e q u a s i - d i f f e r e n t i a b l e f u n c t i o n space.
Chapter 3
70 Let
E
and
F
be two r e a l Banach spaces,
X
a real locally
convex H a u s d o r f f space.
3.2.1 Definition. A function f : E + X is said to be quasi-differentiable at a 6 E if there is an element u 6 L ( E ; X ) such that the foZlowing con dition holds I if s e C([O,l] ,E), s ( 0 ) = a and the Zimit s ' ( 0 ) =
Let x
in
f : E
+R
be a q u a s i - d i f f e r e n t i a b l e f u n c t i o n ,
For a f i x e d
E, t h e l i n e a r f u n c t i o n a l which appears i n t h e above d e f i n i t i o n o f
q u a s i - d i f f e r e n t i a b i l i t y i s unique, and we denote t h e l i n e a r f u n c t i o n a l by f'(x)
.
T h i s d e f i n e d a map
derivative o f
f' : E
-+
El
which i s s a i d t o be t h e q u a s i -
f.
.
Definition A quasi-differentiable function f on a Banach space is of cZass Q' if f ' is bounded in E ' norm and the map ( x , y ) + < f ' ( x ) , ~> is continuous on E x E. 3.2.2 E
Goodman [11
uses c e r t a i n f i n i t e Bore1 measures, which d e f i n e
smoothing o p e r a t o r s a c t i n g on bounded continuous f u n c t i o n s ; and as a conse quence o f t h e f a c t t h a t any f u n c t i o n s a t i s f y i n g a L i p s c h i t z c o n d i t i o n i s smoothed t o a q u a s i - d i f f e r e n t i a b l e f u n c t i o n by these o p e r a t o r s , he proves t h e f o l l o w i n g approximation theorem. 3.2.3 Theorem. Let E be a reaZ separabZe Banach space. The set of bounded functions on E of cZass Q'is dense in the space of bounded uniformZy continuous functions on E , In other words, any bounded un< formly continuous function on E is the uniform limit of quasi-differen tiabZe functions of class Q'
.
3.2.4
CorolZary-. A real separable Banach space admits partitions of Of cZass
Proof.
r
Q
1
.
For a g i v e n Banach space, l e t
centered a t
x
i n t h e space.
above theorem, t h a t f o r any o f class
Q'which
one on t h e s e t
Br(.x)
I t i s an immediate consequence o f t h e
r' < r
there exists a function
vanishes o u t s i d e t h e s e t
Brl(x).
denote an open b a l l o f r a d i u s
Br(x)
on t h e space
and which i s equal t o
The e x i s t e n c e o f p a r t i t i o n s o f u n i t y t h e n f o l l o w s
Polynomial a p p r o x i m a t i o n o f d i f f e r e n t i a b l e f u n c t i o n s
71
from a standard argument.#
CorolZary. If c i and C z are two nonempty disjoint cZosed subsets of a real separable Banach space, then there exists a continuous quasidifferentiable function on the space which vanishes on C I and which is equal to one on C 2 3. 2. 5
.
Sova
111 and t h e n Averbukh-Smolyanov [ l l observed t h a t t h e
quasi-differentiabily
n o t i o n c o i n c i d e s w i t h t h e Hadamard d i f f e r e n t i a b i l i t y .
Hadamard d i f f e r e n t i a t i o n was i n t r o d u c e d by Sova [11 under t h e name o f compact d i f f e r e n t i a t i on.
3 . 2 . 6 Definition. A function f : E - X is said to be Hadamard differ entiabZe (H - differentiabze) at a point a f E , if there exists u e L(E;X) such that f o r every compact set K in E ,
-1
lim E+O
r(f,a,Ex)
E
uniformly with respect to x
= 0 6
where the "remainder" r ( f,a,x)
K,
is defined
by r(f,a,x)
u
= f
i s called the H-derivative
of
f
at
a.
We w r i t e
df(a)
or
f'(a)
u.
instead o f
i s called
f
H-differentiable
if f
is
H-differentiable a t
a E E.
any
Lemma. A function f: E + X is H-differentiabze at a E E if and o n l y if f is quasi-differentiabze at a e E , 3.2.7
Proof. {E-
1
Assume t h a t
[s(E)
-
S(O)
I ,
f
is
H-differentiable a t
~ ' ( 0 ):
E
e
Thus, t h e f o l l o w i n g l i m i t e x i s t s :
R+I
a E E
i s compact i n
. E,
Since
Chapter 3
72
, if f , and
Conversely
i s n o t Hadamard d i f f e r e n t i a b l e a t
i s not either
.
u =O V
a
g
g = f -u
a, then
a g a i n s a t i s f i e s t h e c o n d i t i o n o f t h e lemma w i t h
Then, f o r t h e corresponding remainder, we have t h a t t h e r e e x i s t s
neighbourhood of
such t h a t
0
€;l r(g,a,En Construct
s ( E ~ )= a
,
cn xn
t
in xn)
X, {E,I
c o and
6
K cE
{ x n ) c K,
compact
6 V.
s E C( [0,1l,E)
s(0) = a
such t h a t , s'(0) = 0
and
(see t h e remark below).
Then, -1
En
r(g,a,cn
xn) =
-1
[ g ( a t E ~ x -~ g )( a ) ] =
E~
a contradiction.# Remark:
s
6
If xn
C( [0,1l,E)
In fact
,
xo
-t
,
a E E
such t h a t
s(0) = a
n-€ E nmEn+1 E n t l x n t l t
0
If we denote
X
0
,
and
E ~ =1
s ( E ~ =)
a
, then
t E
n
x
there e x i s t s and
n
s'(0) =xo.
X
-E
ntl
xn
E,
€ =
if
E~~~
E
<
E,
0
i s t h e r e q u i r e d mapping.
t h e space o f a l l continuous
endowed w i t h t h e compact-open t o p o l o g y PC('E;X)
i s complete when
X
n E Iu
n-homogeneous polynomials
o l o g y o f t h e u n i f o r m convergence on t h e compact s e t s o f to verify that
<
i s a r e a l l o c a l l y convex H a u s d o r f f space, f o r each
Pc( 'E;X) to
E
if
= SI(E) + a
E
E~ j.
i f we p u t , E
from
,
,
i.e., E.
t h e top-
I t i s routine
i s complete.
3.2.8 Definition. A function f : E + X is said to be m-times H-differentiable , if f is (m-1) - times H-differentiable and d m - l f : E + P ~ ( ~ - ~ E ; X ) is H-differentiable, where d " f = f . f is said to be m-times H-continuousZy differentiable if f
73
Polynomial a p p r o x i m a t i o n o f d i f f e r e n t i a b l e f u n c t i o n s
i s m-times
m d f : E
H-differentiable and
+
c m P ( E:X)
i s continuous.
We w i l l denote t h e space o f a l l m-times H - c o n t i n u o u s 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 from
E
to
X
by
.
C;,(E;X)
If
m = O ,C;,(E;X)=C(E;X)
i s t h e v e c t o r space o f a l l continuous f u n c t i o n s from By d e f i n i t i o n
, we
g e t t h a t every
E to
X.
H-differentiable function i s
continuous and t h a t t h e H - d i f f e r e n t i a b i l i t y n o t i o n agrees w i t h t h e u s u a l , F r i c h e t d i f f e r e n t i a b i l i t y , when
Definition.
3.2.9
E
i s f i n i t e dimensional.
We endow C,mdE;F)
m
with the
topology. This topology
T~
i s generated by a l l seminorms of the form:
where K i s allowed t o range over the compact subsets o f E and d
O
f ( x ) ( y ) = f ( x ) . (See d e f i n i t i o n n=1
If
l i n e a r maps from
E
,
3.1.1).
P'(~E;x) = L ~ ( E ; x ) to
X
v e c t o r space o f c o n t i n u o u s
equipped w i t h t h e t o p o l o g y o f u n i f o r m con-
vergence on t h e compacts s e t s of
E.
I f f o r each
n
e D1 , n
:1 , we
define L ~nE( ; X ) = L~(E;L,("-'E;X)
Lc(OE;X) = X
where PC('E;X)
where
,it
=Lc('E;X).
K
f o l l o w s t h a t under a n a t u r a l i d e n t i f i c a t i o n
Thus
the sets:
i s a l l o w e d t o range over t h e compact subsets o f
a b a s i s o f neighbourhoods o f 0 of
o
ir:
in
X, form
E
and
V
over
a b a s i s o f neighbourhoods
L~("E;X). I f f o r each compact subset
T(K;c)
=
I f f Cmo(E;F)
K
of
E
: d j f ( K ) c W(Kj;BE)
and
E
> 0
; 0
5
j
2
, we
define:
ml,
BE = { y 8 F : I 1 yII 2 € 1 , i t f o l l o w s t h a t t h e f a m i l y IT(K,E)) N forms a fundamental system o f neighbourhoods o f 0 f o r T: on Cco(E;F).
where
74
Chapter 3
We w i l l use t h e f o l l o w i n g p r o p e r t i e s o f
H-differentiability;
(see e.g. Yamamuro [ l l ) . Let
F,F
dJ(f
be r e a l Banach spaces:
m
If
(a)
G
and
f 6Cco(E;F)
m u E L(G;E), t h e n f o u E Cco(G;F)
and
,... ,ux.)J
u)(x)(xl,. ..¶x.) = dJf(u(x))(uxl J m ( b ) If f eCCo(E;F) and u 6 L(F;G) 0
o
f) = u
(c)
If
,
and
dj(u
and
d(g o f ) ( x ) = dg(f(x))
0
dJf
“A,
f
1
E;F)
5
2
j
and
The space
,
g
f eCA0
0
4;G)
x E E. f : E
+
X
is
H-differen-
,T:
(C;,(E;F)
is compZete.
)
be any Cauchy n e t i n C,,(E;F). For each x E E ( dj fi(x))iEI i s a Cauchy n e t i n L c ( j E ; F ) Since
.
i s complete, i t f o l l o w s t h a t t h e r e a r e
such t h a t each n e t Since
eCFo(E;G)
rn
Proof. L e t (fi)iEI and j , 0 5 j 5 m Lc(jE;F)
o f
, then:
a E E
3.2.10..Theorern.
, then
g 6CAo(F;G)
( d ) (The Mean Value Theorem ) . I f t i a b l e and
u
m.
,
df(x)
o
.
,15 j 5 m
then
and
is K R
E
(dJfi(x))i61
-
, it
space
i n Lc(jE;F)
fj(x)
converges t o f o l l o w s t h a t each
fJ
C(E; Lc(JE;F))
6
. ,
O < j $ m . Now we w i l l show t h a t f o r each j , x E E
, where
f = f’.
For
a l r e a d y been proven f o r given
.
j
=O
fJ(x) = dJf(x), 0 5 j < m t h i s isobvious.Assumethat t h e r e s u l t has
j = p < m
. Let
x
6
E
and
K c E
I t i s s u f f i c i e n t t o prove t h a t :
dpf(x+th)
1i m t+O
-
dPf(x)
t
-
fPtl(x)(th)
= o
compact be
Polynomial a p p r o x i m a t i o n o f d i f f e r e n t i a b l e f u n c t i o n s
u n i f o r m l y on
75
h 6 K.
Let
d(SP;,)
= w(Sp;B
E
be a neighbourhood o f 0
)
in
Lc(PE;F),
and c o n s i d e r t h e mapping: v ( t ) = dPfk(x + th)
-
dPfi(x
+
th).
By a p p l y i n g t h e mean v a l u e theorem we have t h a t
[dPfk(x + th)- dPfk(x) = t-' ( v ( t )
=
IGl
-
~ ( 0 ) )e
{dPtlfk
Since
-
(dPfi(x
t-'l$
+ th) - dPfi(x))]
{ d v ( s ) ( t ) : s E [O,tl}=
: s E [O,t]}.
(x+sh)(h) -dP+'fi(x+sh)(h)
m CcO(E;F)
i s a Cauchy n e t i n
(fi)
t-l =
,
there exists
such t h a t i f
i,k
2 io
and
-
dP+l f k ( y ) Thus, i f
y E x + DK
dP+l f i ( y )
(D = [-ly1I)
E W(K x S p , 4 3 )
,
then
.
-
i , k > i o, we have (v(t)
-
v ( 0 ) ) t - l 8 W(SP,€/3).
Then, i t f o l l o w s :
( 1) [ d P f( x t t h ) - d p f ( x ) -dpf ( x + t h ) -dpf ( x ) ) ] when
i
io ,t B D
and
h e K.
On t h e o t h e r hand, f o r
( 2)
IdP+' f i ( x ) ( t h ) For t h i s
(3)
-
t-' 6 W( S p ,E/3)
h B K
fP+'(x)(th)]
i, t h e r e i s d
,
and some t-'
0 t - l
2 i o, we
W(Sp,~/3),
6
0 < 6 < 1
[dPfi (x+th)-dPfi ( x ) - d P + l f i ( x ) ( t h ) ]
i
6
,
such t h a t
W( Sp , E / 3 )
have
ioe I
76
Chapter 3
when
lt1<6
h 8 K.
and
,
From ( 1 ) , ( 2 ) and ( 3 )
-
[dpf(x t t h ) when It1 < 6 and
dPf(x)
-
i t follows that E W(SP,,)
fPt'(x)(th)l*t-'
h E K.#
For every
5
j, 0
j
5 m y t h e space C(E;Pc(JE;F)) endowed w i t h
t h e compact-open t o p o l o g y i s considered.
m
From t h e d e f i n i t i o n o f t h e
T~
topology i t f o l l o w s t h a t t h e mapping:
and i t s image by 8 Cm (E;F) co i s r e l a t i v e l y compact i f and o n l y i f
i s a t o p o l o g i c a l isomorphism between Consequently
@ c
cTo(E;F)
. e(@)
Since cm (E;F) i s complete , co m e(C;"(E;F)) i s c l o s e d i n t h e producz !d C(E;Pc(jE;F)). From Tychonow's j =O theorem @ i s r e l a t i v e l y compact i n cmco (E;F) i f and o n l y i f each p r o j e c t i o n T I . ( ( P (E;F)) i s r e l a t i v e l y compact i n c ( E ; P ~ ( ~ E ; F ) ) . J co From A s c o l i ' s theorem t h e f o l l o w i n g c o r o l l a r y i s taken. i s r e l a t i v e l y compact i n
(c;~(E;F)).
0 be a f a m i Z y in C" (E;F) . Then, @ co reZativeZy compact i f and onZy if the foZZowing conditions hold: 3.2.11
CoroZZary.
Let
( 1 ) For every
j
,
0
5
j
5
and each K c E , K
m
is
compact , the
set { d j f l K : f E @ } c C ( K ; Pc(JE;F)) (2)
For every x E E
{djf(x) 3.3
: f E
On completion o f
and each
is equicontinuous. j, 0
5
j
is reZativeZy compact i n
5
m
,
the s e t
PC(JE;F).
m ( P ~ ( E ; F ) ,-rC ) .
I n t h i s s e c t i o n t h e r e s u l t s o f s e c t i o n 3.1 and 3.2 prove t h a t t h e space (Pf(E;F)
m
, - c C 1.
(Cm(E;F), -r;)
a r e used t o
i s t h e c o m p l e t i o n o f t h e space
77
Polynomial a p p r o x i m a t i o n o f d i f f e r e n t i a b l e f u n c t i o n s .
S l i g h t m o d i f i c a t i o n s i n t h e p r o o f of lemma 3 . 1 . 3
allow
us t o
prove t h e n e x t 1emma.
3.3.1
Lemma.
E and F be two reaZ Banach spaces, w i t h E having
Let
m
If f ECCO(E;F) a n d A f = { f o u
the appproximation property.
3.3.2
Let E and F be two r e a t Banach space, w i t h E having
Theorem.
the approximation property
(2)
Pf(E) i s T :
(3)
Pf(E;F) is
(2) (3)
m
-
m (E;F). co
( 1 ) i n theorem 3.1.2.
i s s i m i l a r t o C o r o l l a r y 3.1.6.
Let E and F be two Banach spaces, with E Then CFo(E;F) i s the :T
-
having
comptetion of
. m = O ,Co (E;F) = C(E;F) co i s T: - dense i n C(E;F)
If
Pf(E;F) theorem
C
.
i s s i m i l a r t o p r o p o s i t i o n 3.1.5.
Theorem.
Proof.
C;,(E;F C:o(E)
- dense i n
(4)
the approximation property. Pf ( E ;F)
Then
- dense i n
T~
(1) i s similar
Proof.
.
B F i s Tc-dense m in
(1) C!o(E)
3.3.3
-cZosure of A f .
betongs t o the !T
then, f
e E' a El ,
:U
3.2.10
and
Pf(E;F)
is
is
.m I f
T~
-
T~
= T~
m > l
complete and from ( 3 . 1 . 6 )
, Cmco (E;F)
dense i n
C;,(E;F)
i s complete by by
(3)
in
theorem 3 . 3 . 2 .
3.4.
Notes an r e f e r e n c e s . The b a s i c r e f e r e n c e s o f t h i s c h a p t e r a r e Llavona [ I ] , [ 2 1 and
Bombal-LLavona [11 and
Nachbin [31
.
.
A l s o see
P r o l l a [l], [2]
, Prolla-Guerreiro [ l ]
For s i m i l a r q u e s t i o n s w i t h i n t h e c o n t e x t o f d i f f e r e n t i a t i o n i n l o c a l l y convex spaces, see Meise [ l l .
This Page Intentionally Left Blank
79
Chapter 4
WEAKLY CONTINUOUS FUNCTIONS ON BANACH SPACES
4.1
I n t r o d u c t i o n . P r e l i m i n a r y D e f i n i t i o n s . Elementary P r o p e r t i e s . Let
f : A E
+
> 0
F
and
E
i s s a i d t o be weakly continuous
, there
are
A c E
F be Banach spaces and
$ly....$n
in
El
.
A function x E A
i f f o r each
.
We i = l Y . . . , n , t h e n 11 f ( x ) - f ( y ) I I < E space o f a l l f from E t o F which a r e weakly c o n t i n u -
( ~ $ ~ ( x - y <) 6I f o r a l l denote
and
6 > 0 such t h a t i f y E A ,
and
the
ous, by Cw(E;F) ,and by
Cwb(E;F)
a l l f u n c t i o n s from
to
t o bounded s e t s
E
(respectively
Cwk(E;F)) t h e space o f
F which a r e weakly continuous when r e s t r i c t e d
( r e s p e c t i v e l y weakly compact s e t s ) .
C l e a r l y we have
Cw(E;F) c Cwb(E;F) c Cwk(E;F). f : A
A function __ ous
iff o r each
E
F
i s s a i d t o be weakly u n i f o r m l y c o n t i n u in
<6 f o r a l l
t h a t if x,y d A , I $ i ( x - y ) I We denote as
+
there are
> 0,
i = 1,
Cwbu(E;F), t h e space o f a l l
f
and 6 > 0
E'
...,n
from
such
then I p ( x ) - f ( y ) l l E
to
< E .
F which a r e
weakly u n i f o r m l y c o n t i n u o u s when r e s t r i c t e d t o bounded s e t s .
4.1.1 Lemma. Let T 6 cwbu (E;F) and Let Then T(B) is precompact. Proof.
Let
Since
T
6 > 0
and
> 0.
E
x E B
F o r each
let
be a bounded s e t .
BC E
V(X;E) = I t e F
:I[
t -T(x)lI
i s weakly u n i f o r m l y c o n t i n u o u s on bounded s e t s , t h e r e a r e
,.. .
E E'
y@k
,
such t h a t whenever
x,y e 6
l ~ $ ~ ( x - y )
E
...,
@ ( B ) i s precompact i n Rk we can f i n d
xl,.
that
-
I@i(x)
which shows t h a t
..,x
n
E
B
, which
we endow w i t h t h e sup norm. T h e r e f o r e ,
such t h a t g i v e n
I $ I ~ ( x ~ )
U
with
. The mapping i n t o Rk . Hence
... U
x E B Thus
V(X~;E)
,
there i s
1 1 T(x) -
x
j
T(xj)ll <
; that i s
, T(B)
such E
, is
Chapter 4
80
precompac t,
#
,z e
If $ e E l
F
B c E
and
i s bounded
, then
f o r x, y e B
we have :
where
C
m
Pf( E;F) 4.1.2
i s a c o n s t a n t independent o f
x,y
.
e B
It follows that
Cwbu (E;F).
C
Remark.
i s t h e space o f m-hornoqeneous polynomials from E
Pf(mE;F)
t o F which a r e weakly continuous a t 0. I n f a c t , i t i s c l e a r t h a t a n y l i n e a r c o m b l
$m P y
n a t i o n o f polynomials o f t h e form continuous a t O.Conversely,let 6 > 0
f o r some
p
I@I~(x)I < 6
Choose
c
,.. . , x k l
E
A(x
,...,x )
f o r any
X
l$i(x)I <6
,
p(x)ll
5
1
we
.
.
L("E;F)
R , and x
E
E
( i = l,...,k).
p = A
such t h a t f o r any
, we have
x
11
6
E
o
.
Am
.
We f i r s t show t h a t
To see t h i s , n o t e t h a t
p(b(x) t
1x')ll
5
1 whenever
Hence
x;
A(b(x),...,b(x),
m-j A
1)
then
,
E'
such t h a t $ 1. ( x J. ) = 6 i j Thus , any x 6 E can k x = $i(x) xi t x ' , where $ 1. ( X I ) = 0 i=1 k b(x)= $ i ( x ) xi . Now , l e t A be a i=l
= A(b(x),...,b(x))
E
... ,k)
6
1
We l e t
symmetric element o f
,. .. , $k
$1
1
be w r i t t e n i n t h e form ( i = l,...,k).
( i = 1,
i s weakly
e E ' , y e F)
be weakly continuous a t 0. Thus,
P(mE;F)
and l i n e a r l y independent s e t
have t h a t whenever {xl
E
($
... , X I ) X j / I <-
1
j
R , and t h e des r e d e q u a l i t y f o l l o w s r e a d i l y from t h i s . Thus , f o r any x e E for all
E
p ( x ) = A(x,
...,x )
which shows t h a t 4.1.3
Proposition.
= A(b(x)
...,b ( x ) ) =
k
k
1 ... 1 A ( x .
il=l
i,=l
) $if~)...$i(~)
,...,xi
11
m
Ill
p 6 P,(mE;F),
Given a continuous l i n e a r mapping
A : E
-+
F
,A is
compact, if and only if A i s weakly (uniformZy) continuous on bounded
Weakly continuous f u n c t i o n s on Banach spaces
subsets of E Proof.
.
Suppose t h a t
and l e t
K =
A
A(B) , which i s a compact subset o f
t
I J l k ( k ) l = 11 k i s continuous , f o r any
such t h a t
11 xII 2
Qi
o
k
,I1
,I/ GkII = x e E +\I X I \ -
E F'
,
> O t h e r e i s a neighbourhood Uk o f
Thus, s i n c e t h e f u n c t i o n x E Uk.
for all
5
kll
Therefore
e F' ,
sup lJli(k)l l
.
t E
,
1)
then
A(x-y)ll <
since
k K
1
,
lJl,(x)l such i s corn
( i = 1,..., n ) , such Consequently, n o t i n g t h a t
= 1
, we have t h a t i f x, y e B s a t i s f y lJli
A e E'
( i = 1,. ..,n
By t h e
E
,... , Jln
k 6 K
e K , t h e r e i s qk
F.
1)
/ J l k ( x ) l -+ e
p a c t , we can f i n d that for a l l
B be t h e u n i t b a l l o f E
i s a compact o p e r a t o r . L e t
Hahn-Banach theorem, f o r each
that
81
A(x-y)l c
0
E
.
2~
The converse i s c o n t a i n e d i n lemma 4.1.1.# A : E
Mote t h a t a n o n - l i n e a r mapping without
+
F
may be compact
b e i n g weakly u n i f o r m l y continuous o n bounded s e t s .
F o r example,
m
A : l2 +
t h e mapping
R g i v e n by A(x)
1
=
,..,xn,...)e12)
xi
( x = (x1,x2,
N)
f o r t h e usual b a s i s
E
i s f i n i t e dimensional.
n= 1 A(en) = 1 ( n e
i s t r i v i a l l y compact,although {en)
of
l2 , so t h a t
A
6
Cwbu(lp).
,
Cw(E;F) = Cwb(E;F)
i f and o n l y i f
Indeed, i f t h e r e e x i s t s a sequence {@n} o f l i n e a r l y independent c o n t i n u o u s
E
l i n e a r forms on polynomial
Q =
11
such t h a t
1
2-n
= 1 f o r every
i s not o f
n e
(Q B y) f
4.1.2,
Q
i s n o t weakly c o n t i n u o u s
, where y e F , y # 0 .
C,(E;F)
For every
m E l4
, 11 *
any
Q,
> 0 there i s
E
m =
1
2-n
4;
n=l
.
1
-
2-n
n=l
1
Q.
Jo
1 1 Q - Q,I
e Cwbu(E)
c 4 3
.
the
Therefore Thus
show t h a t
, Q E Cwb(E).
m
1
2-n.
Thus
,
for
n=m+l for all
there are
m
2 j , , where
,... , @k 6 E'
, such t h a t i f x,y E E , ( 1 X I ( 2 1 , ( ( y ( (5 1 and I @ i ( x - y ) l < 6 ..., k ) , t h e n 1 1 Qj,(x) - Qj,(y)II < 4 3 . T h e r e f o r e , f o r any such ,I/ Q ( x ) - Q ( y ) I I < E . Then Q E Cwbu(E) c Cwb(E) . Consequently
and 6 > 0 ( i = 1, x and y
<
2-n
n=l
j , such t h a t
Since
L e t us
m
m
.
f i n i t e type
n= .. -1
a c c o r d i n g t o remark
N , then
(Q fl Y ) 4 Cwb(E;f)
, where y
E F
y # 0.
82
Chapter 4 V a l d i v i a [11 shows t h a t a Banach space
o n l y i f e v e r y weakly continuous f u n c t i o n in
E.
By lemma
in
E.
Then we have t h a t
4.1.1,
every f u n c t i o n i n
a f u n c t i o n which belongs t o
E
-+
i s r e f l e x i v e , i f and
R i s bounded on b a l l s
Cwbu(E)
,
Cwbu(E) = Cwb(E)
T h i s suggests t h a t i f
flexive.
f : E
E
i s bounded on b a l l s
i f and o n l y i f
For example , i f
E.
E
g(x)
=[I( $11 -
-'
$(x)l
and i s n o t bounded.
,
E which i s n o t bounded
i s a n o n - r e f l e x i v e separable
We d e f i n e t h e f u n c t i o n it i s
E, which extends
f
i s n o t bounded on t h e u n i t b a l l ;
t h e r e f o r e i t cannot
The
bw
and
bw*
topology 4.2.1
,
on
bw
and
bw*
bounded weak (bw) topology on
a( E;E') a ( E " ; E ' ) ) on bounded s e t s . The space
E
(respectively with the be t h e
r-ball
by
and
)
g, and which
belong t o
CwbU(E).
E
.
It
E i s a l o c a l l y convex
i s reflexive. E", The
topology on E ( r e s p e c t i v e l y the bounded weak* (bw*)
E " ) i s the f i n e s t topoZogy on
p e c t i v e l y by
W C G
Let E be a Banach space with normed bidual
agrees with the weak topology pology
R
,
B1,
topologies are introduced
bw-topology on a Banach space
i f and o n l y i f t h e Banach space
Definition.
-+
E'
.
topologies
I n t h i s section i s proved t h a t t h e
g :B1
6
Thus, a c c o r d i n g t o T i e t z e ' s theorem (0.12.1
t h e r e e x i s t s a weakly continuous
4.2.
$
T h i s f u n c t i o n i s weakly continuous on
Since E i s a separable space,
t h e r e f o r e weakly normal.
re ,
Cwb,(E)
space, t h e James-Klee theorem (0.12.8) s t a t e s t h a t t h e r e e x i s t s which does n o t a t t a i n t o i t s norm.
is
method t o f i n d
and does n o t belong t o
would be t o f i n d a weakly continuous f u n c t i o n over on t h e u n i t b a l l i n
, one
i s not reflexive
Cwb(E)
E
Ellbw* ) . Br(E)
(respectively
E ( r e s p e c t i v e l y on
El')
( r e s p e c t i v e l y w i t h the weak
which
* tg
El') endowed w i t h t h e bw-topology
-
t o p o l o g y ) w i l l be represented by Ebw ( r e ? F o r each r 0 , l e t Br ( r e s p e c t i v e l y 6; )
bw*
(respectively
pology ( r e s p e c t i v e l y w i t h t h e weak*
B r ( E " ) ) w i t h t h e induced weak totopology)
. Then
Ebw ( r e s p e c t i v e l y
ELw* ) i s t h e t o p o l o g i c a l d i r e c t l i m i t o f t h e Br ( r e s p e c t i v e l y o f t h e B;I ) , Ebw = l i m Br , ( r e s p e c t i v e l y E " = l i m B;I ) , and i t i s n o t r +m bw* r-tm d i f f i c u l t t o see t h a t a s e t i s compact i n Ebw , i f and o n l y i f i t i s weakly compact i n
E;
and t h a t a s e t i s bounded i n
E" bw*
,
i f and o n l y i f
Weakly continuous f u n c t i o n s on Banach spaces
bounded, i f and o n l y i f i t i s bounded i n
i t i s weak*
83
El'.
Also i t i s
R easy t o see t h a t Cwb(E) = C(Ebw) , t h e space o f a l l f : Ebw continuous I t i s known t h a t t h e bw* - t o p o l o g y i s a convex l i n e a r t o p o l o -+
.
gY
, see
(Day [ l l , I 1 5 5 ) .
As f o l l o w s f r o m C o l l i n s [l] , t h e
bw
-
topology i s semilinear
i.e.,
a d d i t i o n and s c a l a r m u l t i p l i c a t i o n f u n c t i o n s a r e s e p a r a t e l y c o n t i n u
ous.
Moreover, i t can be shown, F e r r e r a [ l l , t h a t
bw
t o p o l o g y i f and o n l y i f i t i s a l o c a l l y convex one. C o l l i n s [11,
i s a vectorial
A general r e s u l t o f
which can be extended t o t h e complex case,makes t h e f o l l o w i n g
definition valid.
4.2.2
Definition.
The convex
bw (cbw) topology on
E i s t h e unique
l o c a l l y convex topology w i t h a base of a l l convex neighbourhoods of
in the
0
bw -topology. cbw-topology i s t h e f i n e s t l o c a l l y
I t i s easy t o see t h a t t h e
, which agrees w i t h t h e weak t o p o l o g y on bounded s e t s . , characterizing the
convex t o p o l o g y
I n Wheeler [11 t h e f o l l o w i n g r e s u l t may be found Cbw-topol Ogy.
4.2.3
Theorem
g y on E "
to
. E
The cbw -topology i s the r e s t r i c t i o n of the bw* -topolo
. More s p e c i f i c a l l y
,i
f
E i s r e f l e x i v e we have t h a t
the bw -topology on E i s a l o c a l l y convex topology. bw-topology on
Wheeler [11 proves t h a t t h e convex.. Gomez [ l l proves
c o i s not l o c a l l y
t h a t r e f l e x i v i t y i s a l s o a necessary c o n d i t i o n .
I n o r d e r t o p r o v e t h i s r e s u l t i n g e n e r a l , i t w i l l be c o n v e n i e n t t o f i r s t r e s t r i c t t o separable Banach spaces to
and t h e u n i t sphere o f
Bn = nB
E E
c o n t a i n i n g no subspace isomorphic
1'. We denote by
4.2.4
E
B i
= nB"
B
, B"
and
S
t h e closed u n i t b a l l o f
E, r e s p e c t i v e l y , and we w i l l w r i t e and
E
and
E"
f o r each n e w ,
Sn = nS.
, non-reflexive
If contains no subspace isomorphic t o l', there e x i s t s a subset A of uhich i s bw -closed but i s not closed i n t h e r e s t r i c t i o n t o E of t h e
a a .
Let E be a separable
bw* -topology on
E"
.
Banach space.
Chapter 4
84 Proof. From Rosenthal ball B; , i.e. , each tained in Sn for the there exists for each
Sn is weak*-sequentially dense in the
(0.12.7),
z e BL can be approximated by a sequence con-
weak*- topology. Hence if $ e El'\ E and / I $ 1 1 n e N a sequence (x ) contained in Sn kyn keM and converging to E l $ for the weak*-topology. We define A
AnB,={x =({x
k ,n
=
I x ~ , ~k,n : e U I.
For each rn
E
=
1,
N we have
:ksU,n<m}=
k,n
: ksRI , n z m }
u In-'$
:
n
nzm})
B,.
Since the set I x ~ : ,k e~ B , n 5 rn I U {n-'$ : n 2 ml is weak*-compact, is closed for the restriction it is weak*-closed. Then, the set A n B, Since this topology is the same as the restric of o(E";E') to B,. tion of o(E;E') to B, , it follows that A is bw*-closed. On the other hand, let U be a neighbourhood of 0 for the bw*-topology; let W be a bw*-neighbourhood of 0 such that W t W c U. Since W is absorbing , there exists n o E W , such that no1 4 e W. By definition of the bw*-topology , it follows that there exists a weak*-neighbourhood V of 0 such that W n Bin = V fl Bin . Since 0 converges to nil$ for the weak*-topoyogy , there exists ('kYno ) keM koe N such that (x - no-1$ ) E V , Therefore, k0,no X
k o ,no
= x
-
ko ,no
no1+
t
-1$ no
E(v
n B;~
t wc
w
t wc
U,
This proves that 0 belongs to the bw*-closure of A, and since 0 e E, 0 is in the closure of A for the restriction of bw* to E (we denote this topology by rbw*) . Thus, A i s not rbw*-closed.# 4.2.5 Proposition.
Let E
subspace isomorphic t o i f and only i f
be a separable Banaoh space t h a t contains
1'. Then, t h e bw-topology on
r.0
E i s l o c a l l y conveG
E i s reflexive.
Proof. If E is reflexive , we saw in (4.2.3) that bw is a locally convex topology. Conversely, if E i s not reflexive (4.2.3) and (4.2.4) prove that the bw-topology i s not locally convex.#
Weakly c o n t i n u o u s f u n c t i o n s on Banach spaces
E
Let 1'.
to
If
be a Banach space t h a t c o n t a i n s no subspace isomorphic
E
i s not reflexive
E.
i v e subspace F o f From (4.2.5)
85
,
t h e r e e x i s t s a separable n o n - r e f l e x -
O b v i o u s l y , F c o n t a i n s no subspace isomorphic t o 1 ' .
i t f o l l o w s t h a t t h e bw-topolosy on
Since t h e bw-topology o f
F
i s n o t l o c a l l y convex.
i s t h e r e s t r i c t i o n t o F o f t h e bw-topology
F
on E, i t f o l l o w s t h a t t h e bw-topology i s n o t a l o c a l l y convex t o o o l o g y on
E.
T h i s f a c t and ( 4 . 2 . 3 ) prove t h e f o l l o w i n g theorem.
4.2.6
E be a
Let
Theorem.
isomorphic to
Bancrch space that contains no subspace
, i f and
The bw-topology on E i s ZocalZy convex
1'.
onZy i f E i s r e f l e x i v e . Now we a r e g o i n g t o s t u d y t h e problem f o r Banach spaces t h a t 1'.
c o n t a i n a subspace isomorphic t o Lemma. -
4.2.7
There e x i s t s a subset A of
l', which
o f the bw*-topoZagy on
but i s not cZosed f o r the r e s t r i c t i o n t o 1 Proof. en
Let
is a
A.
= {en : n
u(l';lm)
-
a( (ll)",lm)-closed. B (l')"\
1'
A D , and t h e n
compact.
-
closure o f
A.
and
{ n ( e -e ) P 9
n B
f
U
, we
n
2
i t i s n o t hard t o check t h a t
V
Ao.
If
- n1 +
= n(e
e
we have: ( n ( e -e ) + P 9
1
is a A.
N,
z(ll;lm)i s not
ek)
I/ $11 5
A,.Obviously
P
e
P' q'
P # q, P < k
2n+l/n
An
is
be a balanced, convex
Since 4 i s an accumulation p o i n t o f number o f p o i n t s o f
6
$
N
E
e k / n : p,q,k
i s c o n t a i n e d i n t h e sphere o f r a d i u s Let
A.
Consequently
n e
1
define
An
,
For each
.
Now, f o r each
An =
-
o(l';lm)
1'.
(1')".
Therefore, t h e r e e x i s t s a l i n e a r f r o m $ b e l o n g i n g
u((l')";lm)
0
and
t h e usual b a s i s o f
isolated point o f
- c l o s e d s e t which i s n o t t o the
N1
E
bw -cZosed
is
o((l')";lm) Ao,
(I
f
V/3n)
9
1
-
1 .
n 6 U,
closed.
neighbourhood o f 0.
+ V/3n c o n t a i n s an i n f i n i t e
ek e ( 4
3n2
1' and i f
u(ll;lm)
-4) + n($ - e ) +
€ - 1V t - V 1f - 3
of
9 < k
v c v
n A.
with
1 6 (ek - $1 e
p < q < k,
86
Chapter 4
Thus, ( $ / n
V ) n An
t
i s a nonempty s e t . T h i s proves t h a t
, ~ ( ( l ~ ) " , l-~ c)l o s u r e o f
t o the
A =
we d e f i n e
belongs
An.
m
Now
$/n
U n=2
.
An
m
For each
E
El , we have
AnBm= U { A n : n E U , 5 < 2 n 2 t 1 < m . n l . This set i s obviously
-
o(ll;lm)
, and
closed
thus
A
is a
bw-closed
set. On t h e o t h e r hand, s i n c e f o r each o((ll)";lm) 4.2.4,
-
closure o f
i t f o l l o w s t h a t 0 belongs t o t h e
does n o t belong t o to
An ,
A, we have t h a t
1' o f t h e bw*-topology on From lemma 4.2.7
l o c a l l y convex bw
. Hence
if
n
E
N
$/n
belongs t o t h e
reasoning as i n t h e l a s t p a r t o f p r o o f
A
bw*-closure o f
A, and s i n c e 0
i s not closed f o r the r e s t r i c t i o n
(11)",#
i t f o l l o w s t h a t t h e bw-topology on
c o n t a i n s a subspace isomorphic t o
E
i s n o t a l o c a l l y convex topology on
E.
1' i s n o t 1' t h e
T h i s f a c t and ( 4 . 2 . 6 ) prove
t h e f o l l o w i n g theorem.
4.2.8
Theorem.
Let
E be a Banach space
l o c a l l y convex topology 4.3
,i
f and only i f
.
The bw-topology on E i s a
E i s reflexive.
Weakly continuous and weakly u n i f o r m l y continuous f u n c t i o n s on bounded s e t s .
erties of
T h i s s e c t i o n i s d e d i c a t e d t o s t u d y t h e main t o p o l o g i c a l propCwb(E) and Cwbu(E). We w i l l see when t h e y a r e b o r n o l o g i c a l
and b a r r e l 1 ed spaces. We endowed
Cwb(E;F)
w i t h t h e l o c a l l y convex t o p o l o g y o f u n i f o r m
convergence on weakly compact subsets o f note
where
K
T h i s topology, which we d g
i s allowed t o range o v e r t h e weakly compact subsets o f
I f t h e space
C(Ebw)
i t f o l l o w s t h a t t h e spaces
morphic.
E.
i s generated by a l l seminorms o f t h e form:
T~~
Ln
E.
i s endowed w i t h t h e compact-open topology,
Cwb(E)
and
order t o study properties o f
C(Ebw)
are topologically i s 0
CWb(E),
we need t h e f o l l o w i n g
Weakly continuous f u n c t i o n s on Banach spaces
a7
1emma. 4.3.1
Lerruna.
If
is normal and
E is a weakZy normal space, then Ebw
hence completely r e g u l a r . Proof.
E
If
with
i s a weakly normal space, t h e n f o r every
t h e weak
c l o s e d subsets of En
Ebw
,
nD
C
the closed n-ball i n
Urysohn's
.
r e s t r i c t e d t o p o l o g y i s normal E.
.
Since fl :
lemma, we have
6
=
B1
Let
Cn = C
and
C1
Let
D1
n
C
Bn
and
, Dn
n B,
endowed D
=
be D
n Bn ,
a r e weakly c l o s e d , by
a weakly c o n t i n u o u s f u n c t i o n
+[O,l]
such t h a t fl(Cl) Let
f:
:
B1
=
il
C2
101 U Dz
+
11).
and
fl(D1)
[0,1]
be t h e f u n c t i o n d e f i n e d as:
=
T h i s f u n c t i o n i s weakly continuous and i t i s d e f i n e d on a weakly c l o s e d subset o f
62;
f 2 ( c 2 ) = I01 By i n d u c t i o n
, i t can be e x
t h e r e f o r e by T i e t z e ' s theorem (0.12.1)
fz :
tended t o a weakly continuous f u n c t i o n and
f2(D2) =
B2 +
111
[0,11
such t h a t
.
, i t f o l l o w s t h a t a sequence e x i s t s
, fn: 6,
(f,)
+
[0,11
b e i n g weakly continuous, such t h a t fn(Cn) = {03 Let Then Ebw
f : E + R
f
;
fn(Dn) =
Ill and
fnlBn-l
if
be t h e f u n c t i o n d e f i n e d as f ( x ) = f n ( x )
i s c o n t i n u o u s on
Ebw
, f(C)
=
{O} and f ( D )
.
= fn-l
x
6
En.
{I). Hence,
=
i s normal . # Unfortunately
, we
do n o t have a general r e s u l t
t h e h y p o t h e s i s o f weak n o r m a l i t y
, a s s u r i n g t h a t Ebw
p l e t e l y r e g u l a r space, which i s necessary f o r s t u d y i n g theless
,
-
if
Ebw
r e p r e s e n t s t h e space
E
, eliminating
i s always a corn C(Ebw)
.
Never
endowed w i t h t h e c o a r s e s t
88
Chapter 4 Cwb(E) c o n t i n u o u s , i t f o l l o w s i s c o m p l e t e l y r e g u l a r and t h e f o l l o w i n g i n c l u s i o n s
t o p o l o g y t h a t makes each f u n c t i o n i n
-
t h a t t h e space
Ebw
a r e continuous:
g i v i n g e q u a l i t y t o t h e f i r s t i n c l u s i o n , i f and o n l y i f
Ebw
i s complete
l y regular.
F i r s t o f al1,we w i l l see when According t o N a c h b i n - S h i r o t a ' s lent to
4.3.2
Ebw
Cwb(E)
theorem
i s a b o r n o l o g i c a l space.
(0.12.3),
t h i s would be e q u i v g
b e i n g realcompact.
Theorem.
E be a weakly normal Banach space.
Let
Then,
Cwb(E)
is bornoZogica2 if and only if E is weakly reaZcoFact. Proof.
Each c l o s e d s u b s e t o f a r e a l c o i n p a c t
i t i s o n l y necessary t o prove t h a t i f t h e
Bn
weakly realcompact).
The p r o o f f o r
s l i g h t m o d i f i c a t i o n s t h i s c a n be u s e d f o r L e t { ValaeA b e a t i o n p r o p e r t y ; each ous
-1 = fa (0)
U,
E
Ebw
z-ultrafilter
r e a l c o m p a c t . Thus,
b a l l s w i t h weak t o p o l o g y
, t h e n Ebw ( r e s p e c t i v e l y E)
a r e realcompact tively
space i s
i s r e a l c o m p a c t (respecw i l l be c a r r i e d o u t ; w i t h
.
w i t h t h e countable intersec-
, where fa: E - * R i s w e a k l y c o n t i n u -
I
n B # 6 f o r every a E A, no I f t h i s where n o t so, f o r each n 6 N , we w o u ld have an 6 A such t h a t Uan n Bn = 6 , and hence we w o u l d h a v e t h a t n U, = 6 . Thus There i s
no,such t h a t
U,
a,
n=l
n
t h e c o u n t a b l e i n t e r s e c t i o n p r o p e r t y would f a i l . T h e r e f o r e , f o r every
I u, n B ~ I
i s a f i l t e r basis i n
n 2 no,
Bn.
,€A There e x i s t s
nl
2
n o i n s u c h a way t h a t t h e f i l t e r b a s i s
t u a n B ~ ~ I has t h e c o u n t a b l e i n t e r s e c t i o n p r o p e r t y . Supposing t h e ,€A above f a i l s
, f o r every n 2 n o t h e r e would be { a
i n such a way t h a t
n m= 1
1
i n d e x sequence
n y m mew
m
n Bn)
=
6
.
The c o u n t a b l e f a m i l y
Weakly continuous f u n c t i o n s on Banach spaces
{ua n ,m
n ,meN
,
n
no
,
89
has an empty i n t e r s e c t i o n , c o n t r a r y t o t h e
countab e i n t e r s e c t i o n p r o p e r t y . For each
cx
6
,
A
{Ucx
n
1
Bn
i s the basis o f
BnI
c o u n t a b l e i n t e r s e c t i o n pr0pert.y i n i s a continuous f u n c t i o n on
f a ] Bnl ed topology.
IUcx n
BrillCi€A i s a b a s i s
a z - f i l t e r with
, because f o r e v e r y cx
Bnl
6
A
,
endowed w i t h t h e weak r e s t r i c t
n o t , there w o u l d e x
fora z-ultrafilter.If
zero i n B t h a t i s , Z = f-'(O) , f b e i n g weakly nl nl' continuous on B i n such a way t h a t Z n Ua n Bnl # d f o r e v e r y ~1 6 A, n1 , b u t Z n o t c o n t a i n i n g any Ua n Bnl But since B i s a weakly c l o s e d -. n l subset o f E , by n o r m a l i t y a weakly continuous f : E + R e x i s t s , such ist 2 c B
.
7I
that
Bnl
Furthermore
= f.
.Z ! U
therefore
=
z = ua0 n s n1'
a0
Just l i k e
n
(Uol
n
?
, f-l(O)
Bnl
Z =
z"
n Ua # d M A
a6A
compact.
. Since
that
Ebw
f
f o r e v e r y cx e A,
,
and thus
i s realcompact by h y p o t h e s i s , i t f o l l o w s t h a t
# d and thus ,
Bnl)
w i l l be a z e r o s e t o f E ( w i t h
=
.
n B
But flUa # nl f o r some cxo by b e i n g a 2 - u l t r a f i l t e r
t h e weak t o p o l o g y ) , and
i s a l s o continuous on
i s realcompact.
Ebw
. Hence
E
i s weakly r e a l -
, i t can a l s o be i n f e r r e d
#
Weakly realcompact Banach space a r e d e s c r i b e d on Corson Nevertheless, weakly realcompactspaces do e x i s t normal
, as i n t h e case o f
lm, Corson [ l l
.
[ll.
which a r e n o t weakly
I n any case, t h e c l a s s o f
weakly normal and weakly realcompact spaces i s broad; s p e c i f i c a l l y e v e r y W
c
G
space i s weakly L i n d e l g f
( 0 . 1 0 . 1 ) and i s thus so i n o u r hypothesis.
The f o l l o w i n g statement g i v e s a p a r t i a l answer t o t h e problem o f n o t n e c e s s a r i l y normal spaces.
4.3.3 Theorem. If E is the dual of a separable space, then bornoZogicaZ ; i n particuZar Cwb( l a ) is bomoZogicaZ. Proof.
Let
E = F'
and
(x~),,~,,
be t h e f u n c t i o n d e f i n e d as:
be a dense subset o f
Cwb(E) i s
F. L e t f :
Ebw+ R'
Chapter 4
90
T h i s map i s one-to-one being dense i n
because
(
F, and continuous.
x
~ separates ) ~ ~ p~o i n t s o f
RN
Since
i t s subsets a r e as w e l l ( s i n c e p o i n t s
by
i s realcompact and a l l
-
G6
are
E
sets), i t follows
-+
from ( 0 . 8 . 1 ) t h a t
4.3.4 then
CoroZZary.
If
i s realcompact and
Cwb(E)
i s bornological
'#
is the duaZ of a separable space,
E i s W C G or E
i s bornoZogicaZ.
Cwb(E ;F)
Proof.
Ebw
(0.12.3)
I t i s an immediate consequence o f
'#
,"
We a r e going t o see now t h a t
Ebw
is a
NS-space; t h a t i s ,
every r e l a t i v e l y pseudocompact and c l o s e d subset i s compact. Through
( 0 . 1 2 . 3 ) we achieve t h a t 4.3.5
Theorem.
Cwb(E)
i s always b a r r e l l e d .
For every Banach space
E
, Cwb(E)
i s barrelled. 5
Proof.
Let
be a r e l a t i v e l y pseudocompact and c l o s e d subset o f
K
x' E E'
Since f o r a l l
, x'(K)
bounded and thus bounded. hand
Therefore
K
,
it follows that
see r e s l ; l t s
K
Ebw.
i s weakly
E
i s continuous on
.
Ebw,
i s weakly r e l a t i v e l y pseudocompact. From V a l d i v i a [ l 1,
a t t h e end o f ( 0 . 1 2 . 1 1 )
compact and thus compact i n
CoroZZary. Let
E,F
-
,
i t follows that
K
i s weakly
Ebw.#
be Banach spaces. T'hen Cwb(E;F)
I t f o l l o w s from ( 4 . 3 . 5 )
Proof.
K
i s weakly closed. On t h e o t h e r
s i n c e each weakly continuous f u n c t i o n on
we have t h a t
4.3.6
i s bounded
and
i s barreZle,.?.
( 0 . 1 2 . 3 ).#
The problem o f t h e c o m p l e t i o n of
Cwb(E;F)
w i l l be s t u d i e d i n
section 4.4. We endow
Cwbu(E;F)
w i t h t h e l o c a l l y convex topology o f u n i f o r m
convergence on bounded subsets o f
E.
T h i s t o p o l o g y , which we denote
i s generated by a l l seminorms o f t h e form:
f E Cwbu(E;F) where
B
+
supIl( f ( x ) l l
: x E BI
i s allowed t o range over t h e bounded subsets o f
E.
T ~ ,
Weakly continuous f u n c t i o n s on Banach spaces
4.3.7 Proof.
Let
is t h e
Cwbu(E;F)
Theorem.
i f n } c Cwbu(E;F)
E.
on bounded subsets o f
, such t h a t f n
n,
we have
f n E CwbU(E;F), X,Y E B
(1 f(x) -
B c E
there i s
f
, I@i(x - y ) l < 6 ( i = 11 f ( x ) - f ( y ) ( 1 < E
i s weakly
be bounded and
f n ( x ) ( I < ~ / 3( x 6 B ) .
E
@l,..., @ k 8 E ' such t h a t 1, ...,k I y t h e n 1 1 f n ( x ) - f n ( y ) ( I < , which proves t h a t
> 0.
Since
and
6 > 0
Therefore
uniformly
f
-f
I t remains t o be shown t h a t
u n i f o r m l y continuous on bounded s e t s . L e t F o r some
Pf(E;F)
b
be a Cauchy sequence. I t i s easy t o see
f E C(E;F)
that there i s a function
T -compZetion of
91
Cwbu(E;F)
if
~/3.
i s com-
plete. To prove t h e theorem i t s u f f i c e s t o prove t h a t dense i n
Cwbu(E;F)
ed,
E
and
. To
> 0 and l e t
such t h a t i f
x,y 6 B
.
t h i s end, l e t
6 > 0 and
is Pf(E;F) c E be bound
,B
f E Cwbu(E;F)
@k} c E '
, I@i(x-y)/ <6 ( i
= 1,
..., k )
be s e l e c t e d
,
then
Defining 0: E - R k by @ ( x ) = ( @ , ( x ) , ..., @,(x)) , f(x) - f(y)I( < E we choose such t h a t f o r any { x l ,.. ,xm 1 c B x E B ,I1 0 ( x ) - m ( x i ) l l < B / 2 k f o r some i = l , . . .,m (where R i s g i v e n t h e supremum norm). There
(1
.
e x i s t non n e g a t i v e continuous f u n c t i o n s
hl
,. . . ,hm
: Rk
+
R
such t h a t :
m
1
hi(Y i=l supp(hi Choose p o l y n o m i a l s
ql,
c B
(@(xi),&)
for
.. ,qm
: Rk+ R
such t h a t f o r
m
+
(1 1
i=l
i = 1,
R
hi(O(x))(f(xi)
-
f(x))II <
...,m. i = 1,. . .,m
ZF , where we have used t h e
92
Chapter 4
fact that for
x 6 B
,
m
1
hi(o(x))
= l.#
i=1 Let standard norm
r
i n the
E
.
be a Banach space and The
ball
ball
r
Br(E")
Br(E)
of
.
E"
?
can be extended t o a f u n c t i o n
E"
of
i t s strong bidual w i t h the
E
, each
Therefore from
weak*-dense
is
E"
to
f
in
Cwbu(E;F)
F, such t h a t
w*-uniformly continuous when r e s t r i c t e d t o bounded s e t s o f
7
is
E " . Moreover
r > 0
for a l l
Therefore,
t h e space
Cw,bU(E";F)
of all
Cwbu(E;F) f
from
E"
i s t o p o l o g i c a l l y isomorphic t o t h e space to
F which
ous when r e s t r i c t e d t o t h e bounded s e t s o f are
, Cw,bU(E";F)
w*-compact
to
are
continu
E". Since c l o s e d b a l l s i n
i s exactly
F which a r e
w*-uniformly
E " , endowed w i t h t h e t o p o l o g y
o f u n i f o r m convergence on bounded subsets o f E"
(w*-dense)
Cw*b(E";F),
t h e space o f
a l l f u n c t i o n s from
El'
t o bounded s e t s o f
E". Then we have t h e f o l l o w i n g t o p o l o g i c a l isomorphism
representations o f
Cwbu( E;F) :
where to
C(E'Ibw*;F)
F,
w*-continuous when r e s t r i c t e d
i s t h e space o f a l l continuous f u n c t i o n s f r o m
endowed w i t h t h e topology o f u n i f o r m convergence on
Ellbw*
bw*-bounded
E".
sets o f
F i n a l l y i t i s easy t o see t h a t e v e r y r e l a t i v e l y pseudocompact and c l o s e d subset i n
Ellbw*
i s corrpact.
Thus, we have t h e f o l l o w i n g
theorem. 4.3.8
Theorem.
4.3.4 Corollary. is barre Zed. Proof.
For each Banach space Let
E,F
E
, the space Cwbu(E) is Sarrd?,Zcd.
be Banach spaces.
I t f o l l o w s from (4.3.8)
and
(0.12.3).
Then, the space
cwbu( E ;F)
Weakly continuous f u n c t i o n s on Banach spaces
4.4.
93
On c o m p l e t i o n o f spaces o f weakly c o n t i n u o u s f u n c t i o n s . I n t h i s section a useful representation o f the completion o f
C,(E;F),
t h e space o f weakly continuous f u n c t i o n s between two Banach E
spaces
and
F,
i s g i v e n . We i n t r o d u c e t h e space
Cwsc(E;F)
t a i n i n g t h e weakly s e q u e n t i a l l y continuous f u n c t i o n s . s h i p s between t h e spaces
Cw(E;F)
on
are studied; conditions sent the completion o f
Cw(E;F)
i z a t i o n o f Banach spaces
E
are obtained.
containing
E
The r e l a t i o n -
, Cwb(E;F) , Cwk(E;F)
t h e Banach space
con
and
Cwsc(E;F)
so t h a t t h e y repre-
F i n a l l y a character-
l ’ , i n terms o f these classes
o f weakly continuous f u n c t i o n s , i s g i v e n . We c o n s i d e r
Cw(E;F)
, Cwb(E;F)
and
Cwk(E;F)
t o be endowed
w i t h t h e t o p o l o g y o f uniform convergence on weakly compact subsets o f E .
4.4.1
Proposition.
Proof.
Since
Let
i s a dense subspace of
Cw(E;F)
f 8 Cwk(E;F).
(K, o ( E ; E ’ ) I K) fK :
i s compact ( t h r o u g h o u t comDact means compact and
E
-f
CfK) , where
K
A consequence o f t h i s r e s u l t i s t h a t
Now
Cw(E;F)
i s a dense sub-
c~~(E;F).
4.4.2 Corollary. mensicn.
Cw(E;F)
i s compZete if and only i f E has f i n i t e di-
I t f o l l o w s f r o m comments a f t e r p r o p o s i t i o n 4 . 1 . 3 .
A f i r s t answer about t h e c o m p l e t i o n o f
Cw(E;F)
f o l l o w i n g nroposit i o n .
4.4.3
flK.
ranges over t h e weakly
E, converges t o f . #
compact subsets o f
Proof.
, see ( 0 . 1 2 . 2 ) ,
F, a weakly c o n t i n u o u s e x t e n s i o n o f
i t i s easy t o see t h a t t h e n e t
space o f
K c E, we c o n s i d e r
F o r e v e r y weakly compact
H a u s d o r f f ) , i t f o l l o w s f r o m a r e s u l t o f Dowker [ l ] t h a t there exists
C,k(E;F)
Proposition.
The space
c,~(E;F)
i s complete.
i s g i v e n by t h e
94
Chapter 4
Proof.
weakly compact s e t . x
K.
6
F.
Now, i t i s easy t o check t h a t
f
fK : K
and
IfiIieI
that
Definition.
We d e f i n e
f: E
K c E
be a K
,
f(x) = fK(x)
i s w e l l defined, t h a t
converges t o
A function
Let
converges u n i f o r m l y on
+
f E Cwk(E;F)
4.4.4
IfilKlieI
Then, t h e n e t
t o a weakly continuous f u n c t i o n if
.
Ifi}ieI be a Cauchy n e t i n Cwk(E;F)
Let
f.#
F is said t o be weakly sequentially
+
continuous i f it takes weakZy convergent sequences i n E t o convergent
.
F
sequences i n
We denote by
CWsc(E;F)
continuous f u n c t i o n s f r o m 4.4.5
Cwsc(E;F)
and
I t i s obvious t h a t
let
be i n
A c K
Cwsc(E;F)
pair of Banach spaces E,F
the spaces
.
Cwk(E;F) c Cwsc(E;F) and
K
On t h e o t h e r hand,
be a weakly compact subset o f
E.
If
, t h e weak c l o s u r e o f A, s i n c e every Sanach space i s
x E
and
F.
coincide.
Proof. f
t h e space o f a l l weakly s e q u e n t i a l l y
to
For every
Proposition.
Cwk(E;F)
E
a n g e l i c f o r t h e weak t o p o l o g y ( 0 . 6 . 1 ) , i t f o l l o w s t h a t t h e r e i s a sequence
(x,)
A
in
such t h a t
q u e n t i a l c o n t i n u i t y of that 4.4.6.
f(x)
8
(x,)
f, f(xn)
converges weakly t o
converges t o
f(x)
in
By se-
x. F,
showing
?(A).# If E is a Schur space, i . e . , i f the weak convergent
Corollary.
and norm convergent sequences in E are the same, then f o r every Banach space F the space
Cwk(E;F) and
C(E;F)coincide.
Now we a r e going t o study when
Cwb(E;F)
and
Cwsc(E;F)
are
equal. 4.4.7.
Theorem.
A Banach space
E contains no subsrace isomorphic t o
1 ’ i f and only i f f o r every Banaeh space Proof.
Let
f E Cwsc(E;F)
Kaplansky’s theorem
isomorphic t o
B
f o r every
D c A
such t h a t
D. Since Eo
c,~(E;F)
= c,,,(E;F).
be a bounded subset o f
(0.12.4),
e x i s t s a countable set l i n e a r span o f
and
F,
E.
By
A c B and each x e ,there x e l W . L e t E G be t h e c l o s e d
i s separable and c o n t a i n s no subspace
l ’ , according t o a Rosenthal r e s u l t (0.12.7),
there exists
Weakly continuous f u n c t i o n s on Banach spaces
a sequence
-
a(E;E') (f(xn))
c D
(x,)
convergent. converges t o
hence t h a t
which i s
a(E,,;E;)-convergent
to
, s i n c e f e Cwsc(E;F) ,
Now
f ( x ) ; t h i s proves t h a t
95
x
and hence
i t follows that c
f ( A ')
f(A) , and
i s weakly continuous.
flB
The p r o o f o f t h e o t h e r p a r t r e l i e s on P e l c z y n s k i ' s o b s e r v a t i o n
E c o n t a i n s an isomorphic copy o f
that i f
:',
exists
S:, E
let
be a q u o t i e n t map from
q
q
+
such t h a t
S
l', t h e n a l i n e a r o p e r a t o r
i s a b s o l u t e l y 2-summing. To be s u r e , 1'
onto
1 2 , see
( 0 . 1 2 . 1 0 ) . The map
-
i s a b s o l u t e l y summing, ( 0.7 ) , and hence a b s o l u t e l y 2
Grothendieck-Pietsch's
theorem (0.7.2
)
,a
r e g u l a r Bore1 p r o b a b i l i t y
measure p e x i s t s , d e f i n e d on t h e u n i t b a l l such t h a t ,
G : 1'
where
X p denotes t h e c l o s u r e o f
if +
i n a l norm i n t o
1' i n
L2(p)
-
completion o f
t h e unique continuous l i n e a r e x t e n s i o n o f continuous l i n e a r o p e r a t o r t o o . where
T
: L2(p)
( i n i t s weak*-topology)
Blm
L 2 ( p ) , then
X 2 i s t h e n a t u r a l i n c l u s i o n mapping o f X2,the
+
Xz
Let
summing. By
q
l', and to all of
B : L2(p)
+
i s the natural p r o j e c t i o n
1
.
1'
q = P
u
G
i n i t s orig-
P : X p +12 i s XZ. G
d e f i n e d by
is a B = P
3,
G admits a f a c t o r -
i z a t i o n i n t h e form: il
1'-
I
where fore
where ators.
q
=
I
i 2
C(B,m,w*)4
L"(p)
-f
L2(p)
and
L"(p)+
L2(p)
; A = i2 0 i,
il , i p a r e t h e n a t u r a l i n c l u s i o n s . T h e r e
admits a f a c t o r i z a t i o n i n t h e form:
A : 1'
+
Lm(u) and B : L2(p) 1' a r e bounded l i n e a r oper , A extends t o a bounded l i n e a r o p e r a t o r T f r o m E
O f course
-f
96
Chapter 4
to
by
L"(p)
and so
B
E
of
I
o
T = S : E
1'
-t
l 2 (0.7.1 ) .
onto
I i s a b s o l u t e l y 2-summing
i n j e c t i v i t y ( 0.11 ) ;
Lw(p)'s
i s an a b s o l u t e l y
Further
S
2-summing q u o t i e n t map
12, S
b e i n g a q u o t i e n t map o n t o
An a b s o l u t e l y 2-summing o p e r a t o r sends weakly convergent
i s n o t compact.
sequences i n t o norm convergent ones ( 0 . 7 . 2 ) .
S f CwSc(E;l2)
Thus we have
-. Cwb(E;l').# 4.4.8.
Corollary.
, if
1'
E contains no subspace isomorphic t o
A Banach space
and only if f o r every Banach space
Now we a r e i n t e r e s t e d i n those spaces F,
Cwb(E;F)
4.4.9.
and
Proof. 1'
,
Let E,F
Theorem.
phic copy of
1
From
then
C
Cwsc(E;F)
1
, if and
S
a r e always d i f f e r e n t .
. Then
be Banach spaces onlg if
Cwb(E;F)
(E;F) = Cwsc(E;F)
Conversely, l e t and l e t
that
E contains an isomor
!ji Cwsc(E;F).
E
for all
Banach spaces
F.
be any Banach space c o n t a i n i n g a copy o f
be a q u o t i e n t mapping o f
l'onto
t o a continuous l i n e a r , noncompact mapping of t h e
, such t h a t f o r every
E
, i f E does n o t c o n t a i n an isomorphic copy o f
(4.4.7)
wb
is complete.
F, Cwb(E;F)
1 2 . Then S : E
+
S
can be extended
1 2 , and an a p p l i c a t i o n
Grothendieck-Pietsch theorem (see p r o o f o f theorem 4.4.7)
, the
I/ ' 1 1
11 -11
function
yields
, hence weakly s e q u e n t i a l l y continuous.
i s a b s o l u t e l y summing
S
Therefore
l',
6
0s
Cwsc(E).
We show t h i s f u n c t i o n
CWb(E) . S being noncompact , S i s n o t weakly continuous on bounded s e t s (lemma 4 . 1 . 1 ) . T h i s i m p l i e s t h a t E > 0 and a n e t (xlx) S
B
converging weakly t o
1 1 Sx,. -
E.
exist
This, i n t u r n
, w i t h x ,xa
, means
6
u n i t b a l l Bl(E),
t h a t t h e r e e x i s t s an
( x R - x ) converging weakly t o 0 w i t h xu, x
a net
(11 011 zero
Sx(( >
x
0
.
tains
Thus
Let y
, 1) .I/
E. o
S
Hence
6
(11
Cwb(E).
0
Cwb(E)
y
> 0 and
such t h a t
i s n o t convergent t o
S ) (xa- x )
Hence
E
B,(E),
Cwsc(E), i f
E
con-
1'. F i n a l l y , we want t o show E contains 1'.
F when
spaces when
S)(xa- x ) >
6
such t h a t
E
contains f
F with
g(e) = f ( e ) y
for
Cwb(E;F) 5 Cwsc(E;F) f o r a l l Banach As we have shown Cwb(E) CwSc(E) ,
l', choosing an
11 yII =
1
.
e E E
.
Obviously
f 6 Cwsc(E)
but
Consider t h e f u n c t i o n g f Cwsc(E;F)
not i n g : E
.
+
Cw,(E).
F g i v e n by
I f we assume t h a t
Weakly continuous f u n c t i o n s on Banach spaces
C
wb
, then
(E;F) = Cwsc(E;F) n ($i)i=l
there e x i s t
c El
-
f(x)I
1 1 yII < E
,
6 > 0
and i = 1,
I @ i ( x l - x ) I ~ G f o r every If(x')
g 6 Cwb(E;F)
...,n,
a contradiction t o
4.4.10
Let
if
-
n E
and
space
E,F
N.
be
Let Q
Banach spaces.
= { f E P("EE;F)
x,y
if
6
B
= {f e P
6>O
2
l', if and only
be an a r b i t r a r y subset o f
,
such t h a t i f
/I f(x) - fbll<
El
n p4c( E;F) sequences
(x,)
E
in
E
6 o f Q and 6 > 0, such
"weak" ; thus
11
f(x)
-
: for a l l balls
B
in
E,
y 6 B
, I@(x-y)I <
f(y)/l <
6,
(4
6
€1
Q
8) , t h e n
f o r which
0 = El
f o r a l l bounded
:
($(xn))
,
i s Cauchy ( @ E Q)
(f(xn))
FI.
= { f E P('E;F)
f o r which
converges t o When
E;F)
= { f 6 P('E;F)
PQsc ('E;F) in
n
E
( + E 8) , t h e n
,
< 6
in
*
i s a Cauchy sequence i n
(f(xn))
I
B
for a l l balls
:
a1 1 E > 0, t h e r e i s a f i n i t e subset 6 o f
and
x E B
a l l points
16 X -Y
9
pob (";F
(x,)
Then E
,
f E Cwb(E)
PC~E;F).
and f o r a l l E > 0 , t h e r e i s a f i n i t e subset
and
g ( x ' ) - g ( x ) l l < E. That i s
f ( x ) I < E. Hence
E,F be Banach spaces.
with
We w i l l be i n t e r e s t e d i n t h e f o l l o w i n g subspace o f t h e
P4bu (.'E;F)
that
11
x ' e BI(E)
Polynomial case. Let
El
such t h a t f o r a l l
is not complete.
Cwb(E;F)
4.5.
Thus, f o r E > 0, x E Bl(E),
we have
,and so
If(x') the choice o f f.#
Corollary.
.
97
+(x
-
xn)
:
-t
f o r a l l bounded
0
f o r some
x 6 E
(41
E
O),
f ( x ) i n FI.
, we w i l l r e p l a c e
, f o r example PE,bu(nE;F)
@
i n o u r n o t a t i o n by
w i l l be denoted by
w, f o r Pwbu(nE;F)
98
Chapter 4
4.5.1
Remark.
n LGbu( E;F) denotes t h e subspace o f
L('E;F)
c o n s i s t i n g o f those
n - l i n e a r mappings w h i c h c o r r e s p o n d , v i a t h e p o l a r i z a t i o n f o r m u l a ( 0 . 3 . 1 ) , n t o elements o f PQbU(nE;F) ; Lost( E;F) and LQC("E;F) a r e d e f i n e d s i m p 1a r l y
. It i s routine t o v e r i f y t h a t
by
P ('E;F)
, w i t h t h e norm i n d u c e d
Pwbu ('E;F)
i s complete. (See theorem
4.3.7).
I n t h i s s e c t i o n we w i l l show t h a t t h e f o l l o w i n g diagram h o l d s :
where t h e i n c l u s i o n s i g n s mean t h a t s t r i c t i n c l u s i o n can o c c u r , depending on
E
and
F.
m
I n f a c t , t h e same s c a l a r v a l u e d p o l y n o m i a l
, shows
a c t i n g on 1 ' and 1'
the canonical basis vectors f o r a l l n. coincide i n that
Also
,
1'.
t h i s . Indeed, (en) i n
p 6 P (212)
l 2w e a k l y t e n d
\
to
1
x i n=l Pwsc ( 2 1 2 )
p(x) =
0 but
, since
p(en) = 1
p e P w s c ( 2 1 1 ) s i n c e weak and norm convergence sequences
However, an a p p l i c a t i o n o f p r o p o s i t i o n
4.5.8
shows
.
p 6 Pwbu ( ' 1 ' )
The f o l l o w i n g u s e f u l p r o p o s i t i o n i s h e l p f u l i n g i v i n g a g e o m e t r i c i d e a o f some o f t h e above spaces o f p o l y n o m i a l s . 4.5.2
Proposition.
p E P('E;F)
A poZynomia2
beZongs to
PGb(nE;F)
f o r some subset
@ c E ' i f and
onZy i f the foZZorJing condition is s a t i s -
f i e d : f o r any
x
E
such t h a t i f
6
B I ( E ) and
y B Bl(E)
>
satisfies
0
there i s a f i n i t e subset
@ ( x- y ) = 0
(@ e
e) ,
0 c
then
Weakly continuous f u n c t i o n s on Banach spaces
Proof. of
Only t h e s u f f i c i e n c y needs t o be proved. Using t h e homogeneity
p, i t i s c l e a r t h a t t h e c o n d i t i o n h o l d s f o r
Let
99
,x
B = Br(E)
, and
E B
E
,
> 0 be g i v e n
t h e above c o n d i t i o n corresponding t o
2B
Br(E)
and choose 0 c
, and
- z 11
as i n
$
.
z 1 , ...,zm
. Since
p
e
of
I$l,...,q~~
so we may choose p o i n t s
Qi(z.) = 6 , for 1 5 i , j 5 m J ij continuous on 28 , t h e r e i s some c o n s t a n t y, such t h a t
,I]
r > 0.
~ / 2 There i s c l e a r l y
and
no l o s s i n g e n e r a l i t y i n assuming t h a t t h e elements a r e l i n e a r l y independent
f o r any
E
E
i s uniformiy
0 < y <
r , such t h a t
/ I p ( y ) - p ( z ) l ] < E / Z . Now , l e t 6 = y/ (m.max I l l zill : 1 <- i -< m I ) and l e t y e B s a t i s f y I @ i ( x - y ) l < 6 ( 1 2 i 5 m). I f we s e t w = y ~ ~ ( y - x ), zt h~e n I ) w - y l l 4 y and So 11 w / / < 1 ) yIl + r < 2 r . Thus, b;=ltheuniform continuity o f - p , if y,z
6
28
y
1: p(w) - p ( y ) j ) < d 2 . ) I p(w) - p ( x ) 11 < ~ / 2 ,by
< y, t h e n
i = 1,..., m , ~
A l s o f o r each
hypothesis .Therefore
11
p(x)
~ - x() =w 0
- p(y) II<E
, so t h a t
as r e q u i r e d .
We would l i k e t o comment t h a t t h e above p r o o f can be adapted t o any s i t u a t i o n i n which t h e f u n c t i o n i n q u e s t i o n i s u n i f o r m l y c o n t i n u o u s on bounded s e t s .
Proposition. A poZynorniaZ
4.5.3
,if
0 of E '
some subset
p E P('E;F)
beZongs t o Ppbu(nE;F) f o r
and onZy i f f o r any E > 0 there i s a f i n i t e x, y E B,(E) s a t i s f y @ ( x - y ) = 0 (I$ E
0 c 0 such t h a t i f
subset
then
I n p a r t i c u l a r , we have
/ I P(X) -
P ( Y ) / I< E
Theorem.
4.5.4.
integer
I
For any Banach space E and F
, any
@ c E'
,and any
n, pQC(nE;F) = P ~ ~ ~ ( " E ; F ) . For t h e p r o o f , i t w i l l be c o n v e n i e n t t o c a l l a sequence
in E
$-convergent t o
~ ( -y y k ) 4.5.5
e),
3
0
Lemma.
sequences i n
y
in
E (resp.
p-Cauchy) i f f o r a l l
( r e s p . ( @ ( y k ) ) i s Cauchy).
Let E
. Suppose
t o 0 and t h e others are cenverges t o 0 i n
n ~ E;F) ~ and ~
A E L
F
.
- Cauchy
.
(yk) E 0,
We f i r s t need
( (x k ) ,...,(x:)
be n bounded
t h a t a t Zeast one sequence is 0
4
0-convergent
Then the sequence (A(Xf(
,. . . 9Xkn
))
#
Chapter 4
100
.
For Proof. The proof i s by induction Assuming t h e r e s u l t f o r j = l , . . , n - 1 , (i= 1 , ...,n ) be as in the hypothesis; that i s @-convergent t o 0. If some E > 0 , / I ,..., x: ) I ] > E f o r of natural numbers. Now f o r each f i x e d fined by
.
(xi)
AX; ( z '
A(xi
,...,z n- 1)
=
A ( z l ,..., z
n = l , t h e r e s u l t i s immediate. i l e t A a n d t h e sequences (x,) t o f i x t h e notation , assume the r e s u l t i s f a l s e , then f o r in an i n f i n i t e subset J all k k E J , t h e mapping Ax: de-
n-1
n
, xk
)
i s a n element of LoSc("'E;F) , Therefore, by t.he induction hypothesis, f o r some index m ( k ) E J i t follows that11 Ax; (xj' , . . . , x y - ' ) I l < ~/2, m ( k ) ; t h e r e i s c l e a r l y no l o s s in g e n e r a l i t y in supposing whenever j m ( k + l ) > m( k ) f o r a l l k . I n p a r t i c u l a r , f o r each k E J we have
i i i (yk ) ( i = 1, ..., n ) , where y k = x m( k )
Consider now the sequences
n
n
n
f o r i = 1 , . . . , n - 1 and y k = x,,,(~) - x k . These n sequences have the property t h a t a l l a r e O-Cauchy , a n d a t l e a s t two a r e @-convergent t o 0 . By repeating t h e above argument , we can thus obtain n bounded sequences ( z i ) ( i = 1 , ...,n ) which a r e a l l 0- convergent t o 0 , such
n t h a t 1 1 A(ZL ,..., z k sumption t h a t A E L
2 OSC
Proof of Theorem 4 . 5 . 4 .
E/z"-'
('E;F)
.
However
, t h i s c o n t r a d i c t s our as-
, which completes the proof.#
a n d l e t A E LosC('E;F) Let p E PQSc ('E;F) be the associated symmetric m u l t i l i n e a r mapping. By t h e p o l a r i z a t i o n formula ( 0 . 3 . 1 ) , i t s u f f i c e s t o show t h a t i f (x: ) i s a bounded 41- Cauchy sequence in E ( i = 1, ... , n ) then
But
Weakly continuous f u n c t i o n s on Banach spaces
101
I n each o f t h e above terms, a t l e a s t one o f t h e sequences i s vergent t o
0
as
j,k
Q- con-
.
, and t h e o t h e r sequences a r e 0- Cauchy
+,,,
An a p p l i c a t i o n o f lemma 4.5.5
completes t h e p r o o f . #
We now t u r n o u r a t t e n t i o n t o t h e p r o o f t h a t a polynomial which is
, when
+continuous
E, i s i n f a c t
r e s t r i c t e d t o any b a l l i n
uni
f o r m l y 0-continuous on each b a l l . I n o r d e r t o prove t h e e q u a l i t y o f n n Pwb( E;F) and Pwbu( E;F) i n general , r e s t r i c t i o n t o separable Banach spaces w i l l f i r s t be convenient.
Lemma.
4.5.6
.
p E Pwb( E:F)
and F be Banach spaces, E being separabZe , and
E
Let n
Zet
Then there is a countabZe s e t Q c E ' such t h a t
P e PQb(nE;F)( x . ) be a dense sequence i n E , I/ x j / l 5 j J p a i r o f n a t u r a l numbers ( j , m ) , t h e r e i s a f i n i t e subset Proof.
Let
(4
f
m
, then
Qj )
, 11
y E E
that i f a point
11
p(y)
p E POb ("E;F)
show t h a t
-
yII < 2 j
.
To do t h i s
,
For each
~m c E ' , so J @ ( y- x . ) = 0 J we w i l l Letting Q = u j ,m x o f B l ( E ) and E > 0 let
i s such t h a t
< l/m
p(xj)lI
.
,
.
07
be a r b i t r a r y ( i t c l e a r l y s u f f i c e s t o r e s t r i c t o u r a t t e n t i o n t o t h e u n i t ball is
1)
B,(E)).
p
i s u n i f o r m l y continuous on bounded s e t s , t h e r e
,I/
0 < 6 < 1, such t h a t if x,y 6 B 2 ( E )
6,
P(x)
Since
-
p ( y ) \ ) < ~ / 3 . Choose x
such t h a t
j i s such t h a t
1)
xj
x-yll < 6
-
, then
x o \ J < 6, l e t
m >3/~, m 0 j 1.
z o B,(E) $ ( z - x o ) = 0 , ( @f - x o + x j , n o t i n g t h a t 1 1 w I I 2 11 z I I + 6 < 2 . Then @ ( w - xJ. ) = O rn for @ f Q j , so t h a t 11 p ( x j ) - p ( w ) I I < l / m c d 3 . A l s o ,(I w - z l l = = 1 1 x j - x o l l < 6 , so t h a t 11 p(w) - p ( z ) l j < ~ / 3 s i n c e b o t h z and w E B P ( E ) . Therefore, II P ( X O ) - p ( z ) l l 5 1 1 P ( X O ) - p ( x j ) l I + + 11 p ( x j ) - p ( w ) I / t 1 1 p(w) - p ( z ) I j < E , and an a p p l i c a t i o n o f proposiand suppose t h a t
Let
tion
w = z
4.5.3
completes t h e p r o o f . #
102
Chapter 4
{I$.}be any countable s e t i n E ' and l e t ( x j ) J Then ( x ) has a 0 -Cauchy subsequence. be any bounded sequence i n E 4.5.7
Lemma.
Proof.
Let
Let
0 =
N
No =
.
j > 1
and f o r each
n i t e s e t such t h a t t h e s m a l l e s t element and such t h a t 0- Cauc hy
($j(xk))kENj
converges.
,
let
N j c Nj-l
be an i n f i -
i s n o t i n Nj+l n. i n N J j Then t h e sequence ( x n .) i s J
'# be an a r b i t r a r y polynomial w i t h a s s o c i a t e d
p e P('E;F)
Now l e t
A
symmetric n - l i n e a r
.
L('E;F)
E
a s s o c i a t e d l i n e a r mapping
C : E
( n - 1 ) - l i n e a r mappings o f
Ex
To t h i s mapping A, t h e r e i s a uniquely n-1 Ls( E;F) , t h e space o f symmetric
-f
n -1 C ( X )(YI 4.5.8
9 .
- .¶yn-1)
=
Proposition.
A(x ~ Y 3I . .
... x
E
-
)
9Yn-l
into 9
F,
g i v e n by
(x,yi,...
yn-l
6
E i s a separable Banach space and
If
E). p
Pwb('EE;F),
F
then t h e associated mapping C i s a compact l i n e a r mapping. Proof. set
By lemma 4.5.6,
E'
@ c
p
E
n n POb( E;F) c Pas,( E;F)
A
so t h a t t h e n - l i n e a r mapping
.
f o r some c o u n t a b l e
i s an element o f
I f f a c t , we now show t h a t t h e a s s o c i a t e d l i n e a r mapping Losc(nE;F) C i s an element o f LOsc(E;L( n - 1 E ; F ) ) , w h i c h i s equal t o LQ,(E;L("-lE;F))
by theorem
4.5.4.
In fact
,
if
C
6
LoSc(E;L("'E;F))
,
then
f o r some bounded sequence E
> 0
C(xj)/l
y j e B1(E)
>
E.
such t h a t
( x . ) which i s @-convergent t o 0 and some J T h i s means t h a t f o r each j t h e r e i s a p o i n t
(1
C ( x j ( y j,...,yj)\(
>((n-l)! /(r1-1)~-')(~/2) =
t'.
By lemma 4.5.7, we can e x t r a c t a subsequence ( y . ) which i s 0 - Cauchy. Jk Therefore, f o r a l l k , l i A ( x j k 9 Y j k ,...,y. ) I 1 E ' , which c o n t r a d i c t s Jk lemma 4.5.5. Thus , C 8 LQSc (E;L("'E;F)). NOW t o show t h a t C i s
a compact mapping, l e t ( x . 1 E B,(E) be an a r b i t r a r y sequence. Using J (4.5.7) again, t h e r e i s a 0 - Cauchy subsequence ( x ) o f ( x . ) . F i n a l l y , n-1 jk J s i n c e c E LQc(E;L( E;F)) , ( C ( x j k ) ) i s Cauchy i n L("'E;F).# Finally
, we a r e ready t o prove t h a t a polynomial which i s weakly
continuous on b a l l s i s i n f a c t weakly u n i f o r m l y continuous on b a l l s . 4.5.9
Thporern.
For any Banach spacc,;. E and
associated linear mapping
c
: E
-f
L~("-'E;F)
F
, let
p
E
P( 'E;F)
and t h e
be given. Then p e pwb(nE;~)
Weakly continuous f u n c t i o n s on Banach spaces
c
if and o n l y if Proof.
p E Pwb(nE;F)
Let
i s n o t compact. (C(xj))
Consequentzy
pWb(nE;F) = P~~,,("E;F).
and suppose t h a t t h e a s s o c i a t e d mapping ( x . ) c B1(E)
Thus, t h e r e i s a sequence
has no convergent subsequence i n
0, I/C(xj ( j , k ) , where E >
Thus
is compact.
, if
-
xk)/l > j
G
Cxj : j E
NI,
iyjk : j,k 6
i s a non compact l i n e a r mapping.
CIG : G
then
, CIG
On t h e o t h e r hand
u n i f o r i i i l y continuous on
there i s a f i n i t e set 0
(@€
QE)
,
then
11
E
+
L("'G;F)
i s the linear
-
.. , v ) 11 + 11
A(w,v-w,v,.
, each A(v ,...,v-w,w,..
and we conclude t h a t
11
, and so f o r each
, such t h a t i f v,w
c E'
C(v)
B1(E)
C(w)l/ < E/n
.
. ., v ) / I
. ,w)
p ( v ) - p(w)II <
E
=
+
...
E B1(E)
Therefore
i s t h e n - l i n e a r mapping a s s o c i a t e d t o
A(v-w,v,.
generated by
S i n c e i t i s immediate t h a t
i s compact. By (4.1.3) ,C i s weakly C
By symmetry
N 1,
E
P I G B P('G;F). we have o b t a i n e d a c o n t r a d i c t i o n t o ( 4 . 5 . 8 ) . T h e r e f o r e , Conversely, l e t C : E + L s ( n - 1 E;F) be a compact o p e r a t o r .
P I G E Pwb(nG;F)
'E;F)
i s , f o r some
As a r e s u l t , f o r each p a i r E B1(E) such t h a t J ,k
t o be t h e c l o s e d subspace o f
mapping a s s o c i a t e d t o
= 0
such t h a t
. That
Ls(a-'E;F)
C
# k there i s a point y.
we d e f i n e
the vectors
j f k.
whenever
E
103
with
, if
p , we conclude t h a t
+
11 A(w,.. . ,w,v-w) 11
C(v-w)(v,..
,as r e q u i r e d .
.,v
,... ,w,..
.w)
,
#
One consequence o f t h e above r e s u l t i s t h a t f o r e v e r y polynomial n p 6 Pwb( E;F)
p E PcpSc(nE;F) . Indeed. by 4.5.9, there i s a f i n i t e set cpk c E ' and satisfy @ = U
I $ ( x - y ) I < 6k cp
thus of
p e Pcpsc(nE;F) E'
p E Pwbu Ak > 0
( @ E @k) then11 p ( x )
, such
that
, and so f o r each k E N such t h a t i f x,y E B1(E)
-
p(y)ll
< l/k.
Let
I t i s easy t o conclude t h a t i f
k k ' which i s +convergent plication
cp c E '
t h e r e i s a c o u n t a b l e subset
.
( x . ) i s a sequence i n B1(E) J t o a p o i n t x e B1(E) t h e n p ( x . ) -1. p ( x ) , and J I n f a c t , we have a l r e a d y proved t h e converse i m
, namely t h a t i f p
, then p
E Pwbu(nE;F)
E PQSc(nE;F)
.
To see t h i s
f o r some c o u n t a b l e subset
,
note t h a t i f
@
p E PoSc('E;F)
Chapter 4
104
t h e a s s o c i a t e d l i n e a r mapping
then by t h e p r o o f o f p r o p o s i t i o n 4.5.8
C :E
+
i s compact. Thus we have.
L,("'E;F)
. For a poZynomiaZ
CoroZZary
4.5.10
n p 8 P( E;F)
the foZZowing are
equivaZent : (a)
n P e Pwbu( E;F).
(b)
For some countabZe subset n pwSc( E;F) = Pwbu ("E;F)
I n particuZar
n and a7YZ Banach spaces
a22
4.5.11 'wbu
Proof.
S
P;(~E)
=
contains
E'
F
-t
.
whenever E ' is separable , f o r
n,
.
Pf(nE;F)
i s complete , Pwbu(nE;F)x Pf(nE)
and
, we
Since
. For
argue by i n d u c t i o n
n=1
5
F.
, if
i s weakly u n i f o r m l y continuous on bounded s e t s , t h e n i t i s
compact by lemma 4.1.1. A is a limit
of
El
sume t h e r e s u l t f o r C : E
Post( n E;F)
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
To show t h e r e v e r s e i n c l u s i o n
A : E
p 6
F.
F f o r aZZ
Assume f i r s t t h a t
Pwbu(nE;F)
,
El
I f E ' has the approximation property, then
CoroZZarg.
('E;F)
of
0
-t
L,("'E;F)
mapping
D : E
5
Hence, s i n c e F
has t h e approximation p r o p e r t y ,
El
)
elements ( 0 . 5 . 4
,
s = l,.,.,n-1
, so
that
A e E'
p e Pwbu(nE;F)
let
5
F
. As
and
.
t h e a s s o c i a t e d l i n e a r mapping By ( 4 . 5 . 9 ) , t h e n-1 E;F) g i v e n by D ( x ) ( y ) = C ( x ) ( y , ...,y ) , i s
Pwbu(
-+
compact and l i n e a r .
Using t h e a p p r o x i m a t i o n p r o p e r t y o f E ' , i t f o l l o w s
0, t h e r e i s a f i n i t e r a n k l i n e a r continuous mapping k 1 ($i e E l , pi E Pwbu(n- 1E;F)) such t h a t 1 1 D - 1 $i 5 pill < E . i=1 k I n p a r t i c u l a r , s i n c e p ( x ) = D ( x ) ( x ) , f o r 1 1 xi1 5 1 ,llp(x)- J $ i ( x ) ~ i ( x ) l l <
t h a t f o r any k @i 5 Pi i=l
E >
1=1
<
E
.
ci€ Pf(n-l
By i n d u c t i o n
Therefore
.---_ .-___
e Pf(nE)
t o each
E;F ) such t h a t f o r
( i = l,...,k). p
,
5
F.#
,)I
p
11 -
pi 6 Pwbujn-lE;F) x/I5 1
,I]
pi(x)
-
qi(x)]l <
kll
k $i 1=1
corresponds
5
qill < 2~
proving t h a t
@ill
Weakly continuous f u n c t i o n s on Banach spaces
We do n o t know i f t h e a s s e r t i o n n B
IN
(i.e.,
F = R) implies that
fixing
105
Pf( n E ) = Pwbu(nE) E'
for a l l
has t h e a p p r o x i m a t i o n
property. 4.6.
Composition o f weakly u n i f o r m l y c o n t i n u o u s f u n c t i o n s . I n t h i s s e c t i o n homomorphisms between F r 6 c h e t a l g e b r a s Cwbu( E)
are studied.
We prove t h a t these a l g e b r a s a r e f u n c t i o n a l l y continuous and
t h e homomorphisms between them a r e c h a r a c t e r i z e d . F i n a l l y composite subalgebras i n
Cwbu(E) a r e discussed. Necessary and s u f f i c i e n t c o n d i t i o n s a r e
g i v e n so t h a t these Let
E
E' ,
F ' , El' and F"
and
A c E.
tinuous,
subalgebras a r e c l o s e d .
and
F
for all to
,
X
f : A -* X
A function
a r e a l l o c a l l y convex H a u s d o r f f space
i s s a i d t o be weakly u n i f o r m l y conV
i f f o r e v e r y neighbourhood
Q~,..., @ k i n E ' E
be Bandch spaces w i t h normed d u a l s and b i d u a l s
respectively
i = l,...,k
and then
6 > 0
of
in
0
.
there
x, y B A
such t h a t i f
(f(x)-f(y)) 8 V
X,
are
, I ~ $ ~ ( x - y c) l 6
The space o f a l l
f
from
which a r e weakly u n i f o r m l y c o n t i n u o u s when r e s t r i c t e d t o bounded
X
s e t s w i l l be denoted Let
Eiw*
Cwbu(E:X). be t h e space
E"
endowed w i t h t h e
bw*-topology.
An a p p l i c a t i o n o f t h e Grothendieck completeness theorem (Schaefer [ 11,561 yields that
E ' . Also
(Eiw*)'
i t i s immediate t h a t
Eiw*
, from t h e d e f i n i t i o n o f t h e bw*-topology, , ESw,=lim B,",
i s the topological d i r e c t l i m i t
r-+ iu B " = B " ( E ) r b a l l i n E " w i t h t h e induced w*-topology. (See observar r t i o n s a f t e r d e f i n i t i o n 4.2.1). So f o r a g i v e n f u n c t i o n f : El' + X , f
i s continuous f o r t h e bw*-topology i f and o n l y i f f o r a l l bounded subsets B c E",
f l B :(B,w*)
-f
X
i s continuous.
We endow C(Eiw* ; R ) = C(Eiw*) of uniform convergence on bounded s e t s o f denote
T~
B
ELw*
.
T h i s topology, which we
, i s generated by a l l seminorms o f t h e form e C(EbJw*)
where
w i t h t h e l o c a l l y convex t o p o l o g y
-t
s u p { l f ( x ) / : x E B}
i s a l l o w e d t o range o v e r t h e bounded subsets o f
E".
Lemma. A functior. g : F -* E" belongs to t h e space if and on23 ij- @ g B Cwbu( F) for every @ B E' .
4.6.1.
Cwbu(F;Eiw*),
Chapter 4
106
Proof.
Let
B c F
then there e x i s t
bounded, @
,...,I$~ E F'
$1
' Cwbu(F)' o
g
g
6
Cwbu(F;ELw,),
E
> 0.
g
If
Cwbu(F;Eiw*)
6
i t f o l l o w s t h a t f o r every
i s bounded on t h e bounded s e t s .
Thus,in o r d e r t o
i t i s s u f f i c i e n t t o prove t h a t @
0
g
6
, , i.e,
x,y E B
such t h a t i f
\@(g(x)- g(y))\< 6
then
, from (4.1.1)
Conversely
41
and
and 6 > 0
i = 1, ...,k
\ ~ $ ~ ( x - y <) J 6 f o r a l l
@
El
6
Q
E',
6
see t h a t Cwbu(E), which
i s our h y p o t h e s i s . # I t i s known t h a t each
?
way t o a f u n c t i o n
Moreover : t
n E 111
( r e s p . €3;
€3,
.
E C(Eiw*:F)
, supll/ f ( t ) l l ) i s t h e n - b a l l on
l i n e a r and f o r each where
can be extended i n a unique
f E Cwbu(E;F)
E
,
t h e mapping
6
Bn
I=
(resp. i n
e(f) =
?
is
supIl1 ? ( t ) l I : t E B i l , El').
Thus, we have
t h e f o l 1owing : 4.6.2
The mapping e i s a topological isomorphism between
Proposition.
Cwbu(E;F)
and C(Egw*;F).
= R ,e , between
When F
i n She sense of Fre'chet aZgehras 4.6.3
Remark.
e
If
and
r
i s a topological isomorphism,
.
Cwbu(E) and C( EL),
a r e r e s p e c t i v e l y t h e e x t e n s i o n and r e s t r i c
t i o n maps, t h e space o f homomorphisms
A : Cwbu(E)
i f i e d w i t h t h e space o f homomorphisms
A
: C(Eiw,)
Cwbu(F)
-f
-f
can be i d e n t
C(FiWJ,) v i a t h e
f o l l o w i n g commutative diagram:
A Cwb,(E)A
Cwbu(F)
h
h
A = r o A o e 4.6.4
Proposition
.
A = e ~ A o r E
For every
Banach space,
Eiw* i s
a reaZcompact
topologica2 space. Proof. Lindelsf
Since c l o s e d b a l l s i n space and then i t i s
E"
a r e compact i n
realcompact.
#
ELw*
,
Eiw*
is
a
Weakly continuous f u n c t i o n s on Banach spaces
4.6.5.
CoroZZary.
0 : CwbU(E) + R
Let x
e x i s t s a unique point
6
E"
6
e(f)(x)
f o r aZZ
such homomorphism i s automaticalzy
h
L e t a : C(E;jw*) + R i t follows t h a t there e x i s t s Proof. f
be a homomorphism. Then there a(f) =
such t h a t
, every
In p a r t i c u l a r
f E Cwbu(E). continuous ,
107
.
C(Egw*)
4.6.6
Thus,
be defined by g ( f ) = a ( r ( f ) ) . From (4.6.4) x 6 E" such t h a t :(f) = f ( x ) f o r a l l
0(f) =
.
Proposition
e^(e(f)) =
e(f)(x)
f e Cwbu(E).#
for all
R
be a homomophism. Then Let : C(E;w*) + C(Fgw,) i s induced by a f u n c t i o n g E C(Fiw* ; E i w * ) i . e . , i ( f ) = f 0 9 for every f E C(E;jw*). ConuerseZy, each g 6 C(FLw*;E'Ibw*) defines a
homomorphism
^A
Proof.
(4.6.4)
From
i(f)(y) = f(x)
and C( FEW,)
C(),,E;
between
f o r every
f
for a l l
by composition.
y e F", t h e r e e x i s t s
6
C(E;jw*).
and
(4.6.2)
x
E"
6
The r e q u i r e d f u n c t i o n
such t h a t g
i s there
f o r e g i v e n by g ( y ) = x. From
,
B c F"
(4.1.1)
C$
and f o r every
E',
6
T h e r e f o r e , t o prove t h a t g every R > 0 , all E E' and
6> 0
C(FCW* ; E;jw* and a l l E > 0 6
A(0) =
F" g
$
A
(4.6.6)
A^
4.6.8
E
A
there e x i s t
R
,IIYll 5
. However t h i s . The
: Cwbu(E)
8
$1,
. . . , ok
E
IC$i(X-Y)l < 6 i s immediate
+
Cwbu(F)
i s continuous.
i s c o n t i n u o u s . (See g
s u p { l b ( x ) / l : x 6 B} <
6 m,
(4.6.3)).
C(FLw*;Egw,). i t follows
i s continuous.#
Proposition.
way t o a f u n c t i o n
-
Each
g
6
g 6 C(F;,,;E;~*).
CWbU(F;ELw*)
F'
converse i s o b v i o u s . #
i s induced by a f u n c t i o n
Since f o r a l l bounded s e t s B C F",
8
5
C(FGwx)
i s a continuous i f and o n l y i f
From p r o p o s i t i o n that
E
CoroZZary. Euerg homomorphism
Proof.
we need o n l y show t h a t f o r
)
, ,I[ x I I 2 R
1@(g(x) - g ( y ) ) I h
from t h e f a c t t h a t 4.6.7
6
such t h a t if x,y
( i = 1, ..., k ) , t h e n
i t f o l l o w s t h a t f o r a l l bounded sets
can be extended i n a unique
ConuerseZy,if
g
6
c(F;~*;E;~*)
Chapter 4
108
Proof.
Let
@ B El.
g : F"
E"
+
be t h e mapping g i v e n by < g ( y ) , $ .=
e($o g ) ( y ) ,
-
e C(Fiw,;Egw*)
Then
and g i s She u n i q u e e x t e n s i o n o f
g.
The converse i s o b v i o u s . #
CoroZZary. The space of homomorphisms can be i d e n t i f i e d w i t h t h e space Cwbu(F;EEw,) 4.6.9.
Proof. From
g e Cwbu(F;Eiw*)
Given (4.6.3)
f
from (4.6.3)
Cwbu(E) and
, (4.6.6) and (4.6.8)
8
0
0
e(f) =
,
that
there exists
r(e(f)
o
8C(Fiw*).
0;
Therefore, t h e formula Cwbu(F).
follows
it
homomorphism
A : Cwbu(E) -t Cwbu(F) such t h a t A ( f ) = r
Cwbu(F)
o i a t h e formula
g E Cwbu(F).
0
( * ) d e f i n e s a homomorphism between
-t
f e C(EiW*) , e ( f o g ) = f
and
it follows that
A :Cwbu(E)
9) = e ( f )
Conversely,
given
9
8 0
)
C(Fiw*;Eiw* where g,
9 = g l F 8 Cwbu(F;Egw* We a r e now g o i n g t o d i s c u s s 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
so t h a t t h e composite s u b a l g e b r a s
g : f E CwbU(E)), {e(f) Consequently one-to-one , o n t o , e t c
phism 4.6.10
A : Cwbu(E)
+
Cwbu(F)
if f o r each w*- compact s e t
4.6.11 g
o
g(K) = H
g : f
6
C(F:Eiw*),
n
is said t o be semiproper t h e r e e x i s t s a w*- compact s e t
g : FLw* H c E"
,
+
Eiw*
g(F"
Proposition.Let 9 : F i w *
subaZgebra { f
g
are characterized.
Definition. A function
K c F" such t h a t
0
... , homomor-
a r e c l o s e d i n Cwbu(F).
+
B
C(ELw*)I
,
f o r every
ELw* be a continuous map . Then, t h e is closed i n C(Fiw*) if and onZy i f
i s semiproper.
Proof.
Assume t h a t
t h e f i b e r s of
g
g, i.e.
i s s e m i p r o p e r and x E E"
h
8
C(Fgw*)
i s c o n s t a n t on
the r e s t r i c t i o n o f
h
to
{y E F" : g ( y ) = x 1 i s c o n s t a n t . We c o n s i d e r t h e f o l l o w i n g diagram:
Weakly continuous f u n c t i o n s on Banach spaces
I
P
where
p
9
and
with
g
and
Then
g
and
p = h.
o
We endow
+R E"
B;'
K1 c F",
and
f =
h
9-l
0
,
w*-compact
.
Let
B;'
be
such t h a t
K1=
it follows
p ( K 1 ) i s compact i n FGw*/g i s continuous. T h e r e f o r e , f = h
tl
-f
Let
w i t h the quotient topology.
be d e f i n e d by
n g ( F " ) . Since : B;' n g ( F " )
continuous on
FGw*/g
are continuous. f : g(F")
the closed u n i t b a l l i n that
f ?
I
a r e the canonical p r o j e c t i o n (resp. b i j e c t i o n ) associated
h
Let g(K1) = BY
109
--1 g
is
n g(F"). t h e continuous e x t e n s i o n o f f 1 (BY n g ( F " ) ) ,
f,: (B;;w*)+R-be
, t h e 2 b a l l i n E " , l e t K2 be a w*-compact s e t i n F " such t h a t g(K,) = g ( F " ) n B;' . S i m i l a r l y , i t f o l l o w s t h a t f i s c o n t i n u o u s on B;'n g ( F " ) . L e t Q~ : ( g ( F " ) n B Y ) U B;' + R be d e f i n e d by Given
B;1
~ ~ ( =x f () x )
if
x E g(F")
~ ~ ( =x f l)( x )
if
x
E
n B;
BY.
I t i s c l e a r t h a t Q l i s w e l l d e f i n e d and t h a t i t i s continuous. L e t
f 2 : (B;';w*) * R
be t h e continuous e x t e n s i o n o f
f o l l o w s t h a t a sequence e x i s t s
m 5 n
such t h a t f o r each of
.
f,
Therefore
i s well defined
,f
,
and
h = f
+R
: (B;,w*)
f
and
continuous
,
f n i s an e x t e n s i o n
f(x) = fn(x)
g . Then { f
if
x 6 B;
g : f 6 C(Eiw*)I
, assume t h a t g i s n o t a semiproper Then f o r some w*-compact s e t H c E " , g(B;) p H n g ( F " ) f o r a l l
map.
.
U
IN
t h a t (m,)
Conversely
(The o t h e r p o s s i b i l i t i e s a r e e a s i l y t r e a t e d ) . So, t h e r e e x i s t s
xn = g ( y n )
mn 6
, fn
, f n [(B; n g ( F " ) ) = E" + R d e f i n e d by
E C(Eiw*)
i s closed i n C(Fiw*).
n e
f :
Ifn)
Q l . By i n d u c t i o n , i t
such t h a t
such t h a t
mn
xn
6
,
H
ynlI
.
yn
6
F"
and y e t
f l : E;lw*
+
R
6 g(B;).
Let
W i t h o u t l o s s o f g e n e r a l i t y we can assume
i s a s t r i c t l y i n c r e a s i n g sequence. Let
xn
be d e f i n e d as:
Chapter 4
110
1
x = x1
if
fl(X) = continuous everywhere , f2: Eiw* * R
Let
be d e f i n e d as: if
x = x m1
g(B;l) continuous everywhere f,: ELw* + R
Let
I
f3(x) =
. . .etc.
Then
(fn
i s bounded
, we
However,
fnog -ft
an
f,
then
follows that
f2(x)
4.6.12
on
mml
g)
0
i s a Cauchy sequence i n
11 fn
fog g(y)
-
g
0
f,
f o r any
fn
0
f
i s n o t bounded on
+
f
-f
0
f
Let
Coro ZZary
g.
0
R(A)
i s a homomorphism, from
-
i s cZosed i n
Cwbu(F) <=
(2)
R(A)
i s dense i n
Cwbu(F) <=>
(3)
A i s onto <=
i s closed i n
> g
is one-to-one
i s semiproper. i s one-to-one.
and semiproper.
g ( F ) i s dense i n
Cwbu(F)
i s closed i n
i f and o n l y i f
C(FLW*)
g,
e Cwbu(E)l. Then,
R(A)
R(A)
v i a the f o r
be t h e e x t e n s i o n o f
.
I A ( f ) : f e C(Eiw,)>
(4.6.9) i t
CWbu(F;Eiw*)
6
(1)
Proof. (1)
-
g
t h e space I A ( f ) : f
> g
are b i g .
e C(E;jw*). For i f t h e r e e x i s t s such y e F". I n p a r t i c u l a r i t
C(Fiw*;ELw*)
6
, f o r i f B c F"
C(F),;
p r o v i d e d m,n
H.#
Cwbu(F)
Let
gll = 0
for all
g(y)
(4) A i s one-to-one <=>
=
o
i s induced by a f u n c t i o n
A(f) = e ( f )
(see 4 . 6 . 8 ) ) .
x = x mml g(B" )
continuous everywhere
see t h a t
A
follows t h a t
be d e f i n e d as:
if
If A : Cwbu(E) mula
.
. Therefore
ELw*
.
R(i)
=
(1) f o l l o w s from
Weakly continuous f u n c t i o n s on Banach spaces
111
(4.6.11).
( 2 ) R(A) i s dense i n
i f and o n l y i f R(^A) i s dense i n Cwbu(F) Thus, ( 2 ) f o l l o w s from t h e Weierstrass-Stone theorem.
C(F/lw*).
( 3 ) f o l l o w s from (1) and ( 2 ) .
A
(4)
ESSbw* <=> 4.6.13
g(F)
8
<=>
i s one-to-one
i s one-to-one
i=> G(F")
i s dense i n
Egw* .#
i s dense i n
Examples, We g i v e two examples which i l l u s t r a t e t h e c o n c l u s i o n o f t h i s Example (1) g i v e s a s i t u a t i o n i n which t h e homomorphism
section.
A : Cwbu(E)
i s continuous, a l t h o u g h t h e induced mapping
Cwbu(F)
-f
* El' f a i l s t o be continuous ( c o n s i d e r i n g b o t h F and E"
g : F
+
Cwbu(F)
dense i n
such t h a t
A
+
Cwbu(F).
Example 1. For each
n
6
,
W
where
1 1 1 t = - [ - t - I. n 2 rt n + l
Since
g ( t n ) = en g
wbu ( E ) i s n o t c l o s e d and n o t
i s one-to-one, R(A)
an
let
which has s u p p o r t c o n t a i n e d i n
that
with
A :C
t h e i r norm t o p o l o g i e s ) . T h e n e x t example shows a homomorphism
:
+
,
g :R
Let
, t h e usual nth
+
Cm- f u n c t i o n
be a
[0,1]
[l/(n+l) ,l/n]
and such t h a t
c o be d e f i n e d as
t . * t o i n R. J
Then, i f
(tj)
to# 0
$
g
o
C(R)
6
CO, it follows
f o r each
, (4
$ = ( + n ) B 1'
an(tn) = 1
g ( t ) = (@,,(t)).
u n i t basis vector o f
i s n o t continuous. Note t h a t
Indeed, l e t
R
0
$
1 ' = c;.
6
g)(tj) =
m
1
=
an
$n
and so i f
,
is
it
clear
that
n=l m
(4 E
0
g)(tj)
1
-f
(4
$n +n ( t o ) =
n=l
,
> 0
choose
0
g)(to).
If t o = 0
, then
given
m
no
1
such t h a t
c
E.
Therefore
,
n=no m
I 1
,$,
n= 1
no-1
an
5
(tj)l
11
$n ( a n ( t j ) ) l
t E = E
if
j
i s sufficiently
n=l
1arge. Also, n o t e t h a t let
tj
+
$l,..., $k all
to i n R 6
1'
i = l,...,k
and
and
f
o
g
6
C(R)
f o r each
B1 the u n i t b a l l i n
6 > 0
then I f ( x )
such t h a t i f
-
f(y)I <
E.
co.
f
6
CwbU(co).
Indeed,
Then, t h e r e a r e
x,y e B 1 , 1 $ i ( x - y ) I < 6 Since
g ( t ) e B1
f o r each
for
Chapter 4
112
,
t 6R
loi
such t h a t j
o
0
g ( t ) l < 6 when j
T h e r e f o r e , if
ni.
,...,
n o = max(n1 nk) i t follows t h a t I f 0 g ( t j ) - f 0 g ( t o ) ( < E. T h e r e f o r e A ( f ) = f 0 g i s an example o f a homomorphism
2
Example 2. g E 1'
$
Let c
let
B
1" be bounded
5
=
(En)
1$2(X-Y)l <
9E
0
E
+
,
1
-
,
> 0
E
=
Y
m
c(lbw*)
9e
and hence
,g
,
i s one-to-one
i t follows that
R(A)
n E BI
xt
Therefore, i f
G(K) =
such t h a t
.O,.)In p a r t i c u l a r
(4.6.12)
{e(f)
0
. Then
2~
.
Therefore,
, from
4.6.12
there e x i s t s a
.
9
xn = ( a ( n ) , a ( n ) - l , a ( n ) - 2 ,
> n/2. Then
(xn)
composite
R(A)
i s not closed i n
subalgebra CwbU(co).
g ( c o ) i s dense i n
i s one-to-one.
i s not
i s n o t a semiproper mapping.
g : f 6 C w b u ( c ~ ) 1=
A
,
Cwbu(cO) induced by g.
the
F i n a l l y i t i s easy t o see t h a t
4.7.
+
that
Therefore
5
and I h ( x - y ) I < E
5
. Therefore
follows
it
=
By rl g ( l m ) So, f o r each -g ( xn ) = (l,l,l,.~.,l,O,O,O,...).
11 x n l /
T h i s c o n t r a d i c t i o n proves t h a t
From
$2
Cwbu(c0).
= a(n) E R , i t f o l l o w s t h a t
.. .,a(n)-n,O,O,O,.. bounded.
A : CWbu ( c a )
i s semiproper mapping xn E K
g(y)I
0
and
= $
x,y 6 B
$
i s n o t one-to-one
K c 1" such t h a t
there exists
$1
Indeed,
m
i s n o t dense i n
Assume t h a t w*-compact s e t
-
i(x)
G : l m * Im
c(lbw*;lbw*).
D e f i n e t h e homomorphism Then
If
I$ m
x ~ + ~ Since ) .
C(lzw*;l;w*).
6
, E o = 0. 0
-
g 6 C w b u ( c ~ , l ~ w * ) L. e t
$ E 1 ' . Choose
and
it follows that
c o which i s n o t
-t
g(x) = (xn
x ~ + ~ ) Note . that
5,
with
be d e f i n e d as
i t follows that
g(x) = (xn
c
where
53
g : co
Cwbu(co)
be d e f i n e d as
g :R
induced by a f u n c t i o n
A : Cwbu(cO) -t C(R) continuous.
$
-
g(tj)
ni 6 U
there exists
i = l,,..,k,
i t f o l l o w s t h a t f o r each
liw* .
Notes and r e f e r e n c e s . I n t h i s c h a p t e r we have seen t h a t t h e s t u d y o f weakly continuous
f u n c t i o n s on Banach spaces i s s i g n i f i c a n t i n i t s e l f and has i m p o r t a n t c o n sequences i n t h e isomorphic t h e o r y o f Banach spaces. On t h e o t h e r hand, t h i s study i s r e l a t e d t o
differentiable
approximation theory.
To t h i s e f f e c t , i n 1969 Restrepo [l], found an i n f i n i t e dimensional vers i o n o f a B e r n s t e i n ' s theorem f o r a c l a s s o f C i f u n c t i o n s , d e f i n e d on an
E
r e f l e x i v e Banach space, such t h a t t h e y and t h e i r d e r i v a t i v e s a r e weakly
,
Weakly continuous f u n c t i o n s on Banach spaces
continuous on t h e bounded subsets o f Banach spaces and f o r
Cm-functions
E.
113
T h i s r e s u l t was g e n e r a l i z e d t o
, by A r o n - P r o l l a [ l l . They used those
f u n c t i o n s h a v i n g t h e p r o p e r t y o f b e i n g w i t h t h e i r d e r i v a t i v e s weakly u n l f o r m l y continuous on bounded s e t s .
(See c h a p t e r 5 ) .
S i m i l a r types o f functions are appropriate f o r obtaining i n f i n i t e
9
dimensional v e r s i o n s o f N a c h b i n ' s theorem, as can be seen i n c h a p t e r
.
On t h e o t h e r hand, V a l d i v i a [llproves t h a t t h e space o f a l l r e a l weakly c o n t i n u o u s f u n c t i o n s on a Banach space
, w i t h t h e compact-
open t o p o l o g y , i s always b a r r e l l ! a d . I n o t h e r words, i f a Banach space i s n o t r e f l e x i v e , t h e n a weakly continuous f u n c t i o n on which i s n o t bounded on t h e u n i t b a l l o f 4.2,
and theorem 4 . 3 . 5 ) .
E.
E
E always e x i s t s
(See comments b e f o r e s e c t i o n
T h i s i s an i m p o r t a n t c o r o l l a r y o f a s e r i e s o f
r e s u l t s about weakly compactness i n quasi-complete spaces. T h e r e f o r e , t h e space of weakly continuous f u n c t i o n s on
E
i s never complete, when
E
i s i n f i n i t e dimensional , ( C o r o l l a r y 4 . 4 . 2 ) . When s t u d y i n g t h i s completion, new spaces o f weakly c o n t i n u o u s f u n c t i o n s appear as seen i n S e c t i o n 4.4. The r e s u l t s which appear i n t h i s c h a p t e r a r e taken from: F e r r e r a [1I
.
Section 4.1,
Aron-Pro1 l a 111 and
S e c t i o n 4.2,
Gomez 111.
S e c t i o n 4.3,
F e r r e r a [21 and Aron-LLavona [ll.
S e c t i o n 4.4,
Ferrera-Gomez-Llavona
[l I and
A r o n - D i e s t e l -Rajappa
[ll. S e c t i o n 4.5,
Aron-Herves-Valdivia
S e c t i o n 4.6,
Aron-Llavona
[1 I.
11 I.
One f i n a l remark, t h e bw*-topology i s a s t r i c t t o p o l o g y a c c o r d i n g t o C o l l i n s [21
.
This Page Intentionally Left Blank
115
Chapter 5
APPROXI MAT ION OF WEAKLY UNIFORMLY DIFFERENTIABLE FUNCTIONS
5.1
Introduction.
Let P(Rn) be t h e algebra of r e a l polynomials in n v a r i a b l e s w i t h t h e topology of uniform convergence on bounded s e t s of a function and i t s d e r i v a t i v e f ' . A c l a s s i c a l Bernstein theorem says t h a t t h e c l o s u r e of P(Rn) i s t h e algebra C1(Rn) of a l l real-valued functions of c l a s s C'. I n other words, f o r every f B C1(Rn) t h e r e i s a sequence (p,) of polynomials such t h a t pn -+ f uniformly on bounded s e t s and p,! i f 'u n x f o m l y on bounded s e t s . Restrepo [ l l determines t h e P f ( E ) c l o s u r e f o r a r e s t r i c t e d c l a s s of r e f l e x i v e Banach spaces E . 5.1.1 Definition. We say t h a t a Banach space E has property ( B ) if of bounded Zinear operators such t h a t : there i s a sequence IT,, : E -+ E
where
T:I
(i)
n,,(E)
(ii)
IT,?, =
i s f i n i t e dimensionaz for each
n.
ITn *
( i i i ) For every
x
8
E
( i v ) For every
u
8
E ' , l l ili(u)
is the adjoint of
n.
iln(x)
- XI] - 1. 1
* 0
(n
* 0
(n *
+a)
.
m),
Every Banach space with a biorthogonal b a s i s has property ( B ) . In p a r t i c u l a r every H i l b e r t space has property ( 8 ) . Definition. Let E and F be Banach spaces and Let f : E + F be of c l a s s C'. we say t h a t f i s unifomZy d i f f e r e n t i a b z e i n a subset A c Ey i f for every E > 0 there i s some 6 > 0 such t h a t 5.1.2
Chapter 5
116
I/ hi/ 5
whenever
6
f o r a l l a E A.
I n Restrepo [ l ] t h e f o l l o w i n g theorem i s proven.
5.1.3
Theorem.
. Let
e r t y (B)
Let E be a separable Pf( E)
, reflexive
Banach space with prop
E
be the algebra of f i n i t e type polynomials i n
with the topology of uniform convergence on bounded s e t s of a function and i t s d e r i v a t i v e .
Pf( E)
Then the closure of
i s the algebra o f weakly
continuous functions on bounded s e t s which are uniformly d i f f e r e n t i a b l e on bounded s e t s . I n t h i s c h a p t e r we extend t h e work o f Restrepo. We d i s c u s s approximation o f E'
Cm-functions between Banach spaces
E
and
F, where
s a t i s f i e s t h e bounded a p p r o x i m a t i o n p r o p e r t y .
5.2.
U n i f o r m 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 bounded s e t s . Let
E
and
F
be r e a l Banach spaces. I n t h i s s e c t i o n we w i l l
be u s i n g t h e u n i f o r m l y d i f f e r e n t i a b l e f u n c t i o n concept t o d e f i n e a new class o f
C"-functions
from
E
T h i s w i l l be endowed w i t h t h e
to m
F,
which we denote by
topology, i . e . ,
T~
Cmwbu
(E;F).
the topology o f u n i
form convergence o f t h e f u n c t i o n s and t h e i r d e r i v a t i v e s o f o r d e r on t h e bounded subsets o f
5.2.1
Definition. E
> 0
, there is
m
,i
f i s said t o be
f f o r each bounded s e t
0 such t h a t i f
6
. Then
N
m E
f 6 Cm(E;F),
Let
uniformly d i f f e r e n t i a b l e of order each
5 m
E, t h i s being a F r i c h e t algebra.
x E B, y E E with
B c E and
I / yII 5
then
If
we g e t t h e d e f i n i t i o n 5.1.2
m = l
is t h e s e t s a t i s f y i n g the following conditions.
5.2.2.
sets
Definition.
.
(a)
CmwbU (E;F)
f : E
In other words (b)
+
of functions
f E Cm(E;F)
F i s weakly uniformZy continuous on bounded
,f
E Cwbu( E;F).
d J f ( x ) E Pwbu(JE;F)
Y
(J
5
m,
X E
El.
6 ,
117
Approximation of weakly u n i f o r m 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
(Observe that Pwbu(JE;F) = Pwb(JE;F)
by theorem 4.5.9).
is uniformly differentiable of order j , ( j 5 m ) .
(C) f
m
For
m=
m
,
Cibu
(E;F)
Cbu:
fl
=
(E;F).
m=O Remark.
If
E
, r e f l e x i v e Banach space w i t h p r o p e r t y ( B ) ,
i s a separable
t h e n theorem 5.1.3
says t h a t t h e c l o s u r e o f
Pf(E), w i t h the topology o f
u n i f o r m convergence on bounded s e t s o f a f u n c t i o n and i t s d e r i v a t i v e , i s cibu (E). The n e x t p r o p o s i t i o n a l l o w s us t o o b t a i n an easy c h a r a c t e r i z a t i o n o f t h e space
Czbu
i n terms o f p r o p e r t i e s r e l a t e d t o t h e
(E;F)
spaces o f weakly continuous f u n c t i o n s which were s t u d i e d i n c h a p t e r 4.
A l l polynomial spaces b e i n g c o n s i d e r e d i n t h i s c h a p t e r a r e e n dowed w i t h t h e norm topology. 5.2.3
Proposition. 1.- Let
f
Cm(E;F) satisfy the foZZowing condition:
6
9
( j
Then f is uniformly differentiable of order
j
( a ) d j f ECWbU(E;PWbU(JE;F))
2.- Let
5 m)
.
, f o r all
5
j
m.
f B c ~ ( E ; F ) satisfy the following conditions:
( j
( a ) dJf ECWbU(E;Pwbu(jE;F)
2
m - 1).
m.
( b ) f is uniformly differentiable of order
Then dmf B Cwbu (E;P(mE;F)). j <- m, Bn
P r o o f . 1.- L e t
$k} c E l
the
n
6' > 0
and
j
with
2 m, B c E
whenever
11 x - y I I
< 6
x a B
,y
B
E
with
and
then11 d J f ( x )
...,k l ) , n )
I/ djf(x) -
then
be bounded and
E
such t h a t if x,y
( i = 1, ... ,k) and l $ ~ ~ ( x - y<6' )I 6 = m i n ( 6 ' / ( m a x 11 : i = 1, y f: E
ball i n
,
-
E 8
E
. $ 1 1 XI/ 5
> 0
djf(y)l[ <
E
Let 2n
,I1
.
If
yll
5 2n
x B Bn Therefore, i f
i t follows that i f
djf(y)ll < E
.
, i t f o l l o w s t h a t f o r some 6
E
> 0
11
x - y l l < 6, we have t h a t
Thus, by T a y l o r ' s theorem w i t h Lagrange remainder,
11
> 0
,
,
djf(x)-djf(y)[l<E.
Chapter 5
118
provided
11 hi1 5
2.-
B
Let
=E
.
6
be bounded and
contains the closed u n i t b a l l o f and
h
6
E
11 h l l 5
with
11 f ( x + h ) -
f(x)
-
5 3~
.
Thus
11
,
dmf(x)
rn!
Choose
df(x)(h)
-
... -
$k
6
> 0
such t h a t i f
E'
x, y
6
-+I1d m f (
)
drnf(xL m! (h)ll
and
5
€11
B
if
< €
I@i(x - y ) l < 6
dJf(x)
6
hll
B,
rn.
( i = l,...,k)
Pwbu(jE;F)
then
.#
m
(a)
6
6 > 0 such t h a t i f x y y s ( 1 + 6 1 ) B ,
CoroZZary. Cwbu (E;F) is the s e t of functions f satisfying the foZZowing conditions: 5.2.4
x
, then
( i = l,...,k)
for
E.
, then
Now, t h e r e e x i s t s I ~ $ ~ ( x - y <) j 6
,B
> 0 ; without loss o f generality
E
(x
6
E ; j 5 m).
8
Cm (E;F)
Approximation o f weakly u n i f o r m 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
We w i l l now show some b a s i c p r o p e r t i e s o f 5.2.5
Proof.
m
c Cwbu
U,
and
z
8
F
.
T
0
, where
T
m
, f o r a22
(E;F)
I t s u f f i c e s t o show t h a t
$
k
B
z
6
Cbu:
6
w. (E;F)
f o r a given
E',
$ e
B u t t h i s f o l l o w s from t h e f o r m u l a
O f course, g
(E;F).
Cb!u
Proposition. Pf(E;F)
k E
119
Cbu:
(E;F)
c o n t a i n s a l l f u n c t i o n s o f t h e form
i s a f i n i t e rank c o n t i n u o u s l i n e a r o p e r a t o r and
c o n t a i n s no non-zero f u n g Cb!u (E;F) g E Cm(T(E);F) . t i o n s w i t h bounded support, except i n t h e t r i v i a l cases when F = O o r Note a l s o t h a t
dim(E) < and to
m.
Indeed, suppose
I f ( 0 ) l = 2c > 0. E
and
B = {x
:I1
f E Cbu:
Choosing xi1
5 2
}
(E;F)
6> O
and
with
s u p p ( f ) c Cx:ll xi1
a1,. ..,I$
~ ~ ( =x 0) ( i = 1,
...,k ) .
t h e assumption about t h e s u p p o r t o f 5.2.6
Definition.
We endow
Cb:u
Therefore I f ( x ) l >
x, E
1 < 1 1 xII 5 2, contradicting
f.
(E;F)
w i t h the ZocaZly convex topoZogy
m on bounded subsets o f E . This t o p m , i s generated by a21 seminorms of t h e form
of uniform convergence of order
oZogy, which we denote
11
i n t h e d e f i n i t i o n o f a weakly u n i f o r m l y
continuous f u n c t i o n on bounded s e t s , we can e a s i l y f i n d such t h a t
5
corresponding
T,,
where B i s aZZowed t o range over the bounded subsets of
E
.
(Note t h a t
Chapter 5
120
each of these seminorms <s well defined, by Zema m
T
CibU(E;F)
on
If
4.4.1).
The topoZogy
i s defined i n the obvious way.
dim(E) <
, then
m
-rbm
agrees w i t h
m E
W , only
T~
, t h e compact-
open t o p o l o g y o f o r d e r m.
5.2.7
Proposition.
Proof.
We prove t h e r e s u l t f o r
necessary f o r t h e case
m=
.
m
t r i v i a l m o d i f i c a t i o n s being
m Let { f n l c Cwbu (E;F) be a Cauchy sequence. I t i s easy t o see t h a t t h e r e i s a f u n c t i o n f E Cm(E;F) such t h a t d J f n + d J f u n i f o r m l y on bounded subsets o f
E
5
(j
m)
, so t h a t d j f ( x )
E Pwbu(JE;F)
It remains t o be shown t h a t dJf i s weakly u n i f o r m l y ( x 6 E , j 2 m). continuous on bounded s e t s . L e t B c E be bounded and E > 0. For
some
n,
x, y B B (j
5 m).
/I dJf(x) -
we have
m fn B Cwbu (E;F),
,
there i s
6 > 0
and
(i
1,
I$i(x - y ) l < , 6
Therefore
,
11
< 4 3
dJfn(x)II
-
dJf(x)
,...,
$l
...,k )
dJf(y)II <
, j 2 m).
(x B B
$k B E '
Since
such t h a t i f
, then11 d J f n ( x ) - d j f n ( y ) l l < ~ ! 3 E
, which
completes t h e
proof.
5.3
Extension o f B e r n s t e i n ' s theorem t o i n f i n i t e dimensional Banach spaces. Our main r e s u l t i n t h i s s e c t i o n y i e l d s t h a t i f Cb!u
(E;F).
However, f o r
m=O
TI
- c o m p l e t i o n o f Pf(E;F) , t h i s r e s u l t holds f o r a r b i t r a r y E
bounded approximation p r o p e r t y , then t h e is
has t h e
E'
(See theorem 4.3.7). We r e q u i r e two lemmas t o g e t t h e main r e s u l t f o r
5.3.1
Lemma
.
Let E be a real Banach space. Let
a precompact subset, Zet
Bc E
be bounded
Then the s e t L . = tdJf(x) : f J
6
T
,x
B BI
, and
Zet
m > 0.
m T c Cwbu j B
N ,
(E;F) j
2 m.
be
Approximation of weakly u n i f o r m 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 Pwbu(j E;F).
i s a precompact subset o f
Proof.
L e t E. > 0 . S i n c e
f E
such t h a t f o r any By lemma 4.1.1,
Bic
z
6
B Bi
T
f o r each
.
f l ,..., f s
i s precompact, t h e r e e x i s t
there i s
such t h a t f o r any
fi
T
6
w i t h supC/l d J f ( x ) - d J f i ( x ) / I : x e B 1 < c / 2 .
i = l , . . . , ~ t,h e r e i s a f i n i t e s e t o f p o i n t s x E B
,/I
dJfi(x)
-
< ~ / 2, f o r some
djfi(z)jj
Therefore
i s a f i n i t e union o f 5.3.2
T
121
E
balls containing
Lj
, which completes t h e p r o o f . #
Let E be a real Banach space such t h a t E l has the bounded m approximation p r o p e r t y w i t h constant C . Let T c Cwbu (E;F) be a prg compact subset. Given j E U , j < m, E > 0 and B c E bounded, there Lemma.
exists a f i n i t e rank continuous l i n e a r mapping n : E such t h a t
Proof.
Without loss o f g e n e r a l i t y
f o r some
y < rl
r
n/6CJ
2 1. , where
Let
n
> 0
, /I 7 ~ 1 1 5
C
be chosen so small t h a t
chosen i s so small
that
< 3E / ( 3 + ZCJ).
We c l a i m t h a t t h e r e i s 6 > 0 that i f
I/ XI/ 5
M
,I\
yll
Indeed, s i n c e such t h a t
set
Okc 6
5 M , and
T
T c W(f1; y / 3 )
and a f i n i t e subset
I $ ( x - y ) l < 6 (I$6 00) t h e n
i s precompact U ...U
Q O c E ' such
, t h e r e e x i s t fl,...,fs
W(fs;y/3)
E T
, where
5 k 5 s ) , t h e r e e x i s t s a 6k > 0 and a f i n i t e E ' , such t h a t i f 11 X I / 5 M,lly(l 5 M , and I @ ( x - y ) I < 'k
Now f o r each such
(4
E
, we may assume t h a t B = { t s E : Iltll 2 r )
M=rC. Let y > 0
i n turn the
+
@k) then
f k (1
122
Chapter 5
6 = min(bi,...,6s)
Let
1
we choose
11 xi1 2
5
5
M ,IIyII
I @ ( x - y ) I < bk
2
k
, and
M
f e W(fk;y/3).
, and
Ok
/I dif(x) -
i
f e T,
Then, i f
Ok.
We o b s e r v e t h a t if
I $ ( x - y ) I < is ( $ e OO), t h e n i n p a r t i c u l a r
e
$
U
k= 1
such t h a t
s
for all
11 d i f ( x ) - d i f ( y ) I /
and l e t 0 0 =
t h e r e f o r e ( a ) i s t r u e . Hence
difk(x)ll
+
I ( difk(x) -
difk(y)I/
T h i s proves o u r c l a i m (1). By lemma : f e
Li = { d i f ( x )
5.3.1,
each s e t
T ,I1 X I / 2 MI
i Pi c Pwbu( E;F)
a finite set
p e Li
that given
O(pi)c
where
1
. T h e r e f o r e we
( i n f a c t we may assume pi
6
d e f i n e d by can f i n d
Pi c Li)
so
P i such t h a t
, b y C o r o l l a r y 4.5.11, f o r each pi
In addition
pi =
i s p r e c o m p a ct
there exists
a finite set
Li c P wbu (iE;F)
E'
$
i
B
there i s 6 Pi such t h a t f o r a p p r o p r i a t e v e c t o r s b$ e F,
b$
.
W(Pi) Next, s i n c e
E'
has t h e bounded a p p r o x i m a t i o n p r o p e r t y w i t h
5 > 0 t h e r e e x i s t s 7r e E ' B E , 1 1 7111 :C , such 5 f o r a l l $ e O , where 0 = 0 0 U 0; and a; = u { @ ( p i ) : pi e pi , i j I . S i n c e II ~ ~ -0 7 < 5 A for all $ 6 0 and a l l i 2 j , w h e r e A i s a c o n s t a n t , i t f o l l o w s t h a t we may choose 5 > 0 so s m a l l t h a t rg < S and C, f o r any
constant that
1)
@
0
TI
- @ 11
<
Approximation o f weakly u n i f o r m 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 .
Indeed, l e t that for a l l i= O
with
@
x
e a0
we have
e B . Then
we have
1)
f(x)
-
I/ n(x)ll
5
5 gr
< 6
17/6 < E
.
I@(x- n(x))l
f(.rr(x)))) < y <
r C = M. Now r c < 6
.
123
implies
Hence, by (1)
Claim ( b ) .
11
Claim ( a )
and ( b ) t o g e t h e r i m p l y t h e a s s e r t i o n i n t h e lemma.
dif(x)
-
d’(f
0
~ ) ( x ) l
E
,
(x
8
B, f e T, 1 2 i 5j).
To prove c l a i m ( b ) , we f i r s t remark t h a t by t h e c h a i n r u l e and t h e l i n e a r ity of
x,y
e E.
IT
, di(f
o
n)(x)(y) = dif(n(x))(n(y)),
A p p l y i n g t h i s when
f e T, x
6
B
f o r any
, y e E,I/
yII
f e Ci(E;F)
and
2 1, we have
The second term on t h e r i g h t - h a n d s i d e o f ( 5 ) i s e a s i l y estimated. From (1) we have
To e s t i m a t e t h e f i r s t term on t h e r i g h t - h a n d s i d e o f ( 5 ) us w r i t e
The f o l l o w i n g e s t i m a t e s a r e t r u e
,
let
Chapter 5
124
, we g e t from ( 7 ) :
Adding
11
(8)
-
dif(x)(y)
dif(x)(ii(y))ll
< (q/6)(3+2Ci)
F i n a l l y , ( 6 ) and ( 8 ) i m p l y t h a t , f o r a l l
5.3.3
11
we have
1 5 i5 j
-
dif(x)
di(f
ii)(x)ll <
f B T
, x e B and
as d e s i r e d . #
F;
Let E and F be two r e a l Banach
Theorem.
< ~ / 2 .
t h e bounded approximation property w i t h constant m C:bU(E;F). Pf(E;F) i s T~ - dense in
spaces, w i t h E ' having C , and l e t m > 0 .
Then
By W e i e r s t r a s s ' theorem 1.1.2,
Proof.
Ea o f (1)
E
,
i t follows that
P(Eo ;F)
is
NOW, l e t bounded s e t TT
E
f
T
m
-
dense i n
C:bu(E;F)
E
j
B c E, E > 0 and
11 TI/ 5
E' s E with
Let
EO
6 > 0
C
= u(E).
clude t h a t there i s
where
f o r any f i n i t e dimensional subspace
p
E
8
Cm(Eo;F)
be g i v e n
.
By lemma 5.3.2,
,j 5
m
, t h e r e i s a mapping
ll
~ r ( B ) cE O i s bounded, by ( 1 ) we
Since P(Ea;F)
i s chosen so t h a t
( 2 ) and ( 4 )
f o r each
such t h a t
such t h a t
6 <
E/(
2CJ).
By t h e c h a i n r u l e , i t f o l l o w s t h a t
From
.
we g e t
con
Approximation o f weakly u n i f o r m 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 Pf(E;F)
Since
p
5.3.4.
Corollary.
o TI 6
t h e r e s u l t s follows.ii
Let E and F be two r e a l Banach spaces, w i t h E ' having
t h e bounded approximation property w i t h constant i s C:bU(E;F), t h e T mb- completion of Pf(E;F)
Proof.
The a s s u m p t i o n t h a t i s s t r i c t l y weaker t h a n
E
, and l e t
C
m > 0. Then,
and ( 5 . 2 . 7 ) . #
It f o l l o w s from (5.3.3)
fact, property (6)
125
E'
has t h e bounded a p p r o x i m a t i o n p r o p e r t y
has p r o p e r t y ( B ) , see d e f i n i t i o n 5. 1. 1.
implies that
E'
i s separable
In
, w h i l e ( 1 ' ) ' has t h e
bounded a p p r o x i m a t i o n p r o p e r t y .
5.4. N ot e s and r e f e r e n c e s . T h i s c h a p t e r i s based f u n d a m e n t a l l y on A r o n - P r o l l a [ l l and A r o n [1 1
. I t c a n be e a s i l y o b s e r v e d t h a t t h e p r o b l e m o f a p p r o x i m a t i n g
" c o m p l i c a t e d " f u n c t i o n s d e f i n e d on a Banach space
E
by " s i m p l e r " o r
" n i c e r " f u n c t i o n s i s f o u n d s u c c e s s i v e l y t h r o u g h o u t t h i s book.
More
s p e c i f i c a l l y i n c h a p t e r 3 we d e a l t w i t h t h e p r o b l e m o f u n i f o r m a p p r o x i m a t i o n o f u n i f o r m l y c o n t i n u o u s bounded f u n c t i o n s on
E
b y bounded q u a s i -
d i f f e r e n t i a b e f u n c t i o n s , a l o n g w i t h t h e problem o f a p p r o x i m a t i o n o f Cm-functions on
E
by polynomials i n the
-
T:
topology
. Beforehand
in
c h a p t e r 2 t h s p r o b l e m was d e a l t w i t h f o r t h e f i n e t o p o l o g y o f o r d e r m. Th s c h a p t e r has been c e n t e r e d a r o u n d u n i f o r m a p p r o x i m a t i o n o f C m - f u n c t i o n s on bounded s e t s u p t o o r d e r
m w h i l e t h e n e x t c h a p t e r focuses
on u n i f o r m a p p r o x i m a t i o n o n compact s u b s e t s o f N e m i r o v s k i j and Semenov [ 1 1 h a v e mation o f "regular" functions by polynomials.
E
up t o o r d e r
discussed uniform
m. approxi-
This Page Intentionally Left Blank
127
Chapter 6
APPROXIMATION FOR THE COMPACT-OPEN TOPOLOGY
I n t h i s c h a p t e r we w i l l see t h a t t h e space
,
complete
E
and
F
being
-
c o i n c i d e w i t h t h e :T i n chapter 3
, see
in > 2, t h e n
and
completion o f
Pf(E:F).
counterexample 3.0.1, Pf(H)
Cm(E:F)
i s -:T
r e a l Banach spaces. However, i t does n o t
i s n o t :T
-
I n f a c t we have proven
that if
dense i n
H
i s a H i l b e r t space
Cm(H).
Some o f t h e r e s u l t s on weakly c o n t i n u o u s f u n c t i o n s found
in
c h a p t e r 4 w i l l be used here, as t h e y were i n c h a p t e r 5 , t o c h a r a c t e r i z e m t h e T~ c o m p l e t i o n o f Pf(E;F) , when E ' s a t i s f i e s t h e bounded ap-
-
proximation property. E x t e n s i o n o f W e i e r s t r a s s ' theorem f o r i n f i n i t e dimensional
6.1.
Banach spaces. Let
m E
for
E
and
F
be r e a l Banach spaces. We endow
2
m on t h e compact subsets o f
t o p o l o g y c a l l e d compact-open t o p o l o g y o f o r d e r U
,
where
,
, w i t h t h e t o p o l o g y o f u n i f o r m convergence o f t h e f u n c t i o n s
and t h e i r d e r i v a t i v e s o f o r d e r .rm
Cm(E;F)
E. This
m, which we denote by
i s generated by a l l seminorms o f t h e form:
j E D1
,j 5
m
, and
K
i s a compact subset of
E.
A l l polynomial spaces c o n s i d e r e d i n t h i s c h a p t e r a r e endowed w i t h t h e norm t o p 0 OgY
6.1.1
-
Theorem. ( cm(E ;F , 7:
is complete.
Chapter 6
128
be a Cauchy n e t i n
Let
(fa)aeA t h e r e a r e continuous f u n c t i o n s such t h a t E
.
f = l i m fa and
(h,)
in
E,
h,
i t follows that
for a l l f(x)lI 2
a ,B
2
f// hn/l
Therefore
, for
I1 f ( x 211 +
11
f,(X
+
1)
, then
f,(x
and
.
I t i s easy t o see t h a t
g :
E
-L
L(E;F) =
P( 1E;F)
u n i f o r m l y on compact subsets of
i s d i f f e r e n t i a b l e w i t h d e r i v a t i v e df=g.
there exists
E
> 0, x E
E and a sequence
- f,(x) - f g ( x t hn) - f g ( x O AB , so t h a t I / f a ( x + h n ) - f,(x) - f ( x + hn) a 2 a o . Also, f o r a l l a some n18 A ,
t hn)
some
~
for all
n
no
+ hn)
-
-
f
F
T;)
0 , such t h a t
f ( X + hn)
+ h,)
+
g = l i m dfa
I t s u f f i c e s t o show t h a t
I n fact, i f this f a i l s
(C'(E;F),
f : E
m > 1 being s i m i l a r .
the proof f o r
We s k e t c h t h e p r o o f f o r m = 1
Proof.
f(x)
-
f(x
f,(x)-df
which i s a c o n t r a d i c t i o n . Therefore t h e r e s u l t i s e s t a b l i s h e d . # Lesmes [ l ] considered and proved t h e f o l l o w i n g theorem:
C'-
f u n c t i o n s d e f i n e d on a H i l b e r t space
Approximation f o r t h e compact-open t o p o l o g y
129
Theorem. Let H be a reaZ HiZbert space o f inj%nice dimension. The
6.1.2
of t h e potynomials. o f f i n i t e type on H i s T; - dense i n
Pf(H)
aZgebra
, but
C'(H)
not i n
Cm(H)
, for
m
? 2
(3.0.1).
, See
-
I n t h i s c h a p t e r we extend t h e work o f Lesmes. We s t u d y t h e r : completion o f
Pf(E;F)
and we g e t an e x t e n s i o n o f W e i e r s t r a s s ' s theorem
f o r i n f i n i t e dimensional Banach spaces
E
such t h a t
E'
has t h e bounded
approximation property. Let
Pc(JE:F)
t h e norm induced by
be t h e c o m p l e t i o n o f
Pf(JE;F)
w i t h respect t o
A f t e r an i n depth a n a l y s i s o f t h e c o u n t e r
P(JE;F),
example 3.0.1, i t can be e a s i l y seen t h a t t h e reason f o r which Pf(H) i s m n o t T~ - dense i n Cm(H) , f o r m ? 2 H a H i l b e r t space, i s t h a t gen erally if
f 8 C"(E)
,
then
dJf(x)
see ( 4 . 5 . 1 1 ) y we proved t h a t i f property, then
6 Pc(JE) , j
2.
I n c h a p t e r 4,
has t h e Grothendieck a p p r o x i m a t i o n
E'
Pwbu(JE;F) = Pc(JE;F)
( j
polynomial a p p r o x i m a t i o n r e s u l t s i n t h e
2
m). Therefore, i n o r d e r t o g e t
-
T:
topology
,
i t seens l o g i c a l
t o establish the following definition. 6.1.3.
m C k (E;F)
Definition.
, for m
such t h a t for every
f 8 Cm (E;F)
Using theorem 6.1.1,
-
M ,is
B
x B E
t h e space of functions
and j
2
d j f ( x ) B Pwbu(JE;F).
m
i t i s routine to verify that
(C!
(E;F),T:)
i s complete. Note t h a t
m
1
,
CL(E;F) = C(E;F)
and
C i ( E ) = C'(E),
. If
t h e two spaces a r e g e n e r a l l y d i f f e r e n t
a d i s p e r s e d compact H a u s d o r f f space, t h e n f o r e v e r y f E Cm(E)
see ( 0 . 3 . 3 ) . Thus i n t h i s case f o r a l l ( j
5
m
,x
also that
B E).
I n particular
m Cwbu(E;F)
.
such t h a t xn
I$n
-f
0
in
sequences
( n . ) and J
by lemma 4.1.1,
while
E
(p.)
J
I$")
To see t h i s , l e t
o(E';E),II
i n the u n i t b a l l o f
j
2
m
, dJf(x)
C F ( c o ) = Cm(co) f o r a l l
i s always a p r o p e r sub e t o f
i s i n f i n i t e dimensional
and t h a t f o r
E = C(K), w i t h
such t h a t
=
2
,
r$,,(xn
C:(E;F)
, P(JE)
K = Pc(JE)
f P (JE) C-
m 6
M
.
E
provided
be a sequence i n
1,
see (0.12.6
Note E'
and choose
>- 3/2. Then f o r a p p r o p r i a t e
o f integers, f ( x )
f B CF(E).
Our main r e s u l t i n t h i s c h a p t e r i s t h e f o l l o w i n g W e i e r s t r a s s t y p e theorem. Since t h e c l a s s i c Stone-Weierstrass theorem a p p l i e s t o approximation i n
Co(E;F) = C(E;F) = CL(E;F)
with the
T;
-topology
,
Chapter 6
130
we w i l l be c o n s i d e r i n g o n l y 6.1.4
m
2
1 throughout.
Let E and F
be r e a l Banach spaces with E ' having the m bounded approximation property. Then f o r a l l m >_ 1 , Pf(E;F) i s .ru-dense Theorem.
C;
in
(E;F)
.
The p r o o f o f t h i s theorem r e l i e s on t h e f o l l o w i n g lemmas, which a r e s i m i l a r t o those i n c h a p t e r 5, s e c t i o n 5.3.
-.
6.1.5
T c CT(E;F)
Let
be ccmpact, and l e t
be a
N ,j 5
j B
As w i t h lemma 5.3.1,
fl,...,fS
6
T
We l e t
,.. . ,6s)
I t f o l l o w s t h a t f o r any
xk
such t h a t
6.1.6
f E T
( 1 dJf(x) -
11
x -yII <
, i = l,.. .,s.
K, and so t h e r e i s 6i > 0
,I/
Ai
-
and choose a 6
f B T
and
dJfi(xk)(/
dJfj(x)
net {
XI
-
dJfi(y)ll < ~ / 2 .
,.. . ,xnl
for
, t h e r e i s some fi
x B K
K.
and some
< E ,#
Lemma. Let E be a r e a l Banach space such t h a t E ' has t h e bounded
approximation property w i t h constant C compact subsets K c E , Ki c Pwbu (JE;F) f i n i t e rank continuous linear operator that
there i s a f i n i t e set
i s u n i f o r m l y continuous on
x, y E K, w i t h
6 = min(6l
,
: x B K I < ~ / 2, f o r some
sup{/l dJ(f-fi)(x)II
such t h a t f o r
1
given E > 0
such t h a t f o r any
For each i, dJfi
B K
P (JE;F). wbu
i s a precompact subset of Proof.
K c E
. Then the s e t
m
,x
L. = {dJf(x) : f B T J
precompact subset, l e t
T-:
.
Given
j 6
I!
,E > 0
, and
( 0 5 i 2 j ) , there e x i s t s a TI
: E
+
E, with
11 ~ 1 1 51
C
, such
Approximation f o r t h e compact-open t o p o l o g y
Proof.
E'
If
131
C,
has t h e bounded a p p r o x i m a t i o n p r o p e r t y w i t h c o n s t a n t
t h e n i t i s p o s s i b l e t o f i n d a simultaneous a p p r o x i m a t i o n o f compact in
and
operator
K ' c E ' and
TI
11 @
and
, i n t h e f o l l o w i n g sense.
and i n E '
E
K c E
o
: E
n
E
-+
- $11
,
> 0
E
11 nil
(4 E K')
< E
see ( 0 . 5 . 8 ) .
Ki c Pwbu(iE;F)
-
6.1.7
x
11
Lemma.
<
(x
E
E
11
, and
K)
Q
0
compact, there e x i s t s E
11
,
11
with
-
dif(x)
di(f
111
m E
a precompact subset where
+
K),
K c E,
, t h e r e i s a continuous such t h a t /I 5 C ,
> 0
TI
- Q
11 <
j
(Q e u
E
Ki). #
i=O
Let E and F be r e a l Banach spaces such t h a t E ' has the -
n : E
6
has t h e bounded approxi-
bounded approximation property with constant Kc E
(x
T h e r e f o r e , a p p l y i n g an
E'
E
-f
IIir(x)
< E
C, t h e n f o r a r b i t r a r y compact s e t s
( 0 5 i 5 j) and f i n i t e rank l i n e a r o p e r a t o r IT : E E
and
I/ n(x) - XI/
2 C ,
we see t h a t when
mation property w i t h constant
Given a r b i t r a r y compact s e t s
t h e r e i s a continuous f i n i t e r a n k l i n e a r
such t h a t
argument i n lemma 5.3.2
sets
7711
Given
j
E
W,
j
T c C:
5 m ,
(E;F) > 0
E
be
and
a f i n i t e rank continuous linear operator
5
C,
and
n)(x)ll
o
.
. Let
C
( f E T
< E
,x
E
K
,
i 5 j).
f o l l o w s t h e same
We o m i t t h e p r o o f o f t h i s lemma because i t general l i n e s as lemma 5.3.2. Let
P r o o f o f t h e theorem 6.1.4. E
>
0 ,K
c E
compact,
by
continuous l i n e a r o p e r a t o r
TI
By W e i e r s t r a s s ' theorem 1.1.2,
of
E
E
C!
(E;F)
. Given j
E U
,j 5
m
,
lemma 6.1.7 t h e r e e x i s t s a f i n i t e r a n k : E + E , w i t h 11 7711 5 C , such t h a t
f o r any f i n i t e dimensional subspace
Eo
i t follows that
P(Eo;F)
(2) Let
f
E O = u(E)
there i s
is T :
. Since
p E P(Eo;F)
-
dense i n
n(K) c E o such t h a t
Cm(Eo;F) = C!
(Eo;F).
i s compact, by ( 2 )
we conclude t h a t
132
where (4)
Chapter 6
B
1)
E.
i s the u n i t b a l l i n di(f
o
TI)(X)
From (1) and ( 4 )
-
di(p
o
I n o t h e r words,
n ) ( x ) \ l < E/Z
(x E
K , i 5 j).
we g e t
which completes t h e p r o o f s i n c e
p
o
n
E Pf(E;F) .#
CoroZlary. Let E and F be r e a l Banach spaces with E ' having the m 2 1 , !C (E;F) is the bounded approximation property. Then f o r aZZ m T - compZetion of Pf(E;F) 6.1.8
.
6.2.
References.
The b a s i c r e f e r e n c e s o f t h i s c h a p t e r a r e Aron [l], A r o n - P r o l l a i l l , Lesmes [11 , Llavona 1 2 1 , Josefson [13 and Nissenzweig I l l and Aron [ 2 1
.
133
Chapter 7
APPROXI MAT I O N OF WEAKLY DIFFERENTIABLE FUNCTIONS
I n t h i s c h a p t e r we p r e s e n t a new c l a s s o f d i f f e r e n t i a b l e f u n 2
Cm
t i o n s , t h e space o f weakly d i f f e r e n t i a b l e f u n c t i o n s o f c l a s s to
F
, which
c o n t a i n s C:bU(E;F)
i s contained i n
and
C!
from
E
We w i l l
(E;F).
use t h i s c l a s s t o g e t a new c h a r a c t e r i z a t i o n o f r e f l e x i v e Banach spaces. More s p e c i f i c a l l y , we w i l l prove t h a t a Banach space
f : E
and o n l y i f e v e r y weakly d i f f e r e n t i a b l e f u n c t i o n i s bounded on b a l l s o f
E.
E
i s reflexive, i f +
R o f c l a s s Cm
T h i s r e s u l t i s an e x t e n s i o n o f t h e same r e s u l t
f o r weakly continuous f u n c t i o n s t o d i f f e r e n t i a b l e f u n c t i o n s . ( S e e (9.12.11)). We w i l l a l s o i n t r o d u c e a new a p p r o x i m a t i o n p r o p e r t y , a "bounded weak a p p r o x i m a t i o n p r o p e r t y " which we t h i n k c o u l d be o f g r e a t i n t e r e s t i n t h e s t u d y o f i n f i n i t e dimensional a n a l y s i s .
F i n a l l y , some r e s u l t s on poly-
nomial a p p r o x i m a t i o n o f weakly d i f f e r e n t i a b l e f u n c t i o n s a r e g i v e n .
7.1.
Weakly d i f f e r e n t i a b l e f u n c t i o n s . Some r e s u l t s on weak compactness. L o c a l l y convex s t r u c t u r e . Let
7.1.1
E,F
Definition. a
tiabZe a t
8
(i)
be r e a l Banach spaces. A function
(ii)
+
f
+
(F,ll
11
f : E
+
F
E
of
)
is continuous a t
f
such t h a t
a
a
6
B,
a.
.
i s weakly d i f f e r e n t i a b l e , if i t
i s weakly d i f f e r e n t i a b l e a t each p o i n t o f Note t h a t i f
E
6
is Frgehet d i f f e r e n t i a b z e a t
We w i l l say t h a t
However, i f
is said t o be veakly d i f f e r e n
F
f o r each bounded subset
f l B : (B,o(E;E')IB)
the mapping
f : E
i f it v e r i f i e s :
E
E.
i s weakly d i f f e r e n t i a b l e , t h e n
f e Cwb(E;F).
has i n f i n i t e dimension, t h e r e a r e weakly d i f f e r e n t i a b l e
f u n c t i o n s which a r e n o t weakly continuous; f o r i n s t a n c e , i f
(4,)~
E'
Chapter 7
134
i s a sequence o f l i n e a r l y independent l i n e a r forms o f norm 1, t h e f u n c t i o n m
i s a 2-homogeneous, weakly d i f f e r e n t i a b l e polynomial which i s n o t weakly continuous
. All polynomial spaces considered i n t h i s s e c t i o n a r e endowed
w i t h t h e norm topology.
Definition. Let f : E + F be a function and m E M. We wiZ1 say that f is m-times weak23 differentiabze at a point a E E if f is ( m - 1 ) -times weakZy differentiable and the mapping 7.1.2
is weakZy differentiable at a
.
(dof
=
f)
.
i s s a i d t o be m-times weakly d i f f e r e n t i a b l e , i f i t i s m-times weakly d i f f e r e n t i a b l e a t each p o i n t o f E. F o r t h e as t h i s case r e q u i r e s sake o f s i m p l i c i t y , we do n o t t a k e m = t m A function
f
only minor v a r i a t i o n s .
Proposition. Let
7.1.3.
m-times weakly differentiable 1. Then for each x E E and each j E 111 , 1 2 j 5 m ,
ftmct-ion, m
f : E
+
F be an
d J f ( x ) 6 Pwb(JE;F). Proof.
. Given r > 0 and 6 < r , such t h a t
x E E
Let
O< 6 <
1,
Since
f E Cwb(E;F)
W
0
Let
of
h E
E
in
W
,
,I]
which proves t h a t
,
5
> 0, t h e r e e x i s t s
6
,
t h e r e e x i s t s an a b s o l u t e l y convex weak neighbourhood
such t h a t i f
hll
E
r
.
x - y IS W
and
From (1) and ( 2 )
d f ( x ) E Pwb( 1 E;F).#
11
x-yll
we have
5
11
1 then
df(x)(h)ll < c
Approximation o f weakly d i f f e r e n t i a b l e f u n c t i o n s
135
Now l e t 1 2 j 2 m and assume t h a t t h e r e s u l t h o l d s f o r ( j - 1). Reasoning as b e f o r e f o r d j - l f i n s t e a d o f f , g i v e n y 8 E, E > O and r > 0, t h e r e e x i s t s a weak neighbourhood
lly-zll
11 Since
5 r and y - z e W
W
of
in
0
E, such t h a t i f
then
d(dJ-lf)(x
11 d J f ( x ) ( y )
+I1 d ( d J - l f ) ( x ) ( y 5 €12 +
., t h e
€/2 = E
r e s u l t follows.#
Thus f o r each i n t e g e r
E
a mapping from 7.1.4
to
, 1 5 j 5 m, we can c o n s i d e r d J f as
j
P~~(JE;F).
Definition. Let m
6
differentiable of class Cm on and d"f e Cwb(E;Pwb(mE;F)).
111. A function f : E E
, if
7.1.5
Cm
from
E
to
F
by
F is said to be weakly
f is rn-times weakly differentiable
We w i l l denote t h e space o f a l l o f class
-f
weakly d i f f e r e n t i a b l e f u n c t i o n s
C:b(E;F).
Remark. Note t h a t a f u n c t i o n
f : E
F
-f
belongs t o
CEb(E;F)
if
and o n l y i f i t s a t i s f i e s :
where
(1)
f E c~(E;F).
(2)
d J f ( x ) 6 Pwb(JE;F)
(3)
d J f 6 Cwb(E;Pwb(JE;F))
dof = f
.
Thus
(x 6 E (0
, from (4.5.9),(5.2.4)
,
1
2 j2
m).
5 j 5 m), and
(6.1.3)
i t follows that:
Chapter 7
136
Cbu:
I n o r d e r t o s t u d y whether t h e space contained i n bC:
w i l l use t h e f o l l o w i n g lemma.
Lemma. Let A be a subset of
7.1.6
, then
on A
, we
(E;F)
Proof.
z
Every
i n the
separable subspace o f
( E l ; o(E';E))
F , where
it i s restricted t o
E"
n E
W 1
t h e r e e x i s t s a subset E H
)I
is bounded
f 6 C:b(E)
A
such t h a t
i s continuous on every z
of
$(un) = 0
: H +R
be d e f i n e d as:
F
be a
i s n o t continuous when a(E";E')-closure
generated by
satisfies
7
Let
{un :
of
I f t h i s i s n o t so, l e t
belongs t o t h e
z
be t h e l i n e a r subspace of
that i f
If everg
a(E";E')-closure
(El; a(E';E)).
H
.
E
A is weakZy r e l a t i v e Z y compact.
separable subspace o f
Let
is strictly
(E;F)
F, each
un
, then
$ = 0.
where I$ i s a r e a l - v a l u e d f u n c t i o n w i t h r e a l domain, non-negative, class
zero only a t
Cm, = 0
1 7 )
$ ( t ) = exp(v e r i f i e s a l l t h e above c o n d i t i o n s ) . If
1
5
j
f =
flE ,
i t follows that
5 m , and f o r every x dJf(x)
1
n=l
1
i s complete
, we
f
6
Cm(E)
dJ$(
) u nj
# 0 and
and f o r each
j 6 M
,
Pf(JE)
.
i s included i n
(1 2 j
2
Pwb(JE), and
Pwb(jE)
m ; x E E).
O n t h e o t h e r hand, i t i s obvious t h a t f o r every d J f 6 Cwb(E;Pwb(JE)). Thus, f E C!b(E). Since
t
have t h a t
d j f ( x ) E Pwb(JE)
we can d e f i n e
if
E E,
2"
Since t h e norm-closure o f
of
and such t h a t $! and a l l i t s d e r i v a t i v e s a r e
0
bounded ( f o r i n s t a n c e , t h e f u n c t i o n
$(a)
A.
having norm 1, such
n E fd
f o r every
of
and z. By(0.12.11)
E
z d E, i t i s c l e a r t h a t
j,
f(x) # 0
0
5
j
5
,
m
f o r every
x
E
E. Thus
Approximation o f weakly d i f f e r e n t i a b l e f u n c t i o n s
and s i n c e
f
6
,
C:b(E)
i t follows that
g
C:b(E).
6
Finally
137
, let
c A be a o(E";E')-convergent n e t t o z. Then F ( z d ) * f ( z ) = O ('d)d E D and t h e r e f o r e t h e n e t g ( z d ) i s n o t bounded , which c o n t r a d i c t s o u r as-
sumption. Thus, f r o m (0.12.11)
7.1.7
Theorem.
From
Cb:(
The spaces
E ;F )
and
( E ;F)
Cbu:
coincide, i f and
E i s a r e f l e x i v e r e a l Banach space.
only i f Proof.
t h e lemma f o l l o w s . #
m = C:bu(E). t h e n i t i s obvious t h a t C:b(E) If Cwb(E;F) = C:bU(E;F) (4.1.1) i t f o l l o w s t h a t t h e c l o s e d u n i t b a l l o f E s a t i s f i e s t h e
h y p o t h e s i s i n lemma 7.1.6. flexive.
T h e r e f o r e i t i s weakly compact and
E
is
re
The converse i s o b v i o u s . # The f o l l o w i n g lemmas w i l l a l l o w us t o e s t a b l i s h t h e r e l a t i o n s h i p
between spaces
C:b(E;F)
A set
M c
and
R
CL(E;F).
i s s a i d t o have c o n s t a n t siqn, i f e i t h e r a l l i t s
elements a r e non-negative o r a l l a r e n e g a t i v e 7.1.8
If $ 1 ,.. . , I),, E E ' and (x,) i s a bounded sequence i n E , > 0 there e x i s t s an i n f i n i t e s e t A c !I such t h a t :
Lemma.
f o r every
E
{ $ . ( x ) : k E A } has constant sign J k
(i)
-
( i i ) supC/qj(xk)l: k E A1 Proof.
; (1
5 j 5 n).
: k E A I < E ; (1 z j
infIII)j(xk)l
I t i s s u f f i c i e n t t o prove t h e r e s u l t f o r $ E E ' . L e t m
such t h a t f o r each
11 $11 n E
<
m
and
NE > 2
, where M
> 0
i s such t h a t
,N
E
N
I] xn\/ 2
W.
For e v e r y i n t e g e r j l < k < N , l e t A. = J ,k
A. J ~k
, -m 5 j 5
m
and f o r each i n t e g e r
k
,
be d e f i n e d as:
k-1 m s N : j + T
I t i s c l e a r t h a t each
Since
.
A.
J ,k
= A
5
k
$ ( x n ) < j + ~I
s a t i s f i e s ( i ) and ( i i )
. i n lemma 7.1.8
2". be M
Chapter 7
138
i t f o l l o w s t h a t some A
7.1.9. @I,.
=.Let
.. ,I),
exists
p
so t h a t
Proof. that c1
E B
E'
.
must be i n f i n i t e . #
j,k
=2, and
u(E';E)-~uZZ sequcnce in E ' , I l
be a
($,)
Then, there e x i s t s a > 1 so that f o r every k a U
I , p >_ k, rmd x E E with11 xi1 5 1 and 2 a .
II)~(XII< 1 (1
there
5
j 5 n),
$,(x)
M
If
Qjll
= max{ll
1 < a' c 1.
Let
: 1
5
j
2
nl
a' =
and
-
Ei;
(-
1 t y < 2~17
y > 0 be such t h a t
1
)
,
,
and choose
it i s clear
= 1 i- a ' y
For each
k E El, t h e r e e x i s t s
Xk
8
1) XkIl
with
E
= 1 and
.
1 and A p p l y i n g lemma 7.1.8 t o t h e sequence (x,) > 1t It y 2a E = -M 4 t E , i t f o l l o w s t h a t t h e r e e x i s t s an i n f i n i t e subset A o f Iu
$r
k ( xk
such t h a t : ( a ) { +j(xk)
: k
B
has c o n s t a n t s i g n ; ( 1
A?
5
j
5 n).
( b ) s u p I ( @ . ( x ) I : k ~ A l - i n f { \ + . ( x ) I : k s A l <4 m ; ( l J k J k From ( a )
, and i f necessary changing li,j
can assume
Given u(E';E)-null
I f we t a k e
f o r every
Q.(x ) > 0 J k k E 111
,
k
q e A
let
sequence, t h e r e e x i s t s
x = a'x
F i n a l l y , f o r every
P
-
j E
(1
-
N ,
al)x
1
q
,
6
to
A
,
-$
15
be so t h a t
p E A, p
2
, it
j j
n).
f o l l o w s t h a t we
5 n. q
2
k . Since
(Qn)
k, such t h a t
i t i s clear that
5 j 5 n
2 j 5
11 xi1 <
1 and
is a
Approximation o f weakly d i f f e r e n t i a b l e f u n c t i o n s
ilij(X) = a ' q J , ( x 1 J P
-
(1 - a ' ) + j ( x q )
139
2
On t h e o t h e r hand,
?
Qj(Xr)
&'(SUP
rsA
4 - M+3
- (1-a')sup rcA
$.(x ) = J r
Theorem. Let E be a r e a l Banach space of i n f i n i t e dimension. For m , Cwb ( E ; F ) i s a proper l i n e a r subspace o f C r ( E;F).
7.1.10
euery Banach space F From (0.:2.6)
Proof. sequence
(@,)c
E'
i t i s known t h a t t h e r e e x i s t s a
so t h a t
f e C!(E).
It i s c l e a r t h a t q ~ ~ , . . . , y ~E~ E '
such t h a t i f
then
.
k E
lf(x)j< 1
11 $ 1 ~ 1 1
= 2
n 6
Assume t h a t
x G E
lN and x E E w i t h ( 1 x ( [ < 1 and ? a. Thus, we have
.
Let
o(E';E)-null be d e f i n e d as
f
f E C:b(E).
) / I XI( 5
On t h e o t h e r hand, by
U
1, and
(7.1.9)
($j(x)I
5
1
Then t h e r e e x i s t
l $ j ( x ) 1 5 1 (1 5 j 5 n) t h e r e e x i s t s a > 1, (1
5
j < n)
such t h a t
@,(x)
6
C:b(E).
T h i s c o n t r a d i c t i o n proves t h a t
f
prove t h e theorem f o r t h e case
F = R
, the
Since i t i s s u f f i c i e n t t o r e s u l t follows.#
Chapter 7
140
7.1.11 ology
where
m Definition. hie endow Cwb(E;F) with t h e locally convex topm which i s generated by a12 seminorms of the form: T wc '
d o f ( x ) ( y ) = f ( x ) and K
E.
i s a weakly compact subset of
and i s r e f l e x i v e , t h e space C:b(E;F) m T~~ = T~ , see ( 5 . 2 . 6 ) , where K i s now a l l o w e d
Note t h a t i f
E
m
c o i n c i d e and C:bU(E;F) t o range over t h e bounded subsets o f
E.
o f t h e l o c a l l y convex I n o r d e r t o s t u d y t h e main p r o p e r t i e s m t h e f o l l o w i n g lemma w i l l be v e r y u s e f u l . s t r u c t u r e o f (c:b (E;F) , TWC
7.1.12.
m
Let
Lemma.
8
W be an i n t e g e r , m
1
.
Then:
(a)
E ' endowed with the Mackey topology T(E';E), m m i s a complemented linear subspace o f ( Cwb( E) , T ~ ~ ) . (b)
(E',?(E';E)),
For every Banach space
F, (C:b(E;F),~:c) contains a complemented linear subspace isomorphic to ( Cb:( E ) ,T!~ ) .
-
Proof. ( a ) F i r s t a t a l l , n o t e t h a t t h e topology
T ( E ' : E ) on
E'.
I t i s a l s o easy t o check t h a t t h e mapping
i s a continuous l i n e a r p r o j e c t i o n o n t o (b) Let
y c F
y'(y) # 0 spaces
.
Let
be w i t h IIyII = 1
F,
(c:b(E;Fo),
t o p o l o g y induces t h e Mackey
.
Let
El. y ' E F'
, with
I/ y'II
= 1
and
be t h e space generated by y. I t i s obvious t h a t t h e m and ( c W b ( ~ ) ,TJc) a r e isomorphic. On t h e
T:~)
o t h e r hand, i t can be e a s i l y seen t h a t t h e mapping
i s a continuous l i n e a r p r o j e c t i o n o n t o
7.1.13
Theorem.
Let
f o r every Banach space
m Cwb(E;Fo).#
E be a Banach space. F
, the
For every i n t e g e r
m
2 1 and
following conditions are equivalent.
Approximation o f weakly d i f f e r e n t i a b l e f u n c t i o n s
is refZexiue.
(a)
E
(b)
(C:b(E;F),Twc)
(C)
(C:b(E;F)
(d)
(C:b(E;F),
(e)
(Czb(E;F)
Proof. ( a ) => m m T~~ = T~ (see
141
m
is a Fre'chet space.
, T:~)
is a barreZZed space. m T ~ is ~ an) infrabarreZZed space is a bornoZogica2 space.
,T:~)
(b) : I f
E
i s reflexive
m , Cwb(E;F)
m
= Cwbu(E;F)
and
and comments a f t e r d e f i n i t i o n 7.1.11). In m i t i s proven t h a t (Cwbu(E;F), i s complete. F i n a l l y , i t i s
(5.2.7)
(7.1.7)
):T
obvious t h a t t h i s space i s m e t r i z a b l e . ( b ) =>
( c ) : I t f o l l o w s from ( S c h a e f e r [l], 7.1).
( c ) =>
(d)
is trivial.
i s i n f r a b a r r e l l e d , from ( d ) => ( a ) : I f (C:b(E;F),~:c) p. 218) i t f o l l o w s t h a t ( a ) ) , (7.1.12(b)) and ( H o r v i t h [l],
(7.1.12
(E',T(E',E))
i s infrabarreled.
(Horva'th [l], p.218) Finally
,
Since
E
i s b a r r e l l e d , i f and o n l y i f
(E',T(E';E))
(7.1.12(b))
( b ) =>(e)
: (See (Schaefer
( e ) =>(a)
: If
[ll, 5.3)
i s reflexive
and
[ll,8 . 1 ) ) .
, T:~)
(Clb(E;F)
( H o r v a t h [ll,p.222)
b o r n o l o g i c a l , hence i n f r a b a r r e l l e d E
i s quasi-complete
[ l l , 5.5).
(Schaefer
that
( E ' ,T(E';E))
i t follows t h a t i t i s barrelled(Schaefer
.
i s bornological, (7.1.12(a)),
show t h a t
( E l , T(E';E)) i s
NOW, as i n ( d ) =>
(a)
we can prove
must be r e f l e x i v e .# Finally, i f
E,
i t i s n o t hard t o see t h a t
i s a f i n i t e dimensional l i n e a r subspace o f Cm(E1;F)
t o a complemented l i n e a r subspace o f i s not semi-reflexive,
i t follows that
with the (C:b(E;F); (C:b(E;F),
T:-topology -czc)
i s isomorphic
. Since
.zC)
E,
Cm(E1 ;F)
i s n o t semi-re-
f l e x i v e f o r e v e r y Banach space F. The problem o f t h e c o m p l e t i o n o f
(Cmwb(E;F)
, fC)
w i l l be
s t u d i e d i n s e c t i o n 7.4. 7.2.
The bounded weak a p p r o x i m a t i o n p r o p e r t y . I n t h i s s e c t i o n we i n t r o d u c e a new a p p r o x i m a t i o n p r o p e r t y ,"the
bounded weak a p p r o x i m a t i o n p r o p e r t y " . T h i s new p r o p e r t y w i l l be used t o
142
Chapter 7
study polynomial approximation o f weakly d i f f e r e n t i a b l e f u n c t i o n s i n sec t i o n 7.3. Here an i n t r i n s i c s t u d y o f t h i s p r o p e r t y and i t s r e l a t i o n t o t h e usual approximation p r o p e r t i e s i s given. 7.2.1 Definition. A Banach space E i s said t o have the bounded weak approximation property (b.w.a.pl i f f o r every weakZy compact s e t K i n E, there e x i s t s a net (ui) c E l BI E such t h a t :
)
(i) (ui(x
x
converges t o
U {u ( x ) : x 8 K } i s
(ii)
, weakly
uniformly on
a bounded subset of
E
i 7.2.2
K c E
Let
K.
E
.
If E i s a Banach space such t h a t i t s dual E ' has
Proposition.
b. w. a.p.
the bounded approximation property, then E has t he Proof.
x
be a weakly compact subset o f E. Since E ' has t h e , f r o m (0,5.8 ) i t f o l l o w s t h a t t h e r e
bounded a p p r o x i m a t i o n p r o p e r t y exists IT
0
C > 0
E'
E
where
B)
M
E
such t h a t f o r e v e r y f i n i t e subset
with
I( 1 ~ ~ 1 51
(IT@)c E '
T h e r e f o r e we o b t a i n a n e t
=I<
X,@
0
IT@
-$>I 5
x E K
and
there exits
$I
E
Q,
,
BI
E
I
M
f o r every
x
E
K.
satisfying : < IT ( x ) Q,
- x ,@ > 1
=
( T ~ )
converges t o
x
weakly u n i f o r m
11 XI[
2 CM , thus
x E K. (ii)
U
E'
1.
T h i s r e s u l t proves t h a t l y on
11 xII 5
i s a p o s i t i v e c o n s t a n t such t h a t
( i )For every
@ c
and
C
{IT@(x) : x
For e v e r y E
K} i s
x
E
K
, 11
I T ~ ( X 5) ~11~nQII
a bounded s e t . #
Q The n e x t p r o p o s i t i o n shows t h a t most c l a s s i c a l Banach spaces have t h e
b.w.a.p.
7.2.3
If E i s a r e f l e x i v e Banach space, then E has the i f and only i f E has t he bounded approximation property.
Proposition.
b.w.a.p.,
Approximation o f weakly d i f f e r e n t i a b l e f u n c t i o n s
If E
Proof.
has t h e bounded a p p r o x i m a t i o n p r o p e r t y and
has t h e bounded a p p r o x i m a t i o n p r o p e r t y (Kiithe [ l l
E'
by (7.2.2)
E
for
E
that ~ c )E ' ~B E~ such ~
u
x e B(E),
b.w.a.p.,
(ui(x))
converges t o E, and
x
weakly u n i f o r m l y
Iui(x)
; i E I , x B B(E)I
T h i s l a s t f a c t proves t h a t t h e r e e x i s t s a p o s i t i v e
11
M
f o r every
On t h e o t h e r hand, i f
K
i s a compact subset o f
such t h a t
ui/I
c o i n c i d e on bounded subsets o f
6
I. E'
and
E
> 0,
n ( 1 + M) B ( E ) (ui(x))
=
E
converges t o
K"
n (1 + x
i
Thus f o r e v e r y
T h i s proves t h a t
@ 8 K
M) B ( E ) .
weakly u n i f o r m l y on
io E I such t h a t f o r e v e r y
and then, f o r e v e r y
E, i t f o l l o w s t h a t t h e r e
V o f 0 i n E such t h a t
e x i s t s a weak neighbourhood
Now s i n c e
i
and t h e t o p o l o g y o f u n i f o r m convergence on t h e compact
o(E,E') subsets o f E '
V
i s reflexive,
261 ) , and t h e n
there exists a net
2
M
since
exists
has t h e
the closed u n i t b a l l o f
i s a bounded s e t . constant
E
, p.
has t h e b.w.a.p.
Conversely, i f (
143
and e v e r y
i
x E B(E)
, there
2 io
x E B(E)
io
E
has t h e bounded a p p r o x i m a t i o n p r o p e r t y
(KiSthe [IJ,
p. 261).# 7.2.4
CoroZZary.
If
E is a refZexive Banach space, E
i f and only
if E has the approximation property.
Proof.
E
If
has the
b.w.a.p.,
i s a r e f l e x i v e Banach space, t h e a p p r o x i m a t i o n p r o p e r t y
and t h e bounded a p p r o x i m a t i o n p r o p e r t y a r e e q u i v a l e n t ( L i n d e n s t r a u s s Tzafriri [ l ]
, pp. 39-40).#
-
144
Chapter 7
T h i s c o r o l l a r y enables us t o g i v e an example o f a space w i t h o u t t h e b.w.a.p.
E
Example. L e t
be a c l o s e d
approximation p r o p e r t y (see a r e f l e x i v e Banach space and 7.3.
lP (2 < p
subspace o f Lindenstrauss by (7.2.4)
-
E
without the
i m)
T z a f r i r i [ l ] , p.90 ) . E
is
does n o t have t h e b.w.a.p..
Polynomial approximation o f weakly d i f f e r e n t i a b l e f u n c t i o n s . I n t h i s s e c t i o n we w i l l l i m i t ourselves t o p r o v i n q t h e .r:c-den-
sity
of
in
Pf(E;F)
C:b(E;F)
, when E has t h e b.w.a.p. m
(see s e c t i o n ) m 3.21, (Cwb(E;F); TWC ) i s g e n e r a l l y n o t complete. F o l l o w i n g s i m i l a r techm niques t o those used t h e r e , a c h a r a c t e r i z a t i o n o f t h e T~~ - c o m p l e t i o n Analogous t o t h e case o f t h e space
of
Pf(E;F)
can be found E, F
Let
(Cm(E;F);~,
, (see s e c t i o n 7 . 4 ) .
be r e a l Banach spaces. A l l polynomial spaces con-
s i d e r e d i n t h i s s e c t i o n a r e endowed w i t h t h e norm t o p o l o g y .
If A is a precompact subset of Pwb(j E;F) , then f o r every E > 0 and f o r every bounded subset B of E there e x i s t s a weak neighbourhood W of 0 i n E , such t h a t if x , y 6 B and x - y € W 7.3.1
Lemma.
Proof.
Let
M > 0
11 x I / 5 M .
Since
A
be a p o s i t i v e c o n s t a n t such t h a t f o r every
x
E
B
,
We p u t
i s precompact
i t follows t h a t there e x i s t
ply
...,
pk 8 A
such t h a t
=
A For every in
E
p,
k U
(P,+H).
m=1
(1
5 m 5 k)
such t h a t i f
x
t h e r e e x i s t s a weak neighbourhoad W,, y
e B and x - y
6
W,
then
of
0
Approximation o f weakly d i f f e r e n t i a b l e f u n c t i o n s
II PJX) k Put
.
n W,
W =
m'pl-
such t h a t
5
P,(Y)II
42. p E A
Since f o r e v e r y
,15 m 5 k , B and x - y e W
there exists
e H , i t follows that i f x , y
p,
145
E
m
then
7.3.2
and
in
Ifo u : u
Af =
m
(C;b(E;F)
Proof.
f o r m l y on
K,
x E K
Since
(1) that
if
,y j
K U {ui(x)
Let
B
:
x
weakly uni-
1
x e K
i e I
is
a
be an a b s o l u t e l y convex and bounded subset
{z
and t h e s e t
-
,
: z e B1, x E K
ui(x)
c o n t i n u o u s on
K
i e I}.
and weakly
i t follows that
E
B
and
x
be an i n t e g e r
-
y e
1
5
j
i s a compact subset o f
Pwb(jE;F).
U1
of
11 f ( x ) -
then
U1
0
in
f(y)II <
E
, such E.
5 m . The mapping d J f : E I P ~ ~ ( ~ E ; F ) E, hence
i s weakly continuous on t h e bounded subsets o f
(2)
Af
E, there e x i s t s a net
converges t o
i s weakly u n i f o r m l y
,
E CmWb(E;F)
belongs t o the closure of
t h e r e e x i s t s a weak neighbourhood
x E K Let
B1 =
B1
f
B
(ui(x))
E.
containing
c o n t nuous on
f
such t h a t
and
bounded subset o f of
I , then
a weakly compact subset o f
E' I E
in
E' I E
8
If f
b.w.a.p.
).
Twc
9
Given
(ui)ieI
E be a Banach space w i t h t h e
Let
Lemma.
{dJf(x) : x E K
By lemma 7.3.1,
U p o f 0 i n E such
t h e r e e x i s t s a weak neighbourhood
that
f o r every
x e K Moreover
continuous on
that i f
K
,z e
B
, t h e mapping d j f : E
+
and f o r e v e r y
y
and weakly c o n t i n u o u s on
with
y
Pwb(JE;F) B
( 3 ) t h e r e e x i s t s a weak neighbourhood x e K , y E B and x - y e U 3 then
Z E U P .
i s weakly u n i f o r m l y
. Thus U3
of
0
in
E,
such
1
146
Chapter 7
U = U1
Put
,
x E K
7.3.3
-
x
ui(x)
Theorem.
n U p n Us.
E U.
Let
If
Let
f E C:b(E;F)
> 0.
Let
El
,u
= u(E).
i t follows t h a t
E E’ P E
e u(K).
.
Hence i f
Now, s i n c e
p
,
,
o
x
Since
Cwb(E;F).
Kc E
b e w e a k l y compact and
m
Cwb(E1;F)
there exists
,y
E K
u E Pf(E;F)
and
For each
3.w.a.p.
m
Ifwe d e n o t e t h e r e s t r i c t i o n o f
g E C:,(El;F)
theorem ( 1 . 1 . 2 )
Weierstrass’
i 6 I such t h a t f o r e v e r y x,y E K , i t f o l l o w s t h a t
E be a Banach space with the
Proof.
7.4.
u = ui,
m F , Pf(E;F) i s T~~ dense i n
Banach space
E
There e x i s t s
We d e n o t e
p
0
e P(E1;F)
El
to
f
Cm(E1;F)
by g,
, from
such t h a t
5 j I. m
t h e r e s u l t f o l l o w s f r o m lemma
.3.2
‘#
Notes, remarks and r e f e r e n c e s . The b a s i c r e f e r e n c e s o f t h i s c h a p t e r a r e Gomez [ 2 1 , Gomez-L avona
[ l l , V a l d i v i a [ 2 1 , J o s e f s o n 111, N i s s e n z w e i g 111 and L l a v o n a [31. I n t h e p r o o f o f lemma 7 . 1 . 6 , t h e h y p o t h e s i s t h a t
E
i s a real
Banach space i s e s s e n t i a l because i t i s n o t p o s s i b l e t o f i n d an e n t i r e f u n c t i o n w h i c h v e r i f i e s t h e c o n d i t i o n s t h a t we have demanded on @ Dineen 111 and
has
Hwbu(E;F),
p r o v e d t h a t f o r h o l o m o r p h i c f u n c t i o n spaces an a n a l o g y o f theorem 7.1.7
.
Hwb( E;F)
i s f a l s e , showing
that
Approximation o f weakly d i f f e r e n t i a b l e f u n c t i o n s
147
Hwb(cO) = Hwbu(cO). However, t h e q u e s t i o n remains open o f when b o t h spaces Hwb(E;F) and Hwbu(E;F) c o i n c i d e . On t h e o t h e r hand, n o t e t h a t t h e f u n c t i o n s appear i n t h e p r o o f o f lemma 7.1.6,
are actually
and
f
g
m
m
in
, which
Cwb(E) =
fl C:b(E).
m= 1 Thus, lemma 7.1.6 4.3.6
says t h a t
relled for
and theorem 7.1.7 Cib(E;F)
remain v a l i d f o r
m = m
,
Corollary
i s always b a r r e l l e d , w h i l e
rn > 1 o n l y when
E
C:b(E;F) i s bay i s r e f l e x i v e .(See theorem 7.1.13).
We do n o t know an example o f a Banach space w i t h t h e approxima t i o n p r o p e r t y which does n o t have t h e b.w.a.p.. m The space (C:b(E;F); T ~ i~s g)e n e r a l l y n o t complete. F o r i n s tance, l e t
E = 1' and f o r every
I t i s obvious t h a t
E
fN 6 C l b ( l l )
, then there e x i s t s No
i t follows
lxnl < 1
if
E
N
.
111 ( N o
n > No
.
6
U
If
let
K
> m)
fN be d e f i n e d as:
i s a weakly compact subset o f such t h a t f o r e v e r y
Hence if N
, N'
6
U ,N
x = ( x ~ ) EK, > N' > No
we have
Thus ( f N ) c C:b(ll) because t h e f u n c t i o n
i s a Cauchy sequence. However ( f N ) d o e s n o t converge f
d e f i n e d by: ( x = (x,)
does n o t belong t o
C!b(ll)
.
6
1')
(See p r o o f o f theorem 7.1.10).
want t o f i n d t h e c o m p l e t i o n o f t h e space (Pf(E;F), those found i n s e c t i o n s 3.2
and
3.3
i n chapter 3
Thus i f we
rn ) analogous t o T wc f o r (Pf(E;F), :T
),
Chapter 7
148
t h e Hadamard weakly d i f f e r e n t i a b l e f u n c t i o n concept must be i n t r o d u c e d . We w i l l n o t go i n t o d e t a i l on t h i s s u b j e c t because t h e techniques a r e s i m i l a r t o those used i n t h a t c h a p t e r . Let
E
and
F
A c E.
convex H a u s d o r f f space and
weakly continuous i f f o r each in
,...,
X, t h e r e a r e
1 4 1 ~ ( x - y ) I< 6 f o r a l l
X
be two r e a l Banach spaces,
A function
x e A
f
6 > 0
and
X
i s s a i d t o be
E
from
V
. We
to
X
0
of
y e A
such t h a t i f
then ( f ( x ) - f ( y ) ) E V
i = l,Z,...,n,
Cwk(E;X) t h e space o f a l l
-t
and each neighbourhood
E’
I$I~ i n
f : A
a real locally
,
denote as
which a r e weakly continuous
when r e s t r i c t e d t o weakly compact s e t s . For each
j
e N , we d e f i n e Pwk(jE;F)
as t h e space o f a l l
.
j-homogeneous continuous p o l y n o m i a l s which belong t o Cwk(E;F) We endow Cwk(E;F) and Pwk(jE;F) w i t h t h e topology o f u n i f o r m convergence E.
on weakly compact subsets o f
I t i s n o t hard t o check t h a t
endowed w i t h t h i s t o p o l o g y i s complete. (See ( 4 . 4 . 3 ) ) .
, that
(4.4.5)
Pwk(JE;F) = Pwsc(JE;F)
n o t c o n t a i n a copy o f However, Pwb(JE;F)
1’ then
3
Pwb(JE;F)
Pwk(JE;F)
I t i s known,
.
Also, i f
Pwk(jE;F) = Pwb(JE;F)
E
see does
(See ( 4 . 4 . 7 ) ) .
i s i n general p r o p e r l y c o n t a i n e d i n
P ~ ~ ( ~ E .; FF o) r
m
instance
C
p(x) =
xi
n=l polynomial such t h a t
,
x = (x,)
l’, i s a 2-homogeneous continuous
8
p e P w S c ( * l 1 ) = P(’1’)
(See comments b e f o r e p r o p o s i t i o n 4.5.2 7.4.1
Definition.
differentiable tions : (i)
f 6 C,k(E;X).
(ii)
For euery
a e
E
-1
r(f,a,Ex)
=
p JL! Pwb(211) =Pwbu(211)
-+
X
i s said t o be Haa‘amard weakly
if it s a t i s f i e s the following con@
E , there e x i s t s
compact s e t K i n
f o r euery weakly
lim
f : E
A function
(Hw-differentiable)
and
and theorem 4.5.9).
u 6 L(E;X)
such t hat
E
o
O E’
uniformly with respect t o defined by r(f,a,x)
= f(a
x E K
, where
+ x) - f(a)
-
the “remainder”
u(x),
r ( f ,a,x)
is
149
Approximation o f weakly d i f f e r e n t i a b l e f u n c t i o n s
u
i s c a l l e d the Hw-derivative
f'(a)
instead o f
of
f at
We w i l l w r i t e
a.
df(a)
u.
If m o N , f i s s a i d m-times H w - d i f f e r e n t i a b l e , i f (m-1)-times H w - d i f f e r e n t i a b l e and dm-lf : E Pwk(m - 1 E;X) is -+
f e r e n t i a b l e , where
f is Hw-dif-
d"f = f.
i s s a i d t o be
f
or
m-times Hw-continuously d i f f e r e n t i a b l e , i f
i s m-times H w - d i f f e r e n t i a b l e and
dmf : E
+
P (mE;X) wk
f
belongs t o
Cwk(E;Pwk(mE;X)). Condition ( i i ) (E;a(E;E'))
to
says t h a t
i s Hadamard d i f f e r e n t i a b l e f r o m
f
X. (See Yamamuro [ l l ) .
The space o f a l l m-times Hw-continuously d i f f e r e n t i a b l e f u n c t i o n s f : E
-+
X
We endow
7.4.2
m
i s denoted by CFow(E;X)
Ccow(E;X).
with the
T:~
.
I f m = O , CgOw (E;X) = Cwk(E;X) t o p o l o g y d e f i n e d a n a l o g o u s l y t o (7.1.11)
Definition. A Banach space E i s said to have t h e compact weak K
approximation property (c.w. a . p ) , i f f o r every weakly compact subset of E
, there
(
exists a net
(i)
(ui(x))
(ii)
U
converges t o : x B K
{ui(x)
ioI
of
that: ~ c )E l ~B E~ such ~
u
x
, weakly
uniformly on
x
6
K.
1 i s a relatiueZy weakly compact subset
E. C b v i o u s l y t h i s p r o p e r t y i m p l i e s t h e b.w.a.p.
(See
7.2.1).
If
E
i s a r e f l e x i v e Banach space t h e n b o t h p r o p e r t i e s c o i n c i d e .
I t i s easy t o see t h a t f o r Banach spaces E', i f
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 t h e n
E
E
w i t h separable dual
has t h e
c.w.a.p..
(See Gomez 12 I). The p r o o f i s o m i t t e d i n t h e n e x t theorem due t o i t b e i n g s i m i l a r t o those o f theorems 3.2.10 and 3.3.3.
7.4.3
Theorem.
p l e t i o n of
Let
E
,F
be two real Banach spaces.
(1)
(C;,(E;F)
(2)
If E has the c.w.a.p., Pf(E;F).
;rEc)
i s complete. then
Cm (E;F) cow
is t h e
m corn-
This Page Intentionally Left Blank
151
Chapter 8
SPACES OF DIFFERENTIABLE FUNCTIONS THE APPROXIMATION PROPERTY
I n Aron-Schottenloher space
E
[ll
AND
i t i s proven t h a t a complex Banach
v e r i f i e s t h e Grothendieck’s a p p r o x i m a t i o n p r o p e r t y , i f and o n l y
if t h e space
(H(E); T:)
o f a n a l y t i c mappings on
E
w i t h t h e compact-
open t o p o l o g y v e r i f i e 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 . T h i s r e s u l t was proven using the fact t h a t space
(H(E‘);
T;)
E
F = (H(E;F);T:)
f o r any complex Banach
F, and t h e n u s i n g a c h a r a c t e r 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
i n terms of t h e
E-product,
see (0.5 , 5 ) ,
T h i s same q u e s t i o n , i n t h e r e a l case, i s s t u d i e d i n t h i s c h a p t e r f o r the continuously d i f f e r e n t i a b l e f u n c t i o n classes introduced i n chapters
3,5,6 and 7. I n t h e f i n i t e dimensional case t h e problem i s solved. I n f a c t , m n i n t h i s case t h e f o r m u l a (Cm(R n ) ; T): f o r any r e a l E F = (C (R ;F);T:) Banach space F i s known. See ( 0 . 5 . 1 ) . On t h e o t h e r hand, W e i e r s t r a s s ’ m n theorem 1.1.2 t e l l s us t h a t C (R ) 81 F i s r:-dense i n Cm(Rn;F). Conm n s e q u e n t l y ( C (R );T:) s a t i s f i e 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 f o r a l l m 6 i; m m see ( 0 . 5 . 5 ) . Note t h a t i n t h i s case T: = T : = T - ~ Twc *
8.1.
e r o d u c t s o f continuously d i f f e r e n t i a b l e function
spaces.
Agpl i c a t i o n s ,
I n t h i s section the
€-product o f continuously d i f f e r e n t i a b l e
f u n c t i o n spaces i s s t u d i e d . Thus, r e s u l t s about t h e a p p r o x i m a t i o n prope r t y i n such spaces a r e found. Let
8.1.1
E
and
Theorem. The
F E
be two r e a l Banach spaces, and
-.product of
m m ( cwbu( E) ; T b ) and m
is topozogicazzy isomorphic to (c!~~(E;F) ;T b ) .
F,
et
m e
W.
Chapter 8
152
Proof. Let where
$
f
m Cwbu(E;F). D e f i n e
E
F'.
E
B c E
For
From ( 4 . 1 . 1 ) and ( 5 . 2 . 4 ) of
F.
Also i f
$
T
E
, Tf
(x E E
,$
F = L(F;
E
Since on
B(F')
f,
0
let
i s a precompact subset
Lj
5 1 1 5 1.
:llyll
. Conversely
;))!T
;(C:bU(E)
We d e f i n e
; T,,)).
F').
E
= SUP { l $ d J f ( x ) ( y ) l
Tf($) = $
.
1)
e Lj ,
m
L(Fh ; (C:bu(E)
/I Y I I 2
by
5 m ,
i t i s easy t o see t h a t
C:bU(E)
E
C:bU(E)
0
11 d J ( T f ( $ ) ) ( x ) l I Therefore
B
-f
j E I, j
bounded, : x E
Lj = {djf(x)(y)
Tf : F '
fT = f : E
F" by f ( x ) ( $ ) = T ( $ ) ( x ) ,
-+
(unit ball of
, let
F')
the
a(F';F)
topology c o i n c i d e s w i t h t h e compact u n i f o r m convergence topology, by t h e continuity o f
o(F';F)-continuous. t i n u o u s and t h e n For
j =
x e E
i t f o l l o w s t h a t f o r every
T
,for
Therefore
f (x)
E
every
E
E
: E
-f
x
f(x)IB(F') f(x)
is
is
a(F';F)-con-
F.
O,l,...,m
define
g = g
j
P(JE;F)
by
$ ( g ( x ) ( y ) ) = d j T $ ( x ) ( y ) , f o r x,y E E , $ 6 F ' . (Note t h a t when j = O g = f ) . As above, we have t h a t f o r e v e r y x,y E E , g ( x ) ( y ) E F. We f i r s t show t h a t
II g ( x ) l l
x 8 E
i s i n f a c t an element o f
g(x)
= SUP { l @ ( g ( x ) ( y ) ) I :
11 $ 1 1
P(JE;F).
5 1 ,
For
IIYI/
5 1)
m
T 6 Cwbu ( E ) E F , t h e r e i s a compact, convex balanced s e t that i f $ E Lo, t h e n
Since
L c k
t h a t if
1 1 $11 5
and
g(x)
E
B
k > 0
f o r some
l/k P(JE;F).
, then I
By t h e c o n t i n u i t y o f
i s the u n i t b a l l o f
$g(x)(y)I
Now we w i l l show t h a t ed.
(B
5 1 for
11 yII 5
g E Cwbu(E;P(jE;F)).
T, f o r some a b s o l u t e l y
E
.
Since
=
F
F), i t
such
follows
B = E be bound-
convex
K O
every
1. Thus11 g ( x ) l k l / k
Let
, whenever $ E then 11 d J ( T $ ) ( x ) l I 5 1 ( x if x E B , h E E , 11 hi1 5 1 and $ E then
K c F
L
,
compact
B).
set
Therefore
,
K O ,
Thus
,
if x E
B ,h
8
E
,I1
hll
5 1 then g(x)(h)
E
K""=
K.
I n particu-
Spaces of d i f f e r e n t i a b l e f u n c t i o n s
i s compact.
c u l a r , we have shown t h a t
g(x)
o s i t i o n 4.1.3,
,..., $, t h e n [I z / ( 5
such t h a t i f
there exist
z 8 K
,
i.K
6 F'
$1
153
As i n t h e p r o o f o f prop-
, /I
syp ( Q i ( z ) [ +
.
E
1
, dJ(T$i)
qi (i= l,...,n)
For each
u n i f o r m l y continuous on bounded s e t s . Q~ c
6i > 0
and
El
1)
( @ e Q ~ ), t h e n and
6 = min 6
sup hsE llhll:
-
proving that
Then i f
x,y E
SUP
i
n
Let
< 6
$i(g(Y)(h)))
0 = i =U l
(4
'i
E
=
+ E
I dJ(T$i)(x)(h) -
dj(T$i)(y)(h)l
+ E
< 2~
g E Cwbu(E;P(jE;F)).
4.6.1,
g(x)
m T @ E CwbU(E) f o r
(x E E
Pwbu (JE;
6
and ( b )
E F'
,
,j5
m
, 4
E F').
.
But
Therefore
, since g(x)
( x E E ; j 5 rn)
. So
,
i s compact,
we have proven
i n c o r o l l a r y 5.2.4.
dJf(x) = g.(x) f o r x E E J j =O Assuming t h e r e s u l t f o r
Now we w i l l show t h a t The r e s u l t i s t r i v i a l f o r
4
FiW*)
g ( x ) e Pwbu(JE;F)
satisfies (a)
we have t h a t
,[@ ( x - y ) /
1
it follows that
g
i s weakly
1
$ ( g ( x ) ) = dJ T @ ( x ) E Pwbu(JE)
that
E.
I @(x-y)I
B ,
-
Pwbu(JE)
x,y E B
dJ(T q i ) ( y ) l l <
)$i(g(x)(h))
Note t h a t s i n c e
by lemma
-
+
Hence, t h e r e i s a f i n i t e s e t
such t h a t whenever
dJ(TQi)(x)
SUP 1
sup heE llhll:
=
i .
,
: E
...,n )
= 1 ( i =1,
.
,j p
=
0,1,
...,rn.
5 j- 1
<
m,
Chapter 8
154
-
x,h,+)
continuity o f
which tends t o 0 as
gj(x)ll
gj.
Hence we have proven t h a t
f
6
F i n a l l y , i t i s r o u t i n e t o v e r i f y t h a t t h e mapping t o p o l o g i c a l isomorphism between
As a consequence o f 8.1.2 CoroZlary.
T:)
+
0
, by
the
Cmwbu (E;F). f *
Tf
establishes a m m and (Cwbu(E); T ~ E) F,
( 0 . 5 . 5 ) t h e f o l l o w i n g c o r o l l a r y i s immediate.
m E) ; T b)
(c:bu(
onZy if f o r every Banach space
CoroZZary.
has t h e approzimution property, if and , c i b u ( ~ )B F is dense i n ( c mw b u ( ~ ; ~ ) ;mb). T
F
has t h e approximation property, f o r
( Cwbu (E) ; T b ) E ,
every Banach space Proof.
By theorem 4.3.7
Cwbu(E)
E
F
;
hl/
T * fT.#
whose i n v e r s e i s
8.1.3
(C:bU(E;F)
11
Cwbu(E) a F
and theorem 8.1.1,
i s dense i n
and t h e r e s u l t f o l l o w s . #
The p r o o f o f t h e n e x t r e s u l t i s s i m i l a r t o t h a t o f ( 8 . 1 . 1 ) .
8.1.4
Theorem.
The
€-product of
is topoZogicaZZy isomorphic to Therefore only i f
C!(E)
B F
,
m ( C r ( E ) ; T ~ ) and ; T mu )
(c'$E;F)
(CF(E); ):T i s dense i n
F,(C~(E);T:)
E
F ,
.
has t h e approximation p r o p e r t y , i f and m (C!(E;F) ; T ~ )f o r every Banach space F
(see ( 0 . 5 . 5 ) ) . I n t h e case o f t h e space
(Cm(E); T:)
an
E
-
p r o d u c t formula
s i m i l a r t o t h e p r e v i o u s ones i s unknown. For t h i s reason Bombal-LlavonaLll CFo(E;F) , see d e f i n i t i o n 3.2.8, w h i l e t r y i n g t o i n t r o d u c e d t h e space m study when t h e space (Cm(E) ; T ~ )v e r i f i e s t h e approximation p r o p e r t y .
A s was seen i n (3.2.10),
CFo(E;F)
is
r e s u l t was proven i n Bombal-Llavona [ l
: T
I .
-
complete and t h e f o l l o w i n g
155
Spaces o f d i f f e r e n t i a b l e f u n c t i o n s
m
8.1.5. Theorem. The E-product of ( C c o ( E ) ; is topoZogicaZZy isomorphic to ( c:~( E;F) ; T ): m (Cm(E) ; T c )
CoroZZary.
8.1.6.
, if
(hence f o r a l l ) m >- 1 Proof.
m=
f o r some
m > 1
, define
a
has the approsirnation property f o r some
N , only
m e
(Cm(E) ; T ):
, If
E ' c Cm(E)
i f we r e g a r d
F),
E
and only if E has the approximation property.
We prove t h e r e s u l t f o r
necessary f o r
m and F,(Cco(E);~:)
)!T
ir
: Cm(E) + E '
with the
minor modifications being
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 by
n ( f ) = df(0).
induced
E'
i s a continuous p r o j e c t i o n . T h e r e f o r e ,
-tT
-
topology
Note t h a t
, then
n
w i t h t h e induced t o p o l o g y i s
a complemented subspace o f Cm(E) and hence 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 .
E'
However, s i n c e t h e induced t o p o l o g y on
) that
topology, i t f o l l o w s f r o m (0.5.6
i s j u s t t h e compact-open E
has t h e a p p r o x i m a t i o n prop-
erty.
(3.3.'2(1)), f o r e v e r y Banach space F , C:o(E)
Conversely, by
is that
m m - t C - dense i n Cco(E;F) (C:o(E)
;
m
T
)~
.
rn F
T h e r e f o r e , by theorem 8.1.5 i t f o l l o w s
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 . (See ( 0 . 5 . 5
)).
The r e s u l t f o l l o w s f r o m theorem 3.3.3
and t h e general r e s u l t : i f t h e
c o m p l e t i o n o f a l o c a l l y convex space
X
then
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 . # The p r o o f o f theorem 8.1.5
8.2
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 ,
has been o m i t t e d because i n s e c t i o n
an i n t r i n s i c p r o o f o f c o r o l l a r y 8.1.6
i s done w i t h o u t u s i n g t h e
E - p r o d u c t technique.
8.2.
On 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 n spaces o f c o n t i n u o u s l y d i f f e r e n t i a b l e functions. T h i s s e c t i o n i s d e d i c a t e d t o t h e s t u d y o f t h e a p p r o x i m a t i o n prop-
e r t y i n c o n t i n u o u s l y d i f f e r e n t i a b l e f u n c t i o n spaces u s i n g i n t r i n s i c a l l y d i f f e r e n t techniques f r o m those o f t h e E-product.
T h i s approach t o t h e
problem has some advantages, such as i n t h e case o f t h e space (Cm(E); For t h i s case, see s e c t i o n 8.1,
T:).
we have a l r e a d y seen how we had t o use
d i f f e r e n t i a b i l i t y i n o r d e r t o s o l v e t h e problem w i t h t h e
Hadamard E-product technique. T h i s s e c t i o n s begins w i t h 8.1.6. Let
E
be a r e a l Banach space, m E
Aron's proof o f Corollary
.
156
Chapter 8
has the approximation property f o r some ( Cm( E) ; T ): (hence f o r a l l ) m 2 1 i f and only i f E has the approximation property.
8.2.1
Theorem.
Proof.
N, o n l y minor m o d i f i c a t i o n s being m I f ( Cm( E) ; T c ) has t h e approximation p r o p e r t y i t i s easy t o show t h a t E has t h e approximation
f o r some m
.
m =m
2 1 ,
p r o p e r t y . (See p r o o f o f let a
m E
We p r o v e t h e r e s u l t f o r
necessary f o r
(8.1.6)).
To prove t h e converse, l e t
T c (Cm(E) ;):T
be a compact subset o f
Given
K
x,y E
d > 0, such t h a t i f
1 1 Y ' - YII <
6
E.
K
and
and ),!,y(
that for a l l
E E
claim t h a t there i s
with
11
x'
-
xII < 6
in n E
E, (x,)
N , 1)
and
xn-x,!,II
(y,)
j
in
5 m , t h e r e e x i s t sequences K, and
< l / n , ) ) y n-y;\l
(f,)
in
,
< l/n
(X,),
.
T,
such
and
Now, K 1 = K z = (<) 1 L i = 2 E ~ d J f n ( X n ) ( Y n ) - dJf,(X,!,)(y,!,)/ and L Z = a r e compact i n E, so t h a t t h e seminorm p d e f i n e d by
is T :
-
continuous. However, f o r i n f i n i t e
c o n t r a d i c t i n g t h e precompactness o f Let
,
, then
I n f a c t , if t h i s i s f a l s e , t h e n f o r some (x,!,)
, we
E >O
x',y'
be precompact and
u E E' B E
T.
m
and
n
(YJ
values, p ( f n - f m
Thus (1) h o l d s .
such t h a t f o r a l l
x E K
,/I ux
-
XI)<
6
.
Def ne
I$ : Cm(E) -+ Cm(E) by @ ( f ) = f 0 u . Now c o n s i d e r t h e f a m i l y T l u ( E ) o f f u n c t i o n s i n T r e s t r i c t e d t o t h e f i n i t e dimensional space u ( E ) . Since Tlu(E)
i s a compact subset o f
C"(u(E)),
which 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 , t h e r e e x i s t s a continuous l i n e a r mapping o f f i n i t e r a n k @ :
Cm(u(E))
such t h a t f o r a l l (2)
-L
g E T
supIldJg(x)(y)
-
Spaces o f d i f f e r e n t i a b l e f u n c t i o n s
F i n a l l y , t h e mapping t a k i n g
157
e Cm(E) i n t o $ ( f l u ( E ) ) , u
f
i s f i n i t e r a n k , l i n e a r and continuous, and i f
f e T, x,y
f
K
e Cm(E)
then
+ BY m i d d l e term
any
1) and ( 2 ) , t h e f i r s t and t h i r d terms a r e < E, w h i l e t h e m s 0. Thus, (Cm(E) ; T ~ )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
m.# We assume t h a t
E ' has t h e bounded a p p r o x i m a t i o n p r o p e r t y . I n t h i s
case ( 5 . 3 . 3 ) , (6.1.4),(8.1.2) and (8.1.4) i m p l y t h a t (C:bU(E); !T ) and m (C!(E) ; ' r U ) have t h e a p p r o x i m a t i o n p r o p e r t y . The p r e v i o u s f a c t s can be found by t h e f o l l o w i n g r e s u l t s , u s i n g lemmas ( 5 . 3 . 2 ) and ( 6 . 1 . 7 ) , the
E
8.2.2
without
- p r o d u c t technique.
Let E
Theorem.
be a reaZ Banach space w i t h E ' having the bounded m ( C I b U ( E ) ; T ) has
approximation property, and Zet m > 0. Then t h e space t h e approximation property, Proof. E
> 0
with
Let
, and
I ] n.11 5
Let set o f
m T c Cwbu(E)
be a precompact subset.
e U , j 5 m be g i v e n
j C
,
.
lemma 5 . 3 . 2 ,
Since
and c o n s i d e r
T
I
there i s
bounded, TI
e E ' B E,
di(f/E,)(.r(x))
ED
, which
i s a precompact sub-
Cm(Eo) 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 , t h e r e i s $ : Cm(Eo) -+ Cm(Eo) such t h a t
a f i n i t e rank continuous l i n e a r mapping
However,
B cE
such t h a t
E D = u(E)
Cm(ED)
. By
Let
= dif(a(x))I
ED
,
and t h e r e f o r e ( 2 ) i m p l i e s
158
Chapter 8
1) dif(TI(x))
(3)
n TI-
di($(f/EO))(n(x)),
F i n a l l y (1) and ( 3 )
TI\\
< ~ / 2, ( f E T,x E B,i
5
j).
imply
f
L e f t t o be mentioned i s t h a t t h e mapping f i n i t e rank continuous l i n e a r map from
C:bu(E)
-+
into
$ ( f (E,,). TI i s a m Cwbu(E).#
The f o l l o w i n g i s a p a r t i a l converse o f theorem 8.2.2.
m m Theorem. I f (Cwbu(E) ; T b ) some m > 0 , then so does E ' . 8.2.3
has t h e upprorimation property f o r
L e t us c o n s i d e r t h e mapping
Proof.
TI
: C:bu(E)
-+
E' c
C:bu(E)
r ( f ) = d f ( 0 ) , f o r a l l f E CmwbU(E). I t i s n o t hard t o see i s a continuous l i n e a r p r o j e c t i o n o n t o E ' , on which t h e induced m topology of CwbU(E) agrees w i t h t h e norm t o p o l o g y . (See p r o o f o f (8.1.6)).
d e f i n e d by that
T
Hence
(El;
1 1 11
)
(CIbU(E) ;)!T
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 , i f
does.#
S i m i l a r l y we have t h e f o l l o w i n g r e s u l t s . 8.2.4.
Theorem. Let E be a real Banach space w i t h E ' having t h e bounded
approximation property. f o r a7~Z m E Proof. j E
Let
N ,j 2
m
(C!
Then
K. T c C;(E)
m
, and
be a precompact subset, and l e t
> 0
E
be g i v e n .
By lemma 6.1.7,
f i n i t e rank continuous l i n e a r o p e r a t o r (1) Let
11
has the approximation property,
E) ; T u )
di(f
E o = n(E)
Cm(Eo). S i n c e
0
TI)(X)
-
dif(x)Il c
and c o n s i d e r
TT
: E
~ / 2
-+
Kc E
compact,
there exists a
E , such t h a t
(f E T
,x
E K
, i 5 j).
T I E o , which i s a precompact subset o f
Cm(Eo) 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 , t h e r e i s a f i n i t e
rank continuous l i n e a r o p e r a t o r
$ : Cm(Eo)
+
Cm(E,,)
such t h a t
Spaces of differentiable functions
for all f s T , x E K , and i 2 j the chain rule, we get that
. Combining
A final observation is that f continuous linear operator from (C;(E) 8.2.5 Theorem. If (Ct(E) ;):T E’.
159
(1) and (2) and applying
$(flEo). n is a finite ; T m ~ )into itself.#
+
rank
has t h e approximation property, then
so does
Proof. As in (8.2.3) , we prove that f projection of (c:(.E) ; T:) onto E ’ .#
+
df(0)
is a continuous linear
8.3. Notes and references.
In this chapter the results are taken from Aron [13 - [ 2 1 , AronProlla [ l ] and Bombal-Llavona 111 . Also see Bombal 111 . As has already been seen in (8.1.5) the E-product of (CFo(E);r;) and F i s topologically isomorphic to (C:o(E;F) ;T: ). Following techniques similar to those used by Bombal-Llavona [l] m to show (8.1.5) Gomez “21 proved that the E-product of (C:ow(E) ; -rWc ) m and F is topologically isomorphic t o (C;ow(E;F) ;T ~ ~ ) . Because of these results, it i s easy to prove the following. 8.3.1 Theorem. m m 1 .- ( Cco( E) ; T ) for a l l ) m
2 1
2 . - Let
(cFOw(E)
m
;T W C
has the approximation property f o r some (hence if and only if E has the approximation property .
E be a r e a l Banach space u i t h the c.w.a.p. Then m ( r e s p e c t . (C;b(E) ; T ~ has ~ ) t h e approximation property.
This Page Intentionally Left Blank
161
Chapter 9
POLYNOMIAL ALGEBRAS OF CONTINUOUSLY D I F F E R E N T I A B L E FUNCTIONS
As we have a l r e a d y seen i n c h a p t e r 1, i n 1949 Nachbin found t h e d i f f e r e n t i a b l e v e r s i o n o f t h e Stone-Weierstrass theorem. See theorem 1 . 2 . 1 and s e c t i o n 1.8. I n r e c e n t y e a r s many a u t h o r s ( m y s e l f i n c l u d e d ) have worked on t h e a p p r o x i m a t i o n o f c o n t i n u o u s 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 d e f i n e d on Banach spaces. Almost a l l o f these c o n t r i b u t i o n s o r i g i n a t e d from t r y i n g t o extend N a c h b i n ' s theorem 1.2.1 Let
E
and
t o a l g e b r a s o f f u n c t i o n d e f i n e d on Banach spaces.
F
be two r e a l Banach spaces. I n c h a p t e r s 3,5,6
7
we have s t u d i e d t h e c o m p l e t i o n o f t h e space Pf(E;F) m m m m l o g i e s T C , T b , -ru and T W C o b t a i n i n g t h e c l a s s e s
m Ck(E;F)
and
.
C(,E ;F ;)
(See (3.3.3),
(5.3.4),
and
i n v a r i o u s topo-
, c!~~(E;F)
c:~(E;F)
(6.1.8)
and
(7.4.3)).
Each a p p r o x i m a t i o n theorem was g o t t e n f o l l o w i n g t h e same method. Given a
Cm
u : E
-t
strass'
p
f : E
function
0
u
E
+
F
, f was approximated by a f u n c t i o n f
was a f i n i t e r a n k c o n t i n u o u s l i n e a r map. theorem
, where p
1.1.2 8
t o g e t an a p p r o x i m a t i o n o f
,
Pf(En;F)
En = u ( E ) .
As
p
0
u
0
, where
Then, we used Weierf
0
u
by a f u n c t i o n
u 6 Pf(E;F)
t h e re-
s u l t follows. I n t h i s c h a p t e r w i t h t h e aim o f f i n d i n g an e x t e n s i o n o f N a c h b i n ' s theorem we w i l l f o l l o w a procedure s i m i l a r t o t h a t g i v e n i n t h e p r o o f o f (3.1.6),(5.3.3), algebra
A
(6.1.4)
and ( 7 . 3 . 3 ) ,
replacing
and u s i n g N a c h b i n ' s theorem 1.2.1
Pf(E;F)
by a polynomial
instead o f Weierstrass'
theorem 1 . 1 . 2 . Let The s e t
A
be a polynomial a l g e b r a o f
M = {@ h : @
6
M IP F C A , see ( 0 . 4 . 1 ) .
F'
,
Cm
functions
h 6 A } i s a subalgebra o f
If i n addition, A
h : E
Cm(E)
-f
F.
such t h a t
i s a Nachbin polynomial
algebra, i t i s easy t o check t h a t f o r any f i n i t e dimensional subspace of
E, M I E o i s a Nachbin polynomial a l g e b r a i n
Nachbin's theorem 1 . 2 . 1 ,
MIEo
i s dense i n
Eo
Cm(Eo). Thus,applying
Cm(Eo) w i t h r e s p e c t t o t h e
Chapter 9
162
:T
t o p o l o g y o f u n i f o r m convergence o f a f u n c t i o n and i t s f i r s t
t i v e s on compact subsets o f IF
(MIE,)
i s dense i n
Eo.
Cm(Eo;F)
g
o
u
, where g
deriva-
By a standard reasoning, i t f o l l o w s t h a t
.
Thus, a s i m p l e a p p l i c a t i o n o f t h e
c h a i n r u l e w i l l be s u f f i c i e n t t o prove t h a t i n t h i s case approximated by
m
E A.
f
0
u
can be
Hence, t o show t h e r e s u l t we w i l l
need a h y p o t h e s i s l i k e t h i s : "For e v e r y f i n i t e r a n k continuous l i n e a r map u : E
-+
, and
E
closure o f 9.1.
every
g B A
, the
composition
g
u
0
belongs t o t h e
A".
Polynomial algebras. E x t e n s i o n o f Nachbin's theorem t o i n f i n i t e dimensional Banach spaces. R e c a l l t h a t a subspace
a l g e b r a ( 0..4.1
Iy
i f f o r every
A c Cm(E;F) g E A
i s c a l l e d a polynomial p E Pf(F;F)
and
t h e composition
p o g ~ A . The n e x t r e s u l t i s an e x t e n s i o n o f Nachbin's theorem 1.2.1 m t h e c l a s s (Cm(E;F) ; T )~ . 9.1.1
Theorem. Let E and F be two r e a l Banach spaces, with E
t h e approximation property, and l e t
-
A c Cm(E;F) i s !T fa) A
,i
dense
m > 0
. A polynomial
to
having
algebra
f , and only i f , the following holds:
i s a Nachbin polynomial algebra.
map u : E -+ E m u belongs t o the -rc-cZosure o f
( b ) f o r evepy f i n i t e rank continuous l i n e a r and every
g E A
, the
composition
g
0
A. Proof.
If
A
is
rF-dense, t h e n ( b ) i s obvious and ( a )
ified.
f
Conversely, l e t and
K c E
6
Cm(E;F)
.
Given
compact, according t o lemma 3.1.3
l i n e a r operator
u :E
+
E
j o
111
i s e a s i l y verj
5 m ,
E
> 0
a f i n i t e rank continuous
e x i s t s such t h a t
by ( 0.4.1 ) the set Since A i s a polynomial a l g e b r a M = {I$ 0 f : I$ 6 F ' , f 6 A3 i s a subalgebra o f Cm(E) such t h a t M e F c A . Since A i s a Nachbin polynomial a l g e b r a , f o r any f i n i t e dimensional subspace E o o f E, (MIEo) c Cm(E,) i s a subalgebra which
Polynomial a l g e b r a s
s a t i s f i e s ( i ) and ( i i ) Cm(Eo). T h e r e f o r e Let
,
i n theorem 1.2.1.
(MIEo) B F
Eo= u(E)
. Since
is
If
g = p B z
,
Thus, (MIEo)
-im u = cT
n(K) c E O
above remarks t o conclude t h a t t h e r e a r e
i t follows that
g 6 A
163
~dense i n
: is T
-
dense i n
C"(E0;F).
i s compact we may a p p l y t h e p E M
,z
E F
so t h a t
and
By t h e c h a i n r u l e , i t f o l l o w s t h a t
F i n a l l y , by h y p o t h e s i s ( b )
From ( . l ) , ( 3 )
and ( 4 )
,
there i s
h E A
such t h a t
we g e t
as desired.#
9.1.1,
E x a c t l y t h e same reasoning t o t h a t used i n t h e p r o o f o f theorem a l l o w s us t o o b t a i n t h e c o r r e s p o n d i n g i n f i n i t e dimensional v e r s i o n s
o f Nachbin's theorem 1.2.1 and
(CITb(E;F)
f o r classes
CITbu(E;F)
; T:),(C:(E:F)
; :T
m ;Twc).
9.1.2
Theorem. Let E and F be two reaZ Banach spaces, w i t h E ' having the bounded approximation property w i t h constant C , and Zet m > 0. A m m poZynomiaZ algebra A c C w b U ( E ; F ) i s T - dense, i f and onZy i f , t h e following holds: la1 A
i s a Nachbin poZynomiaZ aZgebra.
)
164
Chapter 9
( b ) f o r every f i n i t e rank continuous linear mtp TI : E E , C , and every g 6 A , the composition g o n beZongs t o the -+
with
11 1 ~ 1 1 5
m -cb-cZosure of Proof.
If
A
A. i s dense, c o n d i t i o n s [ a ) and ( b )
m f E Cwbu(E;F)
Conversely, l e t each bounded s e t T
a E ' 81 E
Let
with
Eo= T ( E )
theorem 9.1.1,
where 6 > 0
B eE
,E
11 rlI 2
C
. Since
> 0
, such
be given. By lemma 5.3.2,
j
and
are easily verified.
I , j <-
6
for
m, t h e r e i s a mapping
that
IT(B) c E o i s r e l a t i v e l y compact, reasoning as i n
i t follows that there i s
i s chosen so t h a t
g 6 A
, such t h a t
6 <4 3 Cj
By t h e c h a i n r u l e , i t f o l l o w s t h a t
, by hypothesis ( b ) , t h e r e i s h a A such t h a t
Finally
From ( 1 )
, ( 3 ) and ( 4 ) we g e t
as d e s i r e d . # 9.1.3
A
Remark. C o n d i t i o n ( a ) a l o n e o f theorem 9.1.2
t o be dense, as t h e f o l l o w i n g example by
a n o n - r e f l e x i v e Banach space such t h a t p r o p e r t y (such as
E = co)
and l e t
E'
L.
i s not sufficient f o r
Nachbin shows. L e t
E
be
has t h e bounded a p p r o x i m a t i o n
F = R , Let
A
be t h e subalgebra
165
Polynomial a l g e b r a s
where
a
E
6
and
@
c E"
t h e continuous l i n e a r
@
mapping
4
E.
f
+
A
Then, s i n c e
i s the kernel o f
@ ( d f ( a ) ) , i t i s a c l o s e d subalgebra
o f codimension one. Thus, A i s n o t dense i n
C;bU(E).
although i t i s a
Nachbin polynomial a l g e b r a .
9.1.4
Let E and F be reaZ Banach spaces w i t h
Theorem.
E ' having
m e R. A poZynomiaZ algebra the bounded approximation property, Zst m m i s T - dense i n Ck(E;F) i f , and onZy i f , t h e foZZowing conditions are s a t i s f y e d :
A c C;(E;F)
i s a Nachbin poZynomiaZ algebra. for every f i n i t e rank continuous l i n e a r operator
(a) A fbl TI
: E
E,
-+
w7:th
belongs t o the Proof. j c
11 ITII
m
T~
5
g E A
and every - cZosure o f A . C
the composition
The n e c e s s i t y i s c l e a r . S u f f i c i e n c y : L e t
U
j
5 m ,E
> 0
, and
K c E
f
IT
: E
-+
E
o
IT
e CF(E;F) . Given
compact,by lemma 6.1.7
a f i n i t e rank continuous l i n e a r o p e r a t o r
g
with
there exists
1 1 ITII 2
C
,
such t h a t
Since
A
i s a Nachbin polynomial algebra, an argument s i m i l a r t o t h a t
g i v e n i n t h e p r o o f o f ( 9 . 1 . 1 ) and ( 9 . 1 . 2 )
proves t h a t t h e r e i s
such t h a t
h f A
F i n a l l y , by ( b ) , t h e r e e x i s t s
From ( l ) , ( 2 )
and
I / dif(x) -
(3)
such t h a t
we see t h a t
dih(x)ll
which completes t h e p r o o f . #
<
E
(x c K
,i5
j)
g E A
166
Chapter 9
9.1.5
Remark.
As w i t h theorem 9.1.2,
c o n d i t i o n ( a ) o f theorem 9.1.4
i s n o t i n i t s e l f g e n e r a l l y s u f f i c i e n t t o i m p l y t h e d e n s i t y of A . The most obvious example o f a polynomial algebra s a t i s f y i n g
,
c o n d i t i o n s (a) and ( b ) i n theorem 9.1.4 i s , o f course, A = Pf(E;F) a l t h o u g h t h e r e can be o t h e r
, unrelated
E = c o and G = I f E Cm(co) : f
examples. For i n s t a n c e , i f
has bounded s u p p o r t } ,
i s a polynomial a l g e b r a s a t i s f y i n g c o n d i t i o n s ( a ) and ( b ) tersects
Pf(E;F) Finally
A = G
then
P
which in-
i n o n l y 0.
, using
lemma 7.3.2 i n s t e a d o f lemma 3 . 1 . 3 , t h e p r o o f o f
the next r e s u l t
i s similar t o that o f (9.1.1).
9.1.6
Let E and F be two real Banach spaces, with E having
Theorem.
F
the bounded weak approximation property (b.w.a.p.l, and l e t m > 0. A m m polynomial aZgebra A c Cwb(E;F) i s T~~ - dense , i f and only i f , the following holds:
(a1 A i s a Nachbin polynomial algebra. (bl foi' every f i n i t e rank continuous linear map and every of
g a A
, the
g
composition
o
u belongs t o the
71
T
m
~
: E
+
E,
closure ~ -
A.
9.2
Notes
,
remarks and r e f e r e n c e s .
The r e s u l t s o f t h i s c h a p t e r a r e taken from Llavona 121 , AronP r o l l a [11 and
Gomez-Llavona [ l l
.
Regarding t h e beforementioned e f f o r t s t o extend Nachbin's theorem t o a l g e b r a s o f f u n c t i o n s d e f i n e d on Banach spaces, we would l i k e t o b r i n g o u t t h a t t h e f i r s t i m p o r t a n t c o n t r i b u t i o n was made by Lesmes [ l l , see
(3.0.1) and ( 6 . 1 . 2 ) .
T h e r e a f t e r c o n t r i b u t i o n s by P r o l l a [ 2 1
, Llavona
P r o l l a [ 11 , P r o l l a - G u e r r e i r o [ 11 , Llavona [ 21 , A r o n - P r o l l a [ 13
111,
, Llavona
[31 and Gomez-Llavona [ 11 appear . Along t h e l i n e s , works by Nachbin [2] , [31-[ 5 1-[61-[ 71 , Gomez [ 21 and H o r v i t h [ 2 1 , must be p o i n t e d o u t . Regarding theorem 9.1.:
t o our knowledge t h e f o l l o w i n g c o n j e c t u r e
i s open.
9.2.1
Conjecture.
For every g i v e n r e a l Banach space
conditions are equivalent:
E, t h e f o l l o w i n g
167
Polynomial a l g e b r a s
(C 1)
F o r a r b i t r a r y r e a l Banach space F m polynomial a l g e b r a A i s T~ - dense i n Cm(E;F)
A
, m2 1 , i f (and
then every always o n l y i f )
i s a Nachbin polynomial a l g e b r a . (C 2 )
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 .
( C 1 ) i m p l i e s (C 2 ) , see ( 3 . 1 . 2 ) . The c o n j e c t u r e
I t i s known t h a t
t h a t (C 2 ) i m p l i e s (C 1)
i s an a t t e m p t t o improve theorem 9.1.1.
(See
Nachbin [2 I ) . Along one l i n e o f r e s e a r c h a q u e s t i o n e x i s t s on t h e s t u d y o f App r o x i m a t i o n Theory f o r a l g e b r a s o r modules o f c o n t i n u o u s l y d i f f e r e n t i a b l e v e c t o r valued mappings by u s i n g w e i g h t s
.
This question
, however , i s
s t i l l wide open, i n s p i t e o f t h e a v a i l a b l e r e s u l t s . (See Nachbin [21). Theorem 9 . 1 . 1 p r o x i m a t i o n and
u s i n g Yamabe's theorem 1owing r e s u l t s
.
, see
Yamabe [1 I
, Llavona 1 1 1- [21
found t h e f o l -
Theorem. Let E be a r e a l Banach space w i t h the approximation prop-
9.2.2 erty
can be used t o o b t a i n r e s u l t s on simultaneous ap-
i n t e r p o l a t i o n i n d i f f e r e n t i a b l e f u n c t i o n spaces. I n f a c t ,
, and
m E
N.
(a)
A c C m ( E ) be an aZgebra which s a t i s f i e s
Let
.
i n theorem (9.1.1) Then given K c E a compact s e t , E > 0 , {al . ,a P 1 c E , f E Cm(E ) and E o c E f i n i t e dirnensionaZ subspace, t h e existence o f g E A foZlows such t h a t
and
fbl
). .
9.2.3
Corollary.
p r o p e r t y and {a, ,...,ap) ence of
m E 111 c E
g E Pf(E)
Let
. and
E
be a r e a l Banach space w i t h t h e a p p r o x i m a t i o n
Then f o r any EDc E
K c E
compact
,
f e Cm(E)
a f i n i t e dimensional subspace
f o l l o w s such t h a t
,
,
E
> 0
the exist-
This Page Intentionally Left Blank
169
Chapter 1 0
ON THE CLOSURE OF MODULES OF CONTINUOUSLY DIFFERENTIABLE FUNCTIONS
10.1.
E x t e n s i o n o f W h i t n e y ' s i d e a l theorem t o i n f i n i t e dimensional Banach spaces. I n r e c e n t y e a r s s e v e r a l a u t h o r s have s t u d i e d t h e e x t e n s i o n o f
Whitney's i d e a l theorem, (see ( 0 . 2 . 3 )
, (0.2.4)
and ( 0 . 2 . 5 ) )
, for
algebras
of f u n c t i o n s d e f i n e d on i n f i n i t e dimensional Banach spaces. Along t h e s e l i n e s , i n 1976 f o r t h i s case C.S.Guerreit-o [ l l f o u n d one v e r s i o n o f t h i s theorem. This chapter i s dedicated t o the extension o f Whitney's ideal theorem t o an a r b i t r a r y normed space and s c a l a r f u n c t i o n s t o v e c t o r - v a l u e d f u n c t i o n s . We w i l l prove t h a t t h e c l a s s i c p o i n t v e r s i o n o f W h i t n e y ' s f i n i t e dimensional theorem f a i l s even i n t h e case o f r e a l s e p a r a b l e H i l b e r t spaces. The problem o f W h i t n e y ' s i d e a l theorem e x t e n s i o n was b r o u g h t o u t by G u e r r e i r o [ 3 I i n h e r d o c t o r a l t h e s i s ; these i d e a s w i l l be b r i e f l y o u t l i n e d f o l l o w i n g Nachbin's work (see Nachbin [ 4 1 ) . F i r s t o f a l l , a p r e v i o u s q u e s t i o n w i l l come up, when f u n c t i o n s d e f i n e d on i n f i n i t e dimensional v e c t o r spaces a r e used. I n t h e f i n i t e d l m m m on T mensional case, a l l t h e usual t o p o l o g i e s T~ , 'b , T~ , and wc c o n t i n u o u s l y d i f f e r e n t i a b l e f u n c t i o n spaces c o i n c i d e . (See d e f i n i t i o n s
( 3 . 1 . 1 ) , ( 5 . 2 . 6 ) , comments b e f o r e theorem 6 . 1 . 1 and d e f i n i t i o n ( 7 . 1 . 1 1 ) ; o t h e r w i s e , t h e y a r e d i f f e r e n t f o r t h e i n f i n i t e dimensional case. Thus t h e p r e v i o u s problem o f s e l e c t i n g a t o p o l o g y comes up. We o n l y s e l e c t e d t h e compact-compact t o p o l o g y o f o r d e r m i n t r o d u c e d by P r o l l a and Llavona, see c h a p t e r 3
,
since
,
m T~
i n the algebra
case i t seemstobethe most a p p r o p r i a t e f o r s t u d y i n g some a p p r o x i m a t i o n q u e r t i o n s i n i n f i n i t e dimension. From now on, E and spaces Cm(E;F)
E ' and
F'
F w i l l denote r e a l Banach spaces w i t h dual
respectively
, m a p o s i t i v e i n t e g e r o r i n f i n i t y and
t h e space o f a1 1 c o n t i n u o u s l y m - d i f f e r e n t i a b l e F-valued f u n c t i o n s
,
Chapter 10
170
on
m
E , endowed w i t h t h e
we p u t
C"(E)
and
T~
C"(E)
-
t o p o l o g y . (See d e f i n i t i o n 3 . 1 . 1 ) .
instead
We a r e g o i n g t o l i m i t on
Cm(E;R)
and
If
F= R
C"(E;R).
o u r s e l v e s t o t h e case o f f u n c t i o n s d e f i n e d
E, a l t h o u g h w i t h s m a l l m o d i f i c a t i o n s t h e same r e s u l t s can be found f o r
E.
f u n c t i o n s d e f i n e d on open a r b i t r a r y s e t s of
10.1.1 Theorem. Let M be a Cm(E)- submodule of Cm(E;F . Assume that has the approximation property and that f o r every TI E E ' B E and E g E M
every
g
the composite
o T
m
belongs to t h e
T ~ closure -
of
M.
Also assume t h a t :
(I)
F has t h e approximation property.
(11)
(F' o
F)
if
)I
only i f , for every
.. ,yn)
Proof.
E En
b E F
,f
b)
0
f =
(9 o
f ) B b E My
E M).
m
belongs to the
U , k 5 rn ,
.rc-closure of M i f , and E > 0 and ezlery
n E N , there i s some
g E M
such t h a t
N e c e s s i t y i s c l e a r . We w i l l s u b d i v i d e t h e p r o o f o f s u f f i c i e n c y
i n t o two p a r t s , F i x respect t o
(1)
s a t i s f y i n g t h e assumed c o n d i t i o n s w i t h
f E Cm(E;F)
M.
P-a r t 1. Suppose t h a t k < m and E > 0. By
F =R
. Fix
lemma 3.1.3,
K
c
E
compact and nonempty
there exists
TI
e E'
P
E
,k
E
ftJ
,
such t h a t
IdJf(X)(Y) Let
Eo
= u
E. We can assume t h a t the r e s t r i c t i o n s
E o . Given
Q >
{yl,...,ynl
l o n g i n g t o an if
,
E F'
x E E ,k E
with
(9 11p
M c M , (i.e.,
f E Cm(E;F)
Then (yl,.
o
E ) ; i t i s a f . i n i t e dimensional v e c t o r subspace o f
E o # {O}. L e t M o be t h e i d e a l o f
g I E o = go f o r
g 6 M
.
Let
fo
be t h e
Cm(Eo) formed by
f
restriction to
0 , x E E o , p E N , p 2 m and y E {yl ,...,y n l , where i s t h e s e t o f a l l i n t e g e r combinations o f t h e elements be-
E o c a n o n i c a l b a s i s , from t h e h y p o t h e s i s i t f o l l o w s t h a t
go= glEo then
171
On t h e c l o s u r e o f modules
Taking t h i s r e s u l t and t h e p o l a r i z a t i o n formula ( 0 . 3 . 1 ) i n t o account, i t follows that
V
f o E Mo ( n o t a t i o n as i n theorem 0.2.5)
0.2.5,
there exists
where
yo =
GIE,
-
g E M
. This
and by theorem
such t h a t
implies that
that i s
h e M
By h y p o t h e s i s , t h e r e i s
Then ( 1 ) , ( 2 ) and ( 3 )
Thus
g i v e us
belongs t o t h e
f
P a r t 2 . NOW, l e t
. -
F
such t h a t
7;-
closure
be a r b i t r a r y
of
. Fix
M
in
any
Cm(E)
$ E F'
.
According t o t h e
h y p o t h e s i s i s c l e a r t h a t f o r e v e r y f i n i t e rank continuous l i n e a r map : E
IT
the
+
-
T:
, and e v e r y g
E
B
M
closure o f the ideal
6 .
t h e composition $
M
of
C"(E)
s a t i s f i e s t h e assumed c o n d i t i o n s w i t h r e s p e c t t o $
0
f
n
n
$
0
f
belongs t o
Moreover, once
f
M, i t f o l l o w s t h a t
s a t i s f i e s t h e corresponding c o n d i t i o n s w i t h r e s p e c t
ing t o part 1 (4)
g
M. A c c o r d
$
, we have
belongs t o t h e
The l i n e a r mapping
-rF-closure o f g E C"(E)
+
g
$
0
MI
Ib E
+E
f o r any
C"(E;F)
is
F'
.
m m
T ~ - T ~
172
Chapter 10
, and
continuous
(5)
then
( J ) ) ~ ) b b c ( J , o M ) ~ b
where
(I)
of
o
(J,
m
0
M)
M) = -rc-closure of $ Ib
in
Ib )
(F'
(6)
in
and c o n d i t i o n ( 1 1 )
,
Cm(E)
($
0
m M) I b = -rc-closure
o f t h e statement o f t h e theorem
,b
M cM f o r every $ E F '
0
IF)
M
Cm(E;F).
By ( 4 ) , ( 5 ) we have t h a t
= ($
0
o
f c
M , the
That i s ,
E F.
M
-rF-closure o f
in
Cm(E;F).
NOW, we know t h a t t h e l i n e a r mapping
L(E;F)
i s continuous i f
i s g i v e n t h e compact
-
open t o p o l o g y . Thus
, we
have
f = IF f
(7) in
0
, because
Cm(E;F)
belongs t o t h e
-r;
-
closure o f
IFbelongs t o t h e c l o s u r e o f
(F' B F), f F'
IF
in
L(F;F)
by c o n d i t i o n ( I ) o f t h e statement o f t h e theorem. F i n a l l y , ( 6 ) and ( 7 ) imply t h a t
f If
belongs t o t h e Cm(E;F)
k E N
,
k 5 m
-
closure o f
M
i s endowed w i t h t h e t o p o l o g y
f a m i l y ( w i t h parameters k,x,y)
for
!T
, x,y
E E
o f seminorms
, we
in
Cm(E;F).#
-rm
P
d e f i n e d by t h e
obtain the following c o r o l l a r y .
Corollary. Let E, F and M as i n the theorem 10.1.1, and m assume conditions ( I ) and (11) . Then the closure of M f o r T~ i s equaZ 10.1.2
173
On t h e c l o s u r e o f modules
to t h e closure of M f o r
T
m
P'
10.1.3 Remark. (1) I f F
=
c o n d i t i o n s ( I ) and (11) i n theorem 3.2 v e r i f y
R
trivially. ( 2 ) For e v e r y x
E
E , k
N ,k 5
6
m
- .
I
Mk(x) = i s a closed
Mk(x)
,y
E>O
f e Cm(E;F)
Cm(E)
= (yl,...,yn)
-
with
Cm(E;F)
n E
M ,
I n a s i m i l a r way t o theorems 0.2.3, 0.2.5, M of
,
: dJf(x) = 0
module of
En
E
let j = 0,l
.
x
If
,..., k I . B
,k
E
E
N ,
k
5
in,
let
we denote
, for
e v e r y submodule
c~(E;F) n
M
(M
n
=
t Mk(x))
xeE k <m
With these n o t a t i o n s w h i t n e y ' s theorem f o r i n f i n i t e dimension c o u l d be expressed as: " I n t h e hypotheses o f (lO.l.l), t h e coincides w i t h
'I.
-r:-closure
of
M
T h i s v e r s i o n o f W h i t n e y ' s theorem would be t h e
response t o theorem 0.2.5 f o r f i n i t e dimension.
However, t h e c l a s s i c
f o r m u l a t i o n o f W h i t n e y ' s theorem f o r t h e f i n i t e dimensional case i s theorem 0.2.3. Thus, t h e v a l i d i t y o f t h e f o l l o w i n g v e r s i o n , corresponding t o (0.2.3) of
M
i s b r o u g h t up : " I n t h e (10.1.1) hypotheses, t h e -rF-closure
coincides w i t h A
10.1.4
Example.
dimension. L e t
C
Let
M
H
en : n e
For every Let
k.
The n e x t example shows t h e response t o be
i s n o t g e n e r a l l y c l o s e d , ( s e e comments b e f o r e (0.2.5)).
negative since M
a E
be a separable r e a l H i l b e r t space o f i n f i n i t e
NI H ,
be an ortonormal b a s i s of let
a^
H. $ ( x ) = < x,a> ( x E H),
be t h e f u n c t i o n a l
be t h e i d e a l generated by t h e maps
Gn ,
n E
M
in
C'(H).
174
Chapter 10
I t i s w e l l known t h a t a l l H i l b e r t spaces v e r i f y t h e a p p r o x i m a t i o n
property
, thus
t o check i n t h i s case t h a t t h e (10.1.1)
hypotheses a r e
s a t i s f i e s we o n l y have t o prove:
-
the
T;
@ Ia
e H'
exist
N E
g E M
If
(1)
and
It i s sufficient
I
If
H'
IH
TI E
g
R ,
t o prove ( 1 ) f o r e v e r y
A
@ = b
, let K
c H
$n
be compact and
and every E
> 0 . There
such t h a t
N
11 j =11 where
t h e coiiiposite
M.
closure o f
IH.
IT E
R > 0 and
j
> e
1 1 xi1 5
j
R
-bll
E
2 R(
f o r each
x
I1 a l l +I) e K . We have
that
and
T h i s proves ( 1 ) A
Now
, we a r e g o i n g t o prove t h a t M i s n o t c l o s e d
.
First of
,
there exists
a l l we c l a i m
If
f E C'(H)
n 6 111
, let
such t h a t
a = f(a)
$,(a)
,?
# 0 ; let
= df(a) E H '
.
Since a # 0
On t h e c l o s u r e o f modules
h E C’(H)
It i s clear that
175
a
h(a) =
n A
1
dh(a) =
[ Av - a
~
$,(a)
.
g = h.$,
Let
Thus
f
en
--
1 .
;,(a)
Then
g 6 M
, g(a)
=
a
and
and so f E M + Ml(a). g e Ml(a) L e t Mo= { f E C’(H) : f ( 0 ) = Oj.
-
T h i s proves c l a i m ( 2 ) . M o i s a maximal i d e a l o f
C‘(H).
P
1
f =
, there
f E M
If
Zj
fj
and so
exist
f(0) =
fl,...,fp
f
such t h a t
E C’(H)
f j ( 0 ) $j(0) = 0.
M
Thus
c Mo
and
j=l
j=l
M o we have from ( 2 ) t h a t A = M + M l ( 0 ) c M o . On t h e o t h e r hand, l e t v E H such t h a t v does n o t belong t o M1(0) c
since obviously H o , subspace of
B
However, fl,
...,f P ?
E
M
generated by
H
+
C’(H)
= dC(0) = d(
{en : n E P I
.
It i s clear that
because i f i t i s n o t so
M1(0)
E
Mo.
there e x i s t
such t h a t P
1 fj j=l
Gj)(0) =
P 1 [f.(O): j=l J
+ G.(0)dfj(O) 1 ~
J
=
D =
1
fj(0)Gj
j=l but t h i s implies
v
E
H,.
A
(4)
M
i s n o t closed.
According t o ( 3 ) i t i s s u f f i c i e n t t o prove t h a t
t?
i s dense i n
Mo. Let
176
Chapter 10
n E
N
fn(x) = f ( x )
-
f E Mo; f o r every
let m
1
6
H)
df(0)(ej)Gj
.
df(0)(ej)sj(x)
(x
j=n+l we have t h a t
fn(0) = 0
dfn(0) = d f ( 0 )
and n
m
1
-
df(0)(ej)2j =
j=ntl Thus
fn E M
if
Ml(0)
t
;
n
2
1
1 j=l
.
Moreover
and
Thus (f,) 10.2
converges t o
f
in
C'(H).
T h i s proves t h a t
A
i s dense i n M,.
Notes and r e f e r e n c e s . Chapter 10
i s based f u n d a m e n t a l l y on Nachbin 143
[ll, [21 and Gonez-Llavona
[21
,
Guerreiro
.
S i m i l a r e x t e n s i o n s t o theorem 10.1.1
can be o b t a i n e d f o r l o c a l l y
convex spaces. The e x t e n s i o n o f Whitney's theorem g i v e n i n (10.1.1) i n most cases
, but
i n c e r t a i n cases c o n d i t i o n s ( I ) and (11)
i s enough i n (10.1.1)
a r e n o t easy t o v e r i f y , and sometimes these do n o t h o l d . For t h i s reason some a u t h o r s have o b t a i n e d d i f f e r e n t v e r s i o n s o f W h i t n e y ' s theorem f o r s p e c i f i c cases.
For i n s t a n c e see G u r a r i e 111 where w i t h technique d i f
f e r e n t from t h e one used here, a W h i t n e y ' s theorem f o r f u n c t i o n s on an open s e t of
Rn
t o a r e g u l a r commutative Wlener- Banach a l g e b r a i s found.
177
Chapter 1 1
HOMOMORPHISMS BETWEEN ALGEBRAS OF UNIFORMLY WEAKLY DIFFERENTIABLE FUNCTIONS
L e t E and
F be
r e a l Banach spaces. F o r
m = O,l,...,
m
be t h e space o f a l l f u n c t i o n s from E t o F which a r e u n i l e t C:bU(E;F) f o r m l y weakly d i f f e r e n t i a b l e on bounded s e t s , endowed w i t h t h e -cF-topology. (See ( 5 . 2 . 2 ) ,
(5.2.4)
and ( 5 . 2 . 6 ) .
Our p r i m a r y i n t e r e s t i n t h i s c h a p t e r P m A: CwbU(E)-tCwbU(F) , p, m 6
i.
i s t h e s t u d y o f homomorphisms (See (4.6) f o r t h e case
m = p
0).
We w i l l show t h a t t h e s e homomorphisms a r e "induced" by f u n c t i o n s g:F"+ E " i n a way which w i l l l a t e r be made more p r e c i s e . One o f t h e p r i m a r y purposes of t h i s c h a p t e r i s t o c h a r a c t e r i z e these i n d u c i n g f u n c t i o n s g, i n terms o f a d i f f e r e n t i a b i l i t y property, thereby
c h a r a c t e r i z i n g t h e homomorphisms
A. The b a s i c i n g r e d i e n t s we w i l l need a r e few and r e l a t i v e l y simple. F i r s t , under reasonable hypotheses (such
as E ' h a v i n g t h e bounded appro-
x i m a t i o n p r o p e r t y ) , CEbu (E) can be c h a r a c t e r i z e d as t h e c o m p l e t i o n o f t h e a l g e b r a generated b y E ' under t h e t o p o l o g y o f u n i f o r m convergence o f a f u n c t i o n and i t s f i r s t k d e r i v a t i v e s on bounded subsets o f E, where k ~ f l ,
k< m. (See ( 5 . 3 . 4 ) ) .
Therefore, a c o n t i n u o u s homomorphism @ : C m
xbu(E)
i s determinated by i t s a c t i o n on E l . Second, any f u n c t i o n i n a n e c e s s a r i l y unique e x t e n s i o n t o an element i n Cm(Eiw,). any homomorphism g ( y ) B El'
m A: Ct)rbu(E) -t Cwbu(F]
can be d e f i n e d by
and any p o i n t y
g ( y ) ( @ ) = A($)(y)
, (4
6
E
+
IR
CwbU(E) has Finally, given
F, a f u n c t i o n a l
El).
I n t h i s way,
we g e t a f u n c t i o n g: F-t E" which we w i l l be a b l e t o extend t o
5
: F"+ E "
.
Note t h a t s i n c e CFbU(E) i s a r e a l F r s c h e t a l g e b r a , e v e r y m u l t i p l i c a t i v e l i n e a r f u n c t i o n a l on C:bU(E)
i s a u t o m a t i c a l l y c o n t i n u o u s (Husain-Ng
[Z] ) . From t h i s , we w i l l be a b l e t o deduce t h a t A
111 ,
i s continuous and we
w i l l a l s o be a b l e t o d e r i v e t h e d e f i n i n g d i f f e r e n t i a b i l i t y p r o p e r t i e s o f 9.
Chapter 11
178
E
F w i l l always denote r e a l Banach spaces i n t h i s c h a p t e r .
and
X
F o r any H a u s d o r f f l o c a l l y convex space i s t h e space o f a l l
Cp(X;Y)
mappings from space
Y
X
to
Y
i s omitted
p
and
Y
and f o r
times continuously F r i c h e t d i f f e r e n t i a b l e
(see Yamamuro[ 1 1 ) . Throughout Y =R
then
..ym ,
p = 0,1,.
i f t h e range
i s understood.
A l l polynomial spaces b e i n g considered i n t h i s c h a p t e r a r e ent h e t o p o l o g y o f u n i f o r m convergence on bounded
dowed w i t h t h e .rb-topology, sets.
For each
B L = Cx e E " :
M
n 6
11 X I [
Bn = { x 6 E :
1) xII
5 n I,
and
2 n I.
11.1 Representations o f u n i f o r m l y weakly d i f f e r e n t i a b l e f u n c t i o n s . I n t h i s s e c t i o n we show t h a t f u n c t i o n s i n
C:bu(E;F)
have ex-
t e n s i o n s t o f u n c t i o n s h a v i n g t h e same degree o f d i f f e r e n t i a b i l i t y d e f i n e d on
E".
The importance o f t h i s r e s u l t comes f r o m t h e f a c t t h a t we can
thus o b t a i n a t o p o l o g i c a l and a l g e b r a i c isomorphism between CD(Egw*)
, which w i l l be u s e f u l i n t h e sequel. (See ( 4 . 6 . 2 ) E"
Since closed b a l l s i n
a r e compact i n
CmwbU(E) and
, the
Egw*
m = p = 0).
for
following
i s easy t o prove.
11.1.1 =a.
X be a real Hausdorff Locally convex space. If
Let
g E C(ELw*;X) and
i s bounded , then
B c E"
g(B)
precompact and ,
is
i n p a r t i c u l a r , bounded. From t h e p r o p e r t i e s o f t h e a t e l y deduced t h a t i f
bw*
Remark.
f 6 Cp(EII)
11.1.3
For
p
2
1
CP(ELw,)
E"
t h e n f o r each
f E Cp (ELw* ; FLw,)
d J f ( x ) B P(JE;jw* ; FLw* ) j E PI , j 5 p thus show t h e f o l l o w i n g remark, 11.1.2
topology i n
.
i t can immedi-
x E E " , any
The d e f i n i t i o n s themselves
is the space of a21 functions
which s a t i s f y the foZlowing properties: (a)
For a22 x 6 E "
(b)
For a22 j E 111
Lema. If g
6
and each bounded subset
j 6 111 j
5
p
CP(ELw* ; FLw,) B c E",
, j 5 p , dJf(x)
, dJf e , then
6
P(JE;w,).
C(Eiw* ; P(JELw,)).
f o r each
j 6
sup{II d j g ( x ) ( y ) l l : x,y 6 B }
PI ,
1
< m .
5
j
5
p
Homomorphisms between a l g e b r a s
Proof.
Since
dJg
lemma 11.1.1.
$ E F'
where
CCEi,
6
; P(JEiw*
; FiW,)),
179
dJg(B)
i s precompact by
Let
.
There a r e p o i n t s
yl,.
.. 'y,
6
B
, such t h a t
Theref o r e s u p { l d J g ( x ) ( y ) ( $ ) l : x,y E B3 11.1.4
Cp( Egw,)
D e f i n i t i o n . We endow
5 M + 1 , as r e q u i r e d .# w i t h the ZocalZy convex
o f uniform convergence of order p on bounded s e t s of T :,isgenerated
which we denote by
when
B
P(JE;w,;
F)
for
j
5 p , dJf : E
that
+
+
Pwbu(jE)
d J f 6 C(Eiw* :
11.1.5 L,ema.
bounded subset
Sf
and
C(ELw*;F)
j e [u.
As a consequence
E".
says t h a t t h e f o l l o w i n g p a i r s o f spaces a r e Cwbu(E;F)
t o p o l o g i c a l l y isomorphic : and
El'. This topology,
by a l l seminorms of the form
i s allowed t o range over a l l bounded subsets of P r o p o s i t i o n 4.6.2
tOpGZOgy
, if f
E CibU(E)
can be extended t o
-
and Pwbu(JE:F)
t h e n f o r each dJf : E"
+
j e
P ( j Eiw,)
N such
P(JE;w,)).
P f E Cwbu(E)
, then
B c E " , t h e mapping
j 5 p
,
f o r each
j E 81
0 : B x B
* R , Q(X,Y) = d J f ( x ) ( y ) i s
continuous when B has the induced weak*-topology.
and each
e-
180
Chapter 11
* i s precompact i n
By lemma 11.1.1, d J f ( B )
Proof.
given the
.
Therefore
0-neighbourhood
there exists a f i n i t e s e t { bl
-
k
N
Since each
dJf
> 0
i=1 E
,...,b k l
such t h a t
c B
-
u
dJf(B) c
(1)
and
P(JEGw,,)
+ V).
(dJf(bi)
, we
P(JEcjw,)
such t h a t f o r a l l
can f i n d a f i n i t e s e t x,y
E
B
{$I,...
s a t i s f y i n g IOi(x-y)I<
, $sl
c E'
6 1 ( i =l,...,s),
we have
ru
(2)
-
IdJf(bl)(x)
Y
dJf(bl)(Y)
I
<
(1
E
= l,...,k).
?
On t h e o t h e r hand f i n d a f i n i t e set x,y E B z
E
B
]@Ii(x-y)
< 62
we can
and 6 2 > 0 such t h a t f o r a l l ( i = s+l,
...,t )
, and
for all
have
Letting l@Ii(xl-
d J f 6 C(Egw* ; P(JEBw,)),
{ @ I ~ + ~$ t,l . .=. ~E '
satisfying
, we
, since
-
6 = m i n (61~6,) and choosing
x 2 ) I < 6, l ~ $ ~ ( yy 2~) I- < 6 ( i = l,...,t), rly
IdJf(xi)(Yi)
-
rc.
dJf(x2)(y2)1
x1,x2,y~,y2 E B
so t h a t
we conclude t h a t
5
N
dJf(x E
+
yr
djf(bi)(yz)
-
?
(1) and ( 3 )
d J f ( x 2 ) ( y 2 ) l by
.
Hence
,
cs-'
Yi)
- dJf(xz)(y2)1 5
E
+
E
+
E
+
E
=
4~ , by ( 1 ) and ( 2 ) .#
The f o l l o w i n g r e s u l t i s t h e promised g e n e r a l i z a t i o n o f proposi-
Homomorphisms between a l g e b r a s
181
t i o n 4.6.2.
Every function
Theorem.
11.1.6
-
way t o a function
a22
n e N and
f : E"
-+
R suck t h a t
P r o o f . We p r o v e t h e r e s u l t f o r
-
E C(Eiw,)
p=
E
N
Moreover
f 6 Cp(Eiw*).
p
E
,for
N , o n l y t r i v i a l modifications being
. Proposition
m
f
which extends
d J f E C(Eiwx ; P(JEiw,)) and n
can be extended i n a unique
-
j s N , j < p ,
necessary f o r t h e case
f'
f e Cwbu( P E)
,
and
more
which extend
4.6.1
yields a function
, functions
generally
dJf
.
A l s o , f o r each
j
5 p
, 'Iv
: x E BnI =
supCII d J f ( x ) I I
sup{ll d J f ( x ) l l
: x
E
B;I
I.
rq
The theorem w i l l be proven once we show t h a t
.
and t h i s we show by i n d u c t i o n be a r b i t r a r y all
Let to t
s
.
By (5.2.1)
x e Bn
and
y E E
s e B;
and
t
E
Let
This implies t h a t
d"f
, and
t h e r e i s a r e a l number
,
E"
llyll
5 6 ,
with
11 tll 5
i n t h e weak*-topology and l e t
i n t h e weak*-topology
p = 1
d J f = dj;
, w i t h 11
= df
y,Il
. Let
6
(y,)
(x,)
let
n e
j 2 p and E > 0
N
c Bn
be a n e t c o v e r q i n g
be a n e t i n
5 1 1 tll
.
for
5 p - 1
E
coverging t o
By lemma 11.1.5,
. dJf = d j r
j
,
6 > 0, such t h a t f o r
w
Next, assume t h a t
for all
.
Reasoning as
above and u s i n g t h e i n d u c t i o n h y p o t h e s i s we f i n d t h a t f o r any
Bn
and
Chapter 11
182
E
> 0, t h e r e i s a number
IItll 5
6
11.1.7
, dPf
-
B;
and
,
t B E"
, which concludes t h e p r o o f . #
.
-4
f
The mapping
-+
i s a topological isomorphism,
f
i n the sense of PrBchet algebras, between
11.2
6
N
= dPf
Corollary
f o r a22
s
3
-
Therefore
d > 0 such t h a t f o r a l l
p = 0,l
( Cibu(E);7
.
,...,m
E) and
( Cp(Eiw,)
;T
Homomorphisms between a l g e b r a s o f u n i f o r m l y weakly d i f f e r e n t i a b l e functions. By c o r o l l a r y
11.1.7
e v e r y homomorphism A : CibU(E)
m Cwbu(F)
-+
can be a s s o c i a t e d i n a u n i q u e way t o a homomorphism, s t i l l denoted between
Cp( Eiw,)
and
Cm(Fiw,).
Our o b j e c t
here
A, i s t o characterize
these homomorphisms i n terms o f mappings they induce between
F"
and
E".
Since t h e continuous case has been discussed elsewhere (see s e c t i o n 4.6)
p or m
we w i l l always assume t h a t a t l e a s t one o f
i s bigger than
,
0.
The f i r s t r e s u l t i s b a s l c , a l b e i t e a s i l y proved. 11.2.1 E'
Proposition.
Let
f3:Cp(EGw,)
has the bounded approximation property
x B E " such t h a t
be a homomorphism. Then, i f
+R
e ( f ) = f ( x ) f o r a22
f
, there B
e x i s t s a unique point
Cp(Eiw,).
rn particular
,
every such homomorphism i s automatically continuous. Proof. Thus
0
,
E El.
Cp (E" ) i s a r e a l F r i c h e t a l g e b r a , 8 i s c o n t i n u o u s . wbu bw* t h e r e i s some p o i n t x e E " such t h a t e ( + ) = $(x) f o r a l l Since
As a r e s u l t
B(p) = p(x)
f o l l o w s by t h e d e n s i t y o f
Pf(E)
p
6
Cp(Egw*)
.
for all in
Pf(E)
, and
the result
(See 5.3.3).#
m A : Cp(Eiw*) + C (FbJ,) be a homomorphism. Then, i f E ' has the bounded approximation property , A i s induced by a f u n c t i o n g : F" + E " , i . e . , A ( f ) = f 0 g f o r every f 6 Cp(Eiw*). 11.2.2
CoroZZary.
Let
Homomorphisms between algebras
+R
A : Cp(Eiw,) so t h e r e corresponds a unique p o i n t x 6 E " Proof.
For each
for a l l
y E F",
f 6 (?[ELw*).
6y
o
183
i s a homomorphism, and
such t h a t
The r e q u i r e d f u n c t i o n
6 A(f) = f(x) Y i s t h e r e f o r e g i v e n by
g
g(Y) = X ' # Having e s t a b l i s h e d t h e e x i s t e n c e o f some f u n c t i o n
g
e v e r y homomorphism, we now s t u d y d i f f e r e n t i a l p r o p e r t i e s o f
inducing
g.
Our
p r i n c i p a l r e s u l t here i s t h e f o l l o w i n g theorem.
m * C (FLw*) be a homomorphism. Then, i f E ' has t h e bounded approximation p r o p e r t y , A i s induced by a f u n c t i o n m A ( f ) = f o g f o r ever y f c CP(ELw,). g E C (Fiw*;Eiw*) i.e. 11.2.3
Theorem. L e t
Proof.
The case
.
Define
@ E E'.
Since
$1,
...,
( i = l,...,k), (1)
,
> 0
E
$k E F '
-+
where .x 6
and
P(JFgw,)
by y E F"
6
and so we w i l l
j E
N ,
x
F"
and
6 > 0, such t h a t i f y,z
and
,
f o r each B c F"
bounded s e t
,
each
Sj(XHY) gj(x)
8
](@)I 5
6
I$ E E '
B
j
5
m
, it , there
, I ~ ) ~ ( y - z )<\
y
E
F"
.
E
P(JF;w,;E,'&.)
Next, from t h e d e f i n i t i o n of that for a l l
P(JF";E")
then
I [!4j(X)(Z) -
Consequently
: F"
d j [ A(@)] (x)
f o l l o w s t h a t f o r every exist
g
j dJ[ A(@)] (x)(y),
gj(x)(y)(@) = and
has a l r e a d y been discussed i n ( 4 . 6 ) ,
m=O
m 2 1
assume t h a t
A : CP(E$,,)
= P(JFiw*;E{w,).
d[A(@)l(y)
and a l l bounded s e t s
B
c
F",
, i t i s easy t o see there i s
6 > 0
such
that
i s bounded i n
E",
and t h a t
Combining ( 2 ) and ( 3 ) sets
(4)
, we
conclude t h a t f o r a l l y 6 F " and a l l bounded
B c F", 1i m E+O
g(y
t
Ex)
- g ( y ) - g1 ( y ) ( e x ) ] = o
in
E;~*
uniformly
6
Chapter 11
184
for
x e B
and so
g1
i s the derivative o f
Assuming now t h a t derivatives o f rivative of
g
g.
(j - 1 ) th i s the j
are the f i r s t
gl,gzy...yg(j-l)
, where
g. L e t y
j < m , l e t us show t h a t g dg j e F" be f i x e d , and denote by C(y) t h e unique sym-
m e t r i c j - l i n e a r mapping a s s o c i a t e d t o
g.(y). J
u : FiW*+ P(J-lFiW*; E bw* " )
Let
be t h e l i n e a r mapping g i v e n by u ( x ) ( z ) = C(y)(x,z,.!j:!!.,z). t h a t g i v e n any bounded s e t
We must show
B c F",
As i n ( 2 ) , we f i r s t n o t e t h a t from t h e d e f i n i t i o n o f easy t o see t h a t f o r some 6 > 0,
dJ[A($)](y)
, it i s
i s bounded.
4
Thus, i t w i l l be s u f f i c i e n t t o show t h a t f o r a l l
6
E',
l i m ( E ' l [ g j - l ( y + E x ) ( z ) - g j _ l ( y ) ( z ) - u ( E x ) ( z ) l ( + ) ) = ~u n i f o r m l y f o r x , z E ~ .
(7)
E O '
If 0 dJ[A(+)l of
,
and
i s t h e unique symmetric j - l i n e a r mapping a s s o c i a t e d t o v
i s d e f i n e d i n analogy w i t h
u
D
above, u s i n g
instead
C, then
lim
~ - ~ [ d j - ' [ A ( + ) ] ( y + ~ x ) - d ' - ~ [ A ( ~ ) ] ( y ) - v ( ~ x ) ] u=nOi f o r m l y f o r
x E B.
E O '
I n o t h e r words, (8) l i m
E - [~d j - l [ A ( $ ) ] ( y t ~ x ) ( z ) - d J - l [ A ( I $ )
] ( y ) ( z ) - v ( ~ x )( z ) ] = 0
E-fO
uniformly f o r
NOW , D(~)(EX,Z,.!~:!!.,Z) V(EX)(Z)= u(Ex)(z)(+) . jth d e r i v a t i v e o f g. for all
Finally B c F",
,
x, z
e B. so t h a t
= C(y)(~x,z,!jlt!.,z)(I$)
Therefore ( 8 ) i m p l i e s ( 7 )
i t i s easy t o see t h a t f o r each
, and j
,
so 1
5
g j
j
i s the
2 m , and
185
Homomorphisms between a l g e b r a s
Therefore, t o prove t h a t
g.
J
B c F", a l l @
t h a t f o r a l l bounded s e t s
$I,...,$k
exist
( i = l,...,k),
then
I(gj(x)(z)
immediate f r o m t h e f a c t t h a t
-
> 0, t h e r e
E
, l ~ ) ~ ( x - y )
x,y e B
gj(y)(z))(@)I <
.
E
However, t h i s i s
d J [ A ( @ ) ] B C(FLW, ; P(JF;w,)).#
I t i s o f i n t e r e s t t o n o t e t h a t t h e above p r o o f shows i n f a c t m C (Fiw,;ELw*) c o n s i s t s o f e x a c t l y those f u n c t i o n s g : F" + E "
that
for all
+
E',
@ B
Let
El'
+
be such t h a t f o r a l l
(I$
11.1.7,
0
g)IF
+
i s such t h a t f o r a l l
: F" + E "
for all
g B C"'(F;~,).
g : F" + El'
Then, by c o r o l l a r y g : F
t i m e s c o n t i n u o u s l y d i f f e r e n t i a b l e i n t h e sense t h a t
m
which a r e weakly
4
E ' , and a l l
6
6 > 0, such t h a t i f
F 1 and
6
we need o n l y show
C(F;w,;P(JF;w,;E;w,))
6
by
C~(F;~,;E;,)
,@
0
= { g : F"
= {g : F
+
E" :
(I$
g)(y)
+
EII ; f o r a l l
E'
6
0
g
6
m C (Fiw*).
0
g
6
and we d e f i n e
CEbu(F)
1 1 . 1 . 6 , we see t h a t
Summarizing,
C (FgW*).
for a l l
+
as i n theorem
m
66 -+
0
,
E'
+
On t h e o t h e r hand, i f
C:bu(F).
E
U
G(y)(@)
E'
@ B
B
@ 6 El,
6
E'
,@
g E
m g e Cwbu(F)3
,@
cm (F;~,)}
.
An immediate consequence o f theorem 1 1 . 2 . 3 , lemma 11.1.1
(11.1.4) 11.2.4
=
and
i s the following
Corollary. If
homomorphism
E'
A : Cp(EBw,)
has t h e bounded approximation property ,every m + C ( F i W * ) i s continuous.
Our n e x t r e s u l t w i l l y i e l d t h e complete c h a r a c t e r i z a t i o n o f , i n Corollary homomorphisms between two spaces o f t h e f o r m C:bU(E)
11.2.6. 1 1 . 2 . 5 Theorem. m
Let f
0
g
, and
e c (F~;E;,),;
m g 6 C (FLW*)
Proof.
Assume t h a t E ' has the bounded approximation property. p
m
( [ l l , ii1.8.3 )
,f
let
.
Then f o r every f
E
,
cP(E;,)
.
By Yamamuro
and t h e c h a i n r u l e o f o r d e r
m
holds
o
g
is
m-times d i f f e r e n t i a b l e ,
. By Yamamuro
( [ l l , 5 1.7.2),
it
Chapter 11
186
s u f f i c e s t o prove t h a t
,is
o f course
dm(f
g)
6
R z 0
and
there exists
M _z
such t h a t
By c o r o l l a r y
11.1.7
p
6
Pf(E)
R
and
f e CP(ELw*)
, t h e r e i s a f i n i t e t y p e polynomial
For each $l,...
,$k
11
5 R
11
6
i F'
zzII 5
6
N and
R ,
there are
E" ,
xlrx2 E then
..., S ) ,
are
(4)
be a r b i t r a r y . By lemmas 11.1.1 and 11.1.3,
(5.3.3)
such t h a t whenever
Z111
a v o i d complicated n o t a t i o n s ,
m = 2.
> 0
E
The general case,
such t h a t
Since ( i = I,
, to
proved by i n d u c t i o n . B u t
we w i l l o n l y prove t h e case when Let
C(Fiw,;P(mFgw,)).
,15
)I x11) 5 M
$1,
..., +s 6 1) ~ 2 1 15 M
E'
and
61 > 0
, I$j(x-Y)l
,
< 61
i 5 s , Jli 0 g 6 C(FCW*) , and so t h e r e such t h a t whenever z1,z2 E F" ,
62 > 0
l $ j ( z l - z 2 ) I < 62,
and
(j = l , . . . , k )
then
I +i ( g ( z, Next, f o r
there e x i s t
$ktl
such t h a t I( ZI(/ 'R ,)I z 2 ) (5 R , and ,1. For every f o r i = 1,. . ,k,k+l,.
.
y)
-
d2(f
o
g)(z2)(y)
..
y 6 F",
can be w r i t t e n as a sum
Homomorphisms between a1 gebras
187
o f terms o f t h e f o l l o w i n g t y p e :
( 1 ) , ( 3 ) and ( 4 ) i m p l y t h a t norms o f ( 6 ) and ( 9 ) a r e l e s s t h a n EM; (1) and ( 2 ) y i e l d t h a t ( 7 ) and (10) a r e bounded by EM ; f i n a l l y (5)
i m p l i e s t h a t ( 8 ) and ( 1 1 )
11 d m ( f
Hence fixed
C
, which
0
a r e bounded by
g)(zl)
-
dm(f
.
E
2
g)(z,)ll
, f o r some
(CM)E
concludes t h e p r o o f . #
If E ' has the bounded approximation property, then m the space of homomorphisms A : CEbu(E) + Cwbu(F) , where p >_ m , can CoroZlary.
11.2.6
be i d e n t i f i e d with the space of aZZ functions g : F -+ El' , such t h a t m A ( f ) = 7 0 9. for a22 I$ E E ' , I$ o g E Cwbu(F) , v i a the f o r m l a We have thus f a r excluded t h e case
p < m.
The reason f o r t h i s
i s apparent from t h e n e x t r e s u l t . 11.2.7
Proposition. p
where
m.
If
Let
A : Cp(Eiw*)
m C (Fiw*)
+.
be a homomorphism ,
E ' h s the homded approximation property
induced by a constant function
g : F"
+
, then
A is
E".
The p r o o f depends on t h e f o l l o w i n g elementary lemma. 11.2.8. If
e. Let
p < m and
Proof.
f
0
g : R -+R g E c"'(R)
,g
f o r aZZ
F o r s i m p l i c i t y , suppose t h a t
. Then f L e t f(x)=lxlPt1'2 a t 0 , being a contradiction.
E Cm(R)
o
f
6
.
, and
assume that
cP(R)
, then g i s constant.
,
g (0) # 0 * (pt1)-st derivative
g(0) = 0
m
and t h a t
g does n o t have a The general case f o l l o w s eas l Y . #
2
1
188
Chapter 11
Proof o f P r o p o s i t i o n 11.2.7. m g E C (Fiw*;Eiw*) such t h a t
By C o r o l l a r y
g(0) = 0
and f o r some
L e t us assume t h a t Let
Fa be t h e span o f
v
A(f) =
in
F"
?:
n : E " * E O be t h e p r o j e c t i o n
Let where
J, E E '
g
0
and
11.2.2,
f o r every v E F"
Eo
O($)g(v)
n(@) =
i s chosen t o s a t i s f y g(v)(J,) = 1
theorem 11.2.5
h
E Cp(Eiw,)
II
o
f E CP(EbJw* ) .
, g(v) #
0 in
t h e span o f g ( v ) i n
i s a l i n e a r mapping which l i e s , i n f a c t i n
TI
there i s a function
f o r every
. 6
E"
f o r each
.
0 E E:'
I t i s immediate t h a t
.
Cm(E;w*;EO) h
E".
CP(Eo).
Thus, by
,
Therefore
m n ) = ( h o TI 0 g) E C (FgW*). I n p a r t i c u l a r , ( h n o g ) I F o E Cm(Fo). However, lemma 11.2.8 t e l l s us t h a t ( n o g ) I F O i s c o n s t a n t , a l t h o u g h
A(h
o
IT
g(0) = 0
and
n
0
g(v) = g(v)
# 0
. Thus
, we
have a
contradiction,
and t h e p r o o f i s complete.# 11.3.
Examples. We g i v e t h r e e examples i n t h i s s e c t i o n which i l l u s t r a t e t h e
c o n c l u s i o n s o f t h e preceding s e c t i o n . Example 11.3.1
gives a s i t u a t i o n
i s continuous, a l t h o u g h
Cdbu(F) i n which t h e homomorphism A : CAbu(E) t h e induced mapping g : F"+ E" f a i l s t o be FrGchet d i f f e r e n t i a b l e -+
(considering both
F"
and
E"
w i t h t h e i r norm t o p o l o g i e s ) .
The n e x t example 11.3.2 can be d i f f e r e n t i a b l e Finally
, we
, without
shows t h a t t h e i n d u c i n g f u n c t i o n
g
being continuously d i f f e r e n t i a b l e .
adapt an example o f Bade-Curtis [ l l t o show t h a t
n o t every homomorphism from
C'(R)
i n t o a F r i c h e t a l g e b r a need be auto-
m a t i c a l l y continuous. Let
E
be t h e Banach space
r e a l numbers, and l e t
F
be t h e Banach space o f n u l l sequences o f complex
11.3.1
Example.
c o o f n u l l sequences o f
numbers, considered t o be r e a l Banach space, b o t h w i t h t h e sup norm. For i n Xn each x = (x,) E E , l e t y = (y,) E F be d e f i n e d as yn = E.--for n n
f.
I.
Define
g : E
+
F
i s Hadamard d i f f e r e n t i a b l e , satisfies Indeed
,
zn = i e i n x n yn.
if g
were
by
g ( x ) = y.
By DieudonnG ([l]
with derivative However, g
,
VIII)
,g
g ' ( x ) ( y ) = z , where z = ( z ) n
i s not Frichet differentiable.
Frichet differentiable
,
then i t s F r i c h e t d e r i v a t i v e
would have t o c o i n c i d e w i t h i t s Hadamard d e r i v a t i v e . Thus, f o r each we would have
n e N
Homomorphisms between a l g e b r a s
189
T h i s l a c k o f d i f f e r e n t i a b i l i t y n o t w i t h s t a n d i n g , we now show that
g
A : c;bu(F)
i s induced by a homomorphism
show t h a t f o r a l l
C'-functions
on
f
F
t h a t we need t o show i s t h a t t h e mapping ( f
g)'(x) = f'(g(x))
g'(x),
0
,f (f
0
o
i s continuous
C&,(E).
+
g a C'(E). 9)' : E
.
Let
-+
E'
- i n xn
m
= Re[(-i)
1
e
On t h e o t h e r hand, Since
f
C'(F)
E
and
K
Moreover, t h e sequence
(2)
+
0
as
, where K = I ( B n e F : l B n l 5 l / n , n e N l . , f ' ( K ) i s b o u n d e d i n norm by, say , M. n
-+
00
, where
: x = ( x j ) e f ' ( K ) l , and so g i v e n E > 0 we can f i n d lxjl j>n such t h a t I d n ] < ~ / 6f o r a l l n 2 n o . Choose 6 > 0 such t h a t
I
eiu
-
11 <
E
/ ( 6 noM) whenever
Combining ( l ) , ( 2 ) and ( 3 )
11 yII
- a 1 . n
I
dn = sup no€ N
g(E) c K
i s compact (d,)
be t h e
be t h e element
- i n yn zn(an(y) e
n= 1
,
(an(y))
F' a s s o c i a t e d t o f ' ( g ( x + y ) ) and l e t (a,) a s s o c i a t e d t o f ' ( g ( x ) ) . I f z E E , 11 211 5 1 , t h e n
element o f
(1)
We f i r s t I n fact, a l l
min ( 6 1 , 6 / ( n o + l ) )
then
IuI < 6 .
we see t h a t i f
y
6
E
i s such t h a t
Chapter 11
190
This shows t h a t
(f
0
9)'
i s continuous,
Cibu(E)
I n addition , f o g c f o r every f B C ' ( F ) . I n f a c t , we show more, namely t h a t f o g and ( f o 4 ) ' a r e weakly uniformly continuous on E , and not j u s t on each bounded s e t in E. To see t h i s , l e t E > 0 be a r b i t r a r y Using g ( E ) c K , K being compact, we can find a 6 > 0 such t h a t whenever z 1 , z 2 B K , I/ zl- 2 2 1 1 < 6 , then I f ( z 1 ) - f ( z 2 ) I < E . Let 6' > O such t h a t whenever s , t 6 R , It-sl < then Ieit- e i s / < 6 . Let n o be so l a r g e t h a t 2 / n < 6 f o r a l l n >_ n o and l e t V be the weak ne ghbourhood of 0 in E defined a s
.
V =
{x
6 E
: Ixj
I f x,y 6 E s a t i s f y x-y B V , then 11 g ( x ) - g(y)II < d by an easy c a l c u l a t i o n . Hence , 1) ( f o g ) ( x ) - ( f 0 g ) ( y ) \ \ < E , and so f o g i s weakly uniformly continuous on E. Next, we show t h a t ( f o 9 ) ' i s weakly uniformly continuous on E . Let E > 0 be a r b i t r a r y , and choose d > 0 so small t h a t whenever z 1 , z 2 6 K s a t i s f y 1 1 2 1 - z211 < 6 , then
Let M > 1 be such t h a t be such t h a t
11 f'(z)II 5
M
for all
z e K , and l e t noe N
Homomorphisms between a l g e b r a s
Let
61 > 0
be such t h a t i f
all
x,y Q E
t,s
Re [-i
II z l l 51
)I g'(x)ll 5
Since
i s bounded above
5 n,M
.
,f
0
g
+
ctn(y) ( e
n=l
Then f o r
- i n yn
1
- e
Z n l .
1 for a l l x t h e f i r s t t e r m on t h e r i g h t hand s i d e by E , u s i n g ( 1 ) . The second t e r m i s bounded by
(d4) 2 =
,
E
i s a member o f
Summarizing
A : CAbu(F)
j = l,.. .,no).
for
- i n xn
1
(€/2n0M)) +
Therefore
then
with
m
sup
R , It-sl <
V = { x Q E I x j I < fil/no x - y Q V,
Let
t
Q
191
CAbu(E)
,
u s i n g ( 2 ) and ( 3 ) .
, as
Cdbu(E)
required.
t h i s example shows t h e e x i s t e n c e o f a homomorphism A(f) = f
g i v e n by
automatically continuous
,
0
E
d i f f e r e n t i a b l e between t h e Banach spaces
g
, where
and
F.
g
i s not
Note t h a t
Frichet
A
is
as can be seen a p p l y i n g c o r o l l a r y 11.2.4
or
e l s e by a d i r e c t computation. 11.3.2
Example.
F o r each
n E BI
,let
$,I
which has s u p p o r t c o n t a i n e d i n [ l / n t l , l / n l where
tn =
gn(t) =
IL
1
[ l/n t l/(n+l)
Jln(s)ds
.
I
.
Let
Note t h a t f o r a l l
:R
-+
be a Cm-function
, and such t h a t q n ( t n )
gn : R + R t 6
[0,11
defined
R and a l l
n
6
= 1,
as
W
,
-m
lgn(t)/ 5 l/n(n+l) < l/n2
, and
so t h e f u n c t i o n
g :R
+
co
g i v e n by
Chapter 11
192
.
Moreover , g i s a d i f f e r e n t i a b l e g ( t ) = ( g n ( t ) ) i s well-defined mapping. I n f a c t , i t i s obvious t h a t g i s d i f f e r e n t i a b l e a t any t # 0. I n a d d i t i o n , f o r any if
t
6
t
, 1 1 g ( t ) J J- I t \ - '
[l/(ntl),l/nl
shows t h a t
g ' ( 0 ) = 0.
g ' ( t k ) = ek
,
.
1, I I g ( t ) l l 5 l / n 2
[l/(ntl),l/n
6
2
(n+l)/n2
-+
0
as
Therefore,
n
+ m.
A r o u t i n e c a l c u l a t i o n shows t h a t f o r each
t h e usua,l
u n i t basis vector o f
kth
c o y and so
g'
This k e
H ,
is
n o t continuous a t 0 . Note t h a t
$
g
o
E
C'(R)
$ E
f o r each
l1
= ck
.
Indeed, l e t
co
1
-t u o i n R . Then, if 4 = e l 1, ( + g)'(uj) = $,, q+, ( u j ) , j n=l and so if U O # 0 , i t i s c l e a r t h a t ( + g ) ' ( u j ) -+ $,qn(u0) = ( $ o g ) ' ( u o ) n=l
u
1
u o = 0, then g i v e n
If
E
> 0, choose
n o such t h a t
1 I+,,
< €.Therefore
n=n
s u f f i c i e n t l y large.
D e f i n e a homomorphism
Note t h a t t h e above work shows t h a t
A is
.I;-
T;
Since
A
i s w e l l - d e f i n e d . Furthermore
,
continuous, s i n c e f o r each
C'(R)
1
is
-
, (5.3.3) y i e l d s an e x t e n s i o n
complete
. 1
A : CAbu ( c o ) -,C'(R) as a continuous homomorphism. I t i s s t r a i g h t f o r w a r d t o show t h a t i f a sequence (p,) i n P f ( c o ) converges t o f E CibU(co) for the
8
~b
i s g i v e n by
topology
, then
h(f) = f
o
g
an example o f a homomorphism
(pn
g)
f o r every
-f
f
o
g
i n C'(R)
f e CAbu ( c o )
,
.
Therefore
and so we have
,
Homomorphi sms between a l g e b r a s induced by a d i f f e r e n t i a b l e f u n c t i o n
g :R
+
193
c o which i s n o t
C'.
11.3.3 Example. ( B a d e - C u r t i s [l] ) . Let X = C'[O,l] be t h e Banach algebra o f continuously d i f f e r e n t i a b l e functions x : [O,ll+ R w i t h the usual norm,
II X I 1
=
SUP
O
Ix(t)l +
sup
o
Ix'(t)l
.
I t i s w e l l known (see Gelfand-Raikow-Shilow ( 1 1 1 ,p.23))
X
i d e a l s of
t h a t t h e maximal
correspond t o p o i n t e v a l u a t i o n s i n [ O , l l .
We r e c a l l t h e f o l l o w i n g theorem o f Bade-Curtis [ l ]
X =B
showed t h a t
, who
also
s a t i s f i e s a l l t h e c o n d i t i o n s i n theorem.
Theorem. B be a commutative Banach algebra wi t h i d e n t i f y
Let
, and
Let
M be a maximal id eal i n B such t hat M2 9 M and M 2 i s dense i n M. Then there e x i s t s a Banach algebra Y and a discontinuous homomorphism A of B i n t o Y .
,
Therefore
i f we t a k e
R : C'(R)
+
X
t o be t h e r e s t r i c t i o n
map, we conclude t h a t
A
o
R : C'(R)
+
Y
i s a d i s c o n t i n u o u s homomorphism. 11.4. Notes
,
remarks and r e f e r e n c e s .
T h i s c h a p t e r was drawn up b a s i c a l l y f o l l o w i n g Aron-Gomez-Llavona
[ll
. L e t R' ( r e s p . 0,) be an open s e t o f Rn
p 2 n). algebra
Let
g : R1
Ag = { f
o
-+
R2 be Cm
g : f
B
C"(n,)>
function. c
(resp. Rp)
(assuming
We t a k e t h e composite sub-
Cm(nl). G e n e r a l l y Ag
is
not
Chapter 11
194 W
T
-
U
closed i n
g :R
Example. L e t
.
g(0) = 0
C"(Q1).
-f
f B
As..
R : f E Cw(R)
-+
R 3. The f u n c t i o n f ( x ) , where Kg denotes t h e
j B
Ag
g(x) =
=
,f(t)
= f ( - t ) and
exp(-1/2x2) W
T~
Theorem.
g :
Let
Ql
+
djf(0) = 0
, f(0)
-closure o f
Ag
= 0
in
[21
The f o l l o w i n g theorem can be found i n Glaeser 11.4.1
exp(-l/x2)
,
Then
Ag = { f : R every
R be t h e f u n c t i o n d e f i n e d as
for
satisfies C"(R).
.
complying with the following condi-
Q2
tions :
lil
g i s real a n a l y t i c .
(ii) The rank o f g i s equal t o p on an everywhere dense open
.
s e t o f R1
(iii) g(Q,)
i s closed i n
Q2
For every compact s e t suoh t h a t K 1 ; g(X2) livl
K,
c
nl
m
T
~
K l c g ( R 1 ) t h e r e e x i s t s a compact s e t
~3S is' Tu-closed
I n s e c t i o n 4.6
i n crn(nl). we found t h e necessary and s u f f i c i e n t c o n d i t i o n s ,
so t h a t t h e composite subalgebras spaces
.
A
Cwbu(E).
9
would be c l o s e d f o r t h e case of
T r y i n g t o o b t a i n Glaeser t y p e theorems f o r t h e algebras
m > 0
,
i s one l i n e open f o r r e s e a r c h .
C:bU(E), I n o t h e r words, i t would be v e r y
i n t e r e s t i n g t o be a b l e t o c h a r a c t e r i z e t h e c l o s e d rank homomorphisms
A : C$bU(E) erties.
.+
c:~~(F)
,
A(f) = f
0
g
, using
function
g :
F
+
E"
prop-
F o r general t r e a t m e n t o f composite d i f f e r e n t i a b l e f u n c t i o n s , i n t h e f i n i t e dimensional case, see Bierstone-Milman 111
.
195
Chapter 12
THE PALEY-W I ENER-SCHWARTZ THEOREM I N I N F I N I T E DIMENSION
The Paley-Wiener-Schwartz
theorem c h a r a c t e r i z e s t h e F o u r i e r
t r a n s f o r m s o f d i s t r i b u t i o n s w i t h bounded (compact) s u p p o r t as b e i n g e x a c t l y t h e e n t i r e f u n c t i o n s o f e x p o n e n t i a l t y p e which a r e s l o w l y i n c r e a s i n g .
E
Let
be a r e a l Banach space ; E ' t h e dual space o f
a normed c o m p l e x i f i c a t i o n o f
E
and
E ' , where
complexification o f
(Et)I and
i t s dual; (Et)'
E;
Et
a normed
a r e i s o m e t r i c under t h e
n a t u r a l isomorphism between them. Nachbin-Dineen [ l l d e f i n e d t h e FrEchet space
gNbc(E;F)
of
i n f i n i t e l y n u c l e a r l y d i f f e r e n t i a b l e mappings o f bounded-compact t y p e f r o m
E
valued i n
F, when
E
i s a r e a l Banach space and
F
i s any Banach
space. If
5 i s a c o n t i n u o u s complex l i n e a r f o r m on c N b c ( ~ : t ) = & N b c ( ~ )
i t s Fourier transform
i s t h e complex v a l u e d f u n c t i o n on
E'
d e f i n e d by
A
for
I$ E E l .
The mapping
5
-+
<
t h u s d e f i n e d i s l i n e a r and i n j e c t i v e .
h
Each such 5
may be extended i n a n e c e s s a r i l y u n i q u e way t o an
e n t i r e f u n c t i o n on
30
+
for
I$ E E l
and
by s e t t i n g
i$ 1 =
C
+ti$
c
. Then <
$ E E'
0
i s e n t i r e o f e x p o n e n t i a l t y p e on
i n t h e sense t h a t t h e r e i s an i n t e g e r such t h a t
E'
and s l o w l y i n c r e a s i n g on and a r e a l number
<(e
(Ellt Q
2 0
Chapter 1 2
196
for
JI
C 2 0
for
E ' . More p r e c i s e l y and c 2 0 such t h a t 8
5
, where
6
, there
Im 5
i s an i n t e g e r a ? 0
i s the imaginary p a r t o f A
and r e a l numbers
.
5
Notice that
t h i s more s t r i n g e n t c o n d i t i o n on t h e e n t i r e f u n c t i o n E; on
not
o n l y t r i v i a l l y i m p l i e s , b u t i s a l s o i m p l i e s v i a t h e Phragmen-Lindel6f t h e o r y (Nachbin [ 9 ] t y p e on
El
, 947
)
by t h e f a c t t h a t
and s l o w l y i n c r e a s i n g on
5 i s e n t i r e o f exponential
E'.
I f we c o n s i d e r t h e F r e c h e t space HNb(Et ; E ) o f n u c l e a r l y e n t i r e complex valued f u n c t i o n s o f bounded t y p e on Et , t h e n a complex valued f u n c t i o n on
(Et)'
i s t h e Bore1 ( o r e q u i v a l e n t l y , t h e F o u r i e r )
t r a n s f o r m o f a continuous complex l i n e a r form on
(Et)'
i f i t i s e n t i r e o f e x p o n e n t i a l t y p e on
HNb(EC;t)
,if
( s e e Gupta [ l l ) .
and o n l y I n view
o f t h i s f a c t , one m i g h t n a i v e l y expect t o prove t h a t an e n t i r e complex valued f u n c t i o n o f e x p o n e n t i a l t y p e on on
E'
which i s s l o w l y i n c r e a s i n g
must be t h e F o u r i e r t r a n s f o r m o f a continuous complex l i n e a r form
on ENbc(E)
,a
E
t r u e statement i f
-Schwartz). (Note t h a t when
i s f i n i t e dimensional (Paley-Wiener F = C
i s f i n i t e dimensional and
E
C(E)
ENbc(E)
, the
i s t h e space endowed w i t h t h e space ENbc(E;E) = Schwartz t o p o l o g y (see Schwartz [ 3 1 ) ) . T h i s , however, w i l l be d i s c a r d e d
E
f o r every i n f i n i t e dimensional guaranteed by boundedness on proven t h a t i f
E
, even
i f slow i n c r e a s i n g n e s s on
E' i s
E l . I n f a c t , i n Nachbin-Dineen 111 i t i s
i s i n f i n i t e dimensional
, there
valued f u n c t i o n Of e x p o n e n t i a l t y p e on
i s an e n t i r e complex E'
bounded on
which i s
n o t t h e F o u r i e r t r a n s f o r m o f any continuous complex l i n e a r f o r m on CNbc(E). I n t h i s chapter
, following
f i c i e n t condition i s given of e x p o n e n t i a l t y p e on
, so
(EllE
Abuabara
11
,a
necessary and suf-
t h a t a complex valued holomorphic f u n c t i o n and s l o w l y i n c r e a s i n g on
E'
(when
E
belongs t o a wide c l a s s o f separable Banach spaces) i s t h e F o u r i e r transform of a d i s t r i b b t i o n w i t h bounded s u p p o r t : t h e Paley-Wiener-Schawartz theorem i n i n f i n i t e dimensions. 12.1
The F o u r i e r t r a n s f o r m o f d i s t r i b u t i o n s w i t h bounded s u p p o r t i n
in f in i t e dimensions . I n t h i s s e c t i o n we w i l l i n t r o d u c e t h e space c N b c ( E ; F )
of all
The Paley-Wiener -5chwartz theorem
197
i n f i n i t e l y n u c l e a r 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 o f bounded-compact t y p e
E
from
to
F
and t h e n s t u d y some of i t s main p r o p e r t i e s . S p e c i f i c a l l y ,
Cm
[E) o f a l l i n f i n i t e l y d i f f e r e n t i a b l e c y l i n CYl i s dense i n E N b c ( E ) .
we p r o v e t h a t t h e space
E
d r i c f u n c t i o n s on
The F o u r i e r t r a n s f o r m on an element o f
cINbc(E)
i s a l s o de-
fined. Let
E
F be a r e a l o r complex E by EE and i t s
be a r e a l Banach space and
Banach space; we denote a normed c o m p l e x i f i c a t i o n o f dual by
(Et)I
and ( E l l t
denotes a normed c o m p l e x i f i c a t i o n o f
;
a r e i s o m e t r i c under t h e n a t u r a l isomorphism between them. For
m = 0,1,2,
...,
let
PN(mE:F)
n u c l e a r m-homogeneous p o l y n o m i a l s f r o m
11
IIN
E ' ; (Et)'
E
be t h e Banach space o f a l l to
(See ( 0 . 3 . 4 ) ) .
F.
Let
be t h e n u c l e a r norm ; i t i s t o be d i s t i n g u i s h e d from t h e c u r r e n t P(mE;F) which i s denoted s i m p l y 11 - 1 1
.
norm on
We denote by c N ( E : F ) the vector space of a l l F such t h a t dmf(E) c PN(mE;F) i n f i n i t e Z y d i f f e r e n t i a b l e mappings f : E f o r m = 0,1,2 ,..., and each mapping dmf : E * PN(mE;F) i s differen-
12.1.1
Definition.
-+
, when
t i a b l e of f i r s t order An element of
EN(E;F)
t i a b l e mapping from E 12.1.2
Definition.
al
PN(mE;F) i s endowed with the nuclear norm. i s said t o be an i n f i n i t e l y nuclearly differen-
to
F.
Let c N b ( E ; F ) be the vector subspace o f c N ( E ; F ) F i n c N ( E ; F ) such t h a t f o r m = 0,1,2,~,.,
of a l l mappings f : E dmf : E PN(mE;F) i s bounded on bounded s e t s . An element of E N b ( E ; F ) i s said t o be an i n f i n i t e l y nucZearZy d i f f e r e n t i a b l e mapping of bounded type from E t o F b ) On I N b ( E ; F ) the following countable system of semi-norms i s defined +
-f
.
for
m,n = 0,1,2,.
...
,
Then g N b ( E ; F ) endowed w i t h t h e t o p o l o g y generated by t h a t c o u n t a b l e system o f semi-norms i s a m e t r i z a b l e l o c a l l y convex space.
198
Chapter 12
12.1.3
We denote the ctosure i n cNb(E;F) of
Definition.
CNbc(E;F)
Pf(E;F)
cZearZy d i f f e r e n t i a b z e mapping of bounded-compact type from E t o I t i s easy t o check t h a t
by
i s said to be an i n f i n i t e t y nu-
cNbc(E;F)
An eZernent of
..
ENbc(E;F)
. Then
i s complete
.
F the
following follows. 12.1.4.
equipped d t h the topology induced or, it
Proposition. ENbc(E;F)
by t h a t of 12.1.5
i s a Frdchet space.
ENb(E;F)
Remark,
E
a r e r e a l Banach spaces w i t h E ' m t h e bounded approximation p r o p e r t y , t h e n Pf(E;F) i s .rb-dense If
and
F
.
W
.
in
CwbU(E;F)
ed
support, except i n t h e t r i v i a l cases when
C;~,,(E;F)
AS
c o n t a i n s no non-zero f u n c t i o n s w i t h bouncj
(see comments b e f o r e d e f i n i t i o n t a i n s no
in
(5.3.3)). Hence, s i n c e 11 P I \ 5 1 1 PI1 i t follows t h a t ENbc(E;F) i s contained
f o r a l l m 2 1 (See p 6 PN(mE;F) (m 6 W),
C:bU(E;F) f o r every
having
, it
5.2.6)
F = 0
or
dim(E) <
m
follows that CNbc(E)
n o n - t r i v i a l f u n c t i o n s w i t h bounded s u p p o r t .
, con
More s p e c i f i c a l l y
,
i t i s impossible t o define the support o f a d i s t r i b u t i o n i n i n f i n i t e di-
mension as i t i s i n f i n i t e
dimension.
, there
12.1.6
Proposition. E N b c ( l l ) # gNb(1 ;1that ) is
nitely
nuclearZy d i f f e r e n t i a b t e function of bounded type from 11 to R
is an i n f i -
which i s not of bounded-compact type. Proof.
g :R
Let
+
R be a Cw-function , such t h a t f o r each
there e x i s t
Ck > 0
E R and
Ig(k)(t)I
t
t > 0
and
0
and
5 It\
N
6
with
W
K
=
It1
f : l1
+
5 6k
.
lg(k)(t)I
5 Ck
(For instance
N
k E
f o r every e- l / t
if
R as
l1be a compact
1
,k
E
111 and
E
> 0
with
E
.
5 Ak.
/ x n l < E f o r every x = (x,) 6 K If n>N and ei = (O,O, ...,O,l,O,O ,...) E 1; we have
be such t h a t
m > n > N
if
such t h a t
t <- 0 ) .
if
L e t us d e f i n e
Let
6k > 0
Let
m,n E 11,
199
The Paley-Wiener -Schwartz theorem
f o r every
x = (x,)
6
K. m
Since
( C m ( l ~ ) ; ~ u )i s complete
, ( * ) i m p l i e s t h a t f e C"(l1)
m
and t h a t
dkf(x) =
(0,3,4
and ( * )
)
, since
Finally
k Also from g ( k ) ( x n ) * e n , ( x E 11,k =0,1,2,..+). n= 1 k k i t f o l l o w s t h a t d f ( x ) 6 PN( 1 1 ) , ( x E 11, k =0,1,2.$.
1
k
dkf : l 1
is a
PN( 1 1 )
-f
k
C1(l,;PN(
11))
5
R/6,
Ck9
Thus,
k d f
f
12.1.7
implies that f ~ & ~ ( 1 ~ ) .
i s bounded on bounded s e t s
,
5 R , then
xlll
)
Ck/Ak
f
,
Therefore
.
On t h e o t h e r hand, s i n c e
df(B,)
the u n i t b a l l o f that
R = R (1
f
f €ENb(ll).
follows that
,I/
x E 1
If
(*)
Ti-complete,
C'-function.
We w i l l now see t h a t k = O,l,...,.
is
d f ( e n ) = e-'
i s n o t a precompact subset o f
11. Hence, a c c o r d i n g t o (12.1.5)
1;
en
it
, where
B1
and (4.1.1)
i t follows
b gNbc(l 1) *# Definition. A
mapping
. .,
k ( k = 0,1,2,.
of order
a)
f : E
+
F
i s said t o be a cyZindricaZ
i f there e x i s t a f i n i t e dimensiona2 sub-
space M of E and a k-times continuousZy d i f f e r e n t i a b l e mapping f
such t h a t
=
g
order k from
E
pE : E
pM : E
PM ,where
to
-+
M
-f
PEo(x) =
M
g : M * F,
. We
.
,@ #
0
and
E o t h e v e c t o r subspace o f
0
is a p r o j e c t i o n on
t h e v e c t o r space of all cytindricaZ mappings of k k F In t h e case F = C we denote = Ccyl(E), Ccyl(E;t)
Let @ E E'
Remark.
@ ( x o ) = 1 and define
o
k Ccyl (E;F)
denote by
12.1.8.
is
E o and
g : Ea
@(x)*xO
-+
;
F
b E F E
.
let
x08 E
generated by
as t g ( t x o ) = e .b
with
{ x o ) . I f we
200
Chapter 12
i t follows that
ED.
,
Therefore
,
words
[email protected] = g
@m*b : E
t i o n s o f type
c C
Pf(E;F)
m
o
.
(E;F)
CYl
PE 0
12.1.9
Proposition
top0 zogy
.
F
If
. Analogously
(E;F)
[email protected]:E
+
.
t h e func-
I n other
F.
.
CFyl(E)
2)
The vector subspace of ENbc(E;F)
E N b c ( E ) in theENbc(E)
i s a dense subspace o f
:E * F
e'.b
Cm-function on
i s a complex Banach space t h e same
, where
+€
generated by aZZ mappings
E ' and b
8
F
,is
dense i n ENbc(E;F)
F i s a c o q Z e x Banach space, then the vector subspace of
generated by a22 mappings of the form
b : E
e
+
gNb(E;F)
F;where 4 , $
€
E'
b E F, is dense in ENbc(E;F) ; t h i s is also t m e for the one gen-
arid
erated by a22 mqpings of the form
1) Since
Proof. C:yl(E)c
g : M
M of
E such t h a t
-f
f E Cyyl(E)
If
E
c Pf(M ;
topology
E )
f = g
= P(M ;
k, m e
1.
o
PM
, where
,
i t follows that
PM : E
+
M
i s a projection
t h e r e e x i s t s a sequence
E)
coverging t o
f
i n the I N b ( E )
I t i s easy t o see t h a t
Hence, we have t h a t
Therefore
then t h e r e e x i s t a f i n i t e dimen-
. We c l a i m t h a t t h e sequence
converges t o t h e f u n c t i o n
b e F.
and an i n 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
By N a c h b i n ' s theorem 1.2.1.
M.
E * F,where $ E E ' and
ei'.b:
i t w i l l be s u f f i c i e n t t o prove t h a t
P f ( E ) c Cryl(E)
CNbc(E).
s i o n a l subspace
(9,)
CYl
is a
g
R1 , a r e a l s o i n C:yl(E;F)
1)
of the form
on
Cm
is in
,m e
F
-f
i s t r u e f o r the functions o f type
If
E O and
p E o i s a p r o j e c t i o n on
(f,)
g
i n t h e gNbc(M)=&(M)=C"(M)
, where
fn = g n o P M
topology.
Indeed,
3
fix
The Paley-Wiener -Schwartz theorem
Since t h e sequence
(9,)
converges t o
g
201
i n the C ( M )
topology
, it
follows that
2)
Similar t o
l).#
i s continuous. Proof. point
Let
{($,,bn)1
(+o,bo)
.
be a sequence i n Set
gn = ei4n.b
proves t h a t t h e sequence E N b c ( E ;F)
x F
n (gn)
and
g = ei@'.b0
converging t o t h e
.
A s t a n d a r d argument
converges t o t h e mapping
g
i n the
topol o w . The Paley-Wiener-Schwartz theorem assures t h a t i f
t r i b u t i o n w i t h bounded (compact) s u p p o r t i n Rn then i t s Fourier transform f ( x ) = < Ty
,e
?
T
i s a dis-
( t h a t i s , T E &'(Rn))
,
i s g i v e n by t h e f u n c t i o n d e f i n e d by
-i<x,y>
>
.
T h i s f a c t suggests t h e f o l l o w i n g .
12.1.11
Definition.
(that i s , function on
If 5 i s a continuous linear form on &Nbc(E)
eEibc(E)),
i t s Fourier transform h
E ' defined by
(($1
= t(ei').
i s the complex valued
202
Chapter 12
12.1 .12
Theorem.
of
The Fourier transform
E&ibc(E)
i n a unique way t o a holomorphic function on A
+e
.
i s extended
be s e t t i n g
<($) =
<(ei+)
Proof.
The uniqueness o f t h e e x t e n s i o n f o l l o w s f r o m t h e n e x t f a c t which
f o r every
i s evident : I f f o r every
@
f :
-f
e E ' , then f
t
(
~
1
)
~
i s a holomorphic f u n c t i o n and
f(@) = 0
E 0. A
We have o n l y t o show t h a t t h e complex valued f u n c t i o n
on
(E')t
is
G-holomorphic ; t h a t i s , g i v e n two elements g : E
the f u n c t i o n
+
q1,
5
defined (El)
E d e f i n e d as:
i s a n a l y t i c . (See ( 0.9
)).
Indeed, l e t y be a piecewise c o n t i n u o u s l y
d i f f e r e n t i a b l e curve i n
D
By theorem
A a y + e i('lt
X'2)
.
E CNbc(E)
12.1.10
t h e mapping
i s continuous , Therefore
t h e r e e x i s t s a unique v e c t o r
.Hence
,
@ 5
0
, and
o= 5 Therefore
g
o we have t h a t
(0) =
i s ana y t i c and theorem 12.1.12
E'
i s proven.#
, by
(0.12.12)
The Paley-Wiener -Schwartz theorem 12.2
203
Characterization o f the Fourier transform o f d i s t r i b u t i o n s w i t h bounded s u p p o r t i n i n f i n i t e dimensions. I n t h i s s e c t i o n a Paley-Wiener-Schwartz theorem i n i n f i n i t e
dimensions i s presented. space
E
sequenee
I t s h o u l d be remembered t h a t a r e a l Banach
E
-+
E
pn(E)
i s f i n i t e dimensional, p,(x)
vi
* x
ei
a)
@
If
: E
+
a
by
5 6&ibc(E)
+E
and
pi(@)+
,
Pn.
Y
E N b c ( E ) generated by alZ functions
, where 9
, where
Im $
2 ) The scq’xence (Sn) c Y ’ where
5,
E E’.
* E
and i f f :
i s the function
then : I)
defined by f ( $ ) = S(ei ’) for $ E rnorphic f u n c t i o n on ( E ’ l E and t h e r e e x i s t
for every
if a
E be a separabZe Banach space with property ( B ) .
Let
Let us denote t h e vector space of o f t h e form
x c E
f o r every
denotes t h e a d j o i n t o p e r a t o r o f
Theorem.
,
o f continuous l i n e a r p r o j e c t i o n e x i s t s such t h a t
where 12.2.1
( B ) , (See ( 5 . 1 . 1 ) )
s a t i s f i e s t h e Restrepo p r o p e r t y pn :
c > 0
,m ,v
f is a h o k E
111 such t h a t
denotes the imaginary p a r t o f
$.
i s defined by
is equicontinuous. b) 1)
m d Zl
,i
ConverseZg
, then
f
thew exists
f :
-+
t
i s a function satisiying
c ( @ ) =f ( @ )
such t h a t c ( e i@ ) --
5 “Abc(E)
f o r every @ E E l .
Proof.
a ) 1) That
f
i s a holomorphic f u n c t i o n on
proven i n theorem 12.1.12. follows t h a t there e x i s t
was a l r e a d y
From t h e c o n t i n u i t y o f E, on E N b c ( E ) c > 0
, m ,v
E
111 such t h a t
it
204
r;
=
Chapter 12
, then
I$ t i$ E
dk ( e i r ;) ( x ) = ( i ) k e i d x )
.
Therefore,
'
I( dk(ei
e- $
= It follows that
qm,v (e")
Then we have t h a t
Sn(g) =
S(gn)
n = 1,2,
for
5 on Y , i t f o l l o w s t h a t , g i v e n 6 > 0 , m l , k l e N such t h a t n u i t y of
9
and qm,,k,(g)
, where
Set 6 , = 6 / akl >O
I
5
supll p n l I < n have proven t h e f o l l o w i n g i n e q u a l i t y ki
(.i i) for
1
< 6 =>
E
qm,,k,(gn) n = lyZ,
...,
' f o r every
n.
and Thcis
qml,k
g
6
~1
L e t us suppose t h a t we
Y , Then
( 9 ) < 61=>1
(Sn) c Y '
i s equicontinuous.
Next we show t h e i n e q u a l i t y ( i i ) .
from t h e c o n t i -
5(g
qml ,k, ( 9 )
5 a
and f o r e v e r y
... Now,
> 0 there are constants
205
The Paley-Wiener -Schwartz theorem
1
For
g =
representation o f
1 j=l
ct
j
k d g(y)
e i$j E Y
, where
Then
d kg ( y ) =
,let
8J. c E , Jlj
S
1
B. qJk J
be any
j=l J
E El, j =
1,2,
...,s .
It follows that
Hence,
f o r n = 1,Z ,... , and f o r e v e r y b)
Conversely
,
f o r each
g E Y
n E IN
.
we d e f i n e t h e f u n c t i o n
f,
: ( P ~ ( E ) ' ) ~C+
by
where
5 =
0 + iJl
I ) c
.
Therefore
,
(pn( E )
E (P,(E)')~
.
Then
f,
i s a holomorphic f u n c t i o n on
Moreover,
by t h e Paley-Wiener-Schwartz
theorem, t h e r e e x i s t s
A
T,
cE'(pn(E!)
such t h a t
fn($) =
Tn($)
f o r every $
6
pn(E)I
,
that is,
206
Chapter 12
.
f o r every @ E p n ( E ) '
aj
, j
E E'
. Now
n E 111
f o r each
=
5,
We d e f i n e
, if
Now
1
"j f ( + j
j=l
, by h y p o t h e s i s ( t n ) -- rJ(Y',Y)
1
K = ( t n
is
'From theorem 12.1.10, induced on i t by
E Y
, where a. E J
O
Pn)
,
.
i s equicontinuous
c Y'
u(Y',Y)
-
. Therefore,
compact.
i s s e p a r a b l e and, t h e r e f o r e
a(Y',Y)
C
i s metrizable
.
K w i t h t h e topoloqy
Hence, t h e r e e x i s t s a sub-
-
) o f (Sn) a(Y',Y) c o v e r g i n g t o some t 1 E KC Y ' . k i s dense i n E N b c ( E ) , p r o p o s i t i o n 12.1.9, t h e r e e x i s t s
sequence Since
Y
e
we have t h a t
1
Cn(d =
by
iOj
1
1 aj j=l
g =
, then
1,2,...,1
E Y'
(5,
Y
5
(E)
SIY =
such t h a t
= lim
k-
f(p;
k
@ 8 E'
12.3. Notes
,
Thus
. Then we
have t h a t
(@))= f($) A
f o r every
51
5
= f
.
Hence
, 2)
i s proven.#
remarks and r e f e r e n c e s .
The b a s i c r e f e r e n c e s o f t h i s c h a p t e r a r e Abuabara [ll and Nachbin
-
Dineen 111
.
The Paley-Wiener-Schwartz
theorem i n t h e f i n i t e dimensional
case can be seen i n Hormander [ll, Rudin 111, Schwartz 131 and
Yosida
[ 11.
I t i s well-known t h a t i f
E
i s f i n i t e dimensional,, t h e n
207
The Paley-Wierier -Schwartz theorem
ENbc (E)
i s a semi-Monte1
n u c l e a r space ( H o r v a t h [11, p . 239
and
Lesmes 1 2 3 , p . 98 ) . If
E
i s an i n f i n i t e dimensional Banach space
i s n o t a semi-Monte1 space. E N b c ( E )
,
then ENbc(E)
i s n o t a n u c l e a r space e i t h e r
(Abuabara [ll) . F o r a s t u d y on c y l i n d r i c a l mappings see Kr6e [l]. Abuabara [11 f i n d i n g s where adapted t o t h e case o f n u c l e a r spaces i n Ansemil-Colombeau [ll. Other r e s u l t s were o b t a i n e d by d e f i n i g a s u i t a b l e space o f
C"-
f u n c t i o n s i n Colombeau
-
Ponte L.11 f o r t h e
case o f n u c l e a r spaces and i n Colombeau- Paques [ l I spaces.
Other v e r s i o n s o f Paley
seen i n Colombeau [1 1
Colombeau
-
Wiener
-
-
Schwartz
Paques [ 2 1 - [ 3 1
f o r t h e case o f Banach theorems can be and Gal6 [11
.
This Page Intentionally Left Blank
209
Appendix I
WH ITNEY '3 SPECTRAL THEOREM
§I. If denote
k = (kl,...,
Ikl = kl t
... + kn
kn) E
N n and x
; k! = k l !
...
( x i , ...,x n ) e R n we will kn! ; x k = x ki i ... xnk n . =
In Illn an order i s given by : " k 5 1 i f , and only i f , for every j , l! 1 5 j 5 n , k . 5 1 'I. We will write ( k1 ) = i f k 5 1 and J j I ( k ) = 0 if 1 < k. Let A be a compact cube of Rn, E a finite-dimensional vector space over R. C m ( A ; E ) (resp. C m ( A ) ) , m E I , will denote the space of a l l E-valued (resp. real-valued) functions on A of class Cm endowed with the topology induced by the norm
where
I
I
denotes the norm on E (resp. the absolute value on R ) and
I t i s easy t o see that Cm(A;E) and Cm(A) norm, which i s equivalente t o 1) )I , is
On C m ( A ) ,
I ,I
verifies t h a t
If*gl, 5
Ifl,
are Banach spaces. Another
l g l m for every f , g e Cm(A),
210
Appendix I
and thus
Cm(A)
i s a Banach a l g e b r a .
I n t h e general case
, Cm(A;E)
is
Cm(A ) -modul e.
a
I n this
a E
A
i s the cardinal o f the set
N
If
note, a l l t h e modules considered a r e
{k
E
Cm(A)-modules.
Wn : I k / 5 m 1 , f o r each
we d e f i n e t h e map
T h i s i s o b v i o u s l y a continuous l i n e a r map, when c o n s i d e r i n g t h e p r o d u c t Hence, i f M i s a submodule o f Cm(A;E) , T!(M) N a v e c t o r i a l subspace o f E , and s i n c e EN has f i n i t e dimension
t o p o l o g y on
EN.
EN
i s closed i n
is
, T:(M)
and so i s closed.
(T,)m -1 (T!(M))
The p r o o f of Whitney's theorem t o be g i v e n i s a s i m p l i f i e d v e r s i o n B. Malgrange o f t h e o r i g i n a l . (See Malgrange "1 I
by
1.1. Definition.
a submodule M such t h a t
A function
of
c"'(A;E)
We w i l l denote by wise i n
M.
f e c ~ ( A ; E ) is said t o be a pointwise i n if f o r every a E A there exists g E M
the s e t o f a l l functions
i s closed 6
f e Cm(A;E) p o i n t -
Hence
I t i s easy t o check t h a t
a
1.
A
i s a submodule o f
, because as we saw b e f o r e (T:)-l
(T!(M))
Cm(A;E)
which
i s c l o s e d f o r every
A. If
d e R
,0
< d < 1
, l e t Cd
having s i d e s w i t h a l e n g t h o f 2d (jld,..
.,jnd)
when
j
. . ,jn
be t h e f a m i l y of a l l open cubes
and t h e c e n t e r b e i n g t h e p o i n t s are integers.
Appendix I
1.2.
Lemma. There e x i s t s a
t o c d such t h a t f o r
211
( + c ) CBCd subordinate
Cm-partition o f u n i t y
/ k l _ <m
where A i s a p o s i t i v e constant depending onZy on m and
Proof. L e t Y be a lxji
1 57 ,j
= I,...,n,
.
such t h a t
1 x . l >3/4 J
where
i s the center o f
xc
Yc(x) = 0 that
if
x
Yc(x) z 0 For every
It i s c l e a r t h a t
cd .
where
Moreover
,
Let
C
For every
C.
E
(there are a t most C
8
Cd , we
(bc)cscd
is a
Cd
Zn
m
we d e f i n e Yc
Y(x) = 1
,
1
Yc(x) = is a
Rn , t h e r e e x i s t s
5 Y(-
d
Cm-function,
C 6
Cd
such
cubes l i k e t h i s ) .
, Ikl 5
m
centered a t 0
,
k
m
and n) so t h a t
i t i s clear that there
and
if
j 5 n x - x
Cm-partition o f u n i t y subordinate t o
(depending o n l y on
A 1 (depending o n l y on
,
define
since f o r every
A.
Cd 6
1
i f there exists j
It i s c l e a r t h a t
6 C and f o r e v e r y x
C o i s t h e cube o f
e x i s t s a constant
Y(x) = 0
and
5
0 5 Y
Cm-function which v e r i f i e s
n
n ) so t h a t
9
By L e i b n i t z ’ s
of
formu a i t f o l l o w s t h a t
x B Rn
f o r every 1.3
I
Appendix
212
,
k
and
5
Ikl
.
m
Lermna. Let M be a sub-moduZe of
A such t h a t
a constant.
F
Let
@ 6 Cm(A) such t h a t
Proof. L e t
m ITa fl,
a E K t h e rank of
f o r aZZ
a 6 K.
..., Tma fP
6
K be a compact subset
Cm(A;E). Let
over R i s equaZ t o
T:(M)
. Then , f o r
M
@ = 1
every E > 0 , there e x i s t on a neighbourhood of K and
By hypothesis t h e r e e x i s t
i s a basis o f
.
T(M :
f l y ...,f
There e x i s t s
P
6
M
a > 0
f
6
p, M and
so t h a t so t h a t
for all
f
(a~,...,a ) 6 Rp w i t h P of
a
, so
l a j / = 1 ; t h e r e e x i s t a neighbourhood
j=l that i f
x E Va
fj(a)
- a kf j ( x ) I
1 ak Therefore i f
.
(al,. . ,a )
a 2N
<-
f Rp
and
P
> a
1
-
l a <m
T h i s shows t h a t
T:
LL = > 2N 2
..., T:
fly
9
laj
j=l
o , f
P
5 m
Ikl
f o r every
I
I
Va
j = 1,...,p.
= 1
,
x e
v,.
we have
a r e l i n e a r l y independents and so s i n c e
Appendix I
(MI)
dim (T:
, it
= p
..., TmX fP
IT;
1
fl,
Hence d e f i n e d on
213
follows that
Tz(M)
i s a basis o f
x E Va.
f o r every
i+bl,...,
there e x i s t real-valued continuous functions
Va
such t h a t
By compactness o f
Va.
neighbourhoods
bounded f u n c t i o n s
, we
K
Hence $l,...y
can cover i t by a f i n i t e number o f
,there exist
i+b,
fly
...,fl
c M
and r e a l - v a l u e d
so t h a t
1
1
TY F =
$P
,
$j(x) T z f j
K.
x e
f o r every
j=1 a E K
For every
,
let
f a be d e f i n e d as
1
Since t h e p a r t i a l d e r i v a t i v e s o f E
> 0
there exists
F
6 > 0 ( 6 < 1)
a r e u n i f o r m l y continuous such t h a t i f
, x'
x
E A
, f o r every )x - x'I < 6
then
(On A we c o n s i d e r t h e E u c l i d e a n norm). F
- fa
Let
and a l l t h e i r d e r i v a t i v e s v a n i s h on
every
k
where
B
,
Ikl
5 m and every
i s a constant
x E A
c 6
C' be t h e f a m i l y o f C' and l e t a, 6 C
, independent o f a,x
a l l cubes o f
n K.
If
Cd
and
x E
K
;
since
a, by T a y l o r ' s f o r m u l a f o r
with
W i t h t h e same n o t a t i o n s as lemma 1.2, let
x E A Ix
-
a[ < 6
and taking
c
i t follows that
. 6
d = ___ 2 JT which i n t e r s e c t s K
.
Let
Appendix I
214
@ ( x ) = 1 on a neighbourhood o f
then
By L e i b n i t z ' s
E
, lemma 1.2. and ( 1 )
formula
A ' = A B Zm
where
,f
K
o n l y depends on
m
M
and
i t follows
and n
, but
n o t on 6
and
E.#
We a r e now ready t o g i v e t h e p r o o f o f W h i t n e y ' s s p e c t r a l theorem. 1.4.
Let M
Theorem.
be a sub-module of
+ coincides with the module M
'Proof.
Let
B
of aZ1 functions pointwise i n
Va
of
so t h a t f o r e v e r y
a
.
From t h i s i t f o l l o w s immediately t h a t A sequence t h a t
B
claim:
.
p } ; reasoning l i k e a t t h e begindim (T:(M)) i t can be proven t h a t i f = r then there
e x i s t s a neighhourhood r
M
M
of
2
= { x E A : dim(T:(M))
P n i n g o f lemma 1.3,
d.im(T1 ( M ) )
Cm(A;E). The closure
i s closed. I f
P
p
2 0 , let A
-. B
P
P
,
x E Va
= Bp
i s open and as con-
'Bp-l
.
F i r s t we
h
(Hp) and
f
and
l@F
M
6
F
If
6
such t h a t
- flm
5
.
E
The statement
M
and
E
> 0, t h e r e e x i s t
@(x) = 1 f o r a l l
(H,)
x
@ E
Cm(A)
i n a neighbourhood o f
i s t r u e by lemma 1.3
because
B
P
Ao= B o i s
.
H i s t r u e f o r some p 2 1 Hence , P- 1 there exist function +p-l E Cm(A) and fp-l 6 M such t h a t C $ ~ - ~ ( =X )1 f o r a l l x i n an open neighbourhood V of P-1 5 ~ / 2 Let K = B V ; thus K i s a Bp- 1 and I @ p - l F fp-llm P P-1 compact s e t and so a p p l y i n g lemma 1.3 t o ( 1 - @ )F instead o f F , P-1
So, l e t us suppose t h a t
closed. given
E
> 0
,F
h
E M
-
.
Appendix I
f
there e x i s t
E
M
4
and
6
Cm(A)
,4
215
= 1 i n a neighbourhood o f
K,
such t h a t
Let
, f c M ,4
Cm(A)
E
+ $ ( l - $p-l)
$p = $p-l P
P
,
f
bPF - fplm 5 bP-lF - fp-lIm (H ) . I n particular, i f P The r e v e r s e i n c l u s i o n i s immediate . #
T h i s proves
(resp.
s2
Cm(R))
of all
I
+
+
.
f
.
We have and
P
$41 - 4p-1)F
-
flm 2
E.
proves t h a t
M^ c M .
We c o n s i d e r t h e space
Cm(R;E)
p = N
L e t 9 be an open subset o f Rn in
= f
P P- 1 = 1 i n a neighbourhood o f B
, (Hp)
E-valued ( r e s p . r e a l - v a l u e d ) f u n c t i o n o f
Cm c l a s s
, endowed t h e t o p o l o g y induced by t h e f a m i l y o f seminorms
I k l 9 i s a compact subset of
R . I n a s i m i l a r way, we d e f i n e Cm(R;E) ( r e s p . C"(R)) as t h e space o f a l l Cm-functions i n R w i t h v a l u e s i n E ( r e s p . R ) endowed w i t h t h e t o p o l o g y generated by t h e f a m i l y o f seminorms
when
1 Imk
K
where now
K
and
m
a r e a l l o w e d t o range over t h e compact subsets
o f R and t h e n a t u r a l numbers r e s p e c t i v e l y .
!T
I n t h e same way as above we d e f i n e
M
Also, i f
i s a sub-module o f
contained i n
1.5
Theorem. rf
Cm(!2;E)
, we
denote by
,
i.e.,
if
M
i s an Cm(R)-module
the intersection
(Whitzey ' s spectra2 theorem)
M is a sub-module of h
coincides w i t h
C"(R;E)
by
M
.
c"'(~;E)
m
, the closure i;i of M in c ( n ; ~ )
Appendix I
216
be a C m - p a r t i t i o n o f u n i t y i n R ($i)isI (Ai)iEI t o a l o c a l l y f i n i t e c o v e r i n g o f R by open cubes
Proof.
Let
xi
A
Let
; a p p l y i n g theorem 1.4
f E M
to
by t h e d e f i n i t i o n o f t h e topology on
$ifl Cm(R;E)
we have
, subordinated so t h a t TicQ. that $if E ,
, and thus
A
Hence
1.6
M cw. The r e v e r s e i n l c u s i o n i s immediate.
be ox open subset of R
Let R
Theorem.
. The cZosure w of M
of Cm(R;E)
m
n
and Zet M be a sub-rnoduk i n Cm(Q;E) is t h e modute of aZZ functions
m
(MI for every
i n c ~ ( R ; E ) such t h a t m -> 0 .
T,
Proof.
M the intersection
f
#
f E T,
x e R
and every
A
We a l s o denote by
msBi A
We o n l y have t o prove t h a t obvious t o (1.5)
.
Let f
f
A
E
,
M
K c R
On t h e o t h e r hand,
,...,
Y1
,
because t h e r e v e r s e i n c l u s i o n i s
rn any p o s i t i v e i n t e g e r .
and
According
belongs t o t h e c l o s u r e o f t h e module generated by
Cm(n) and so t h e r e e x i s t
y1
M cw
eC"'(n)
,
$1,
...
J o i n i n g b o t h i n e q u a l i t i e s we g e t
T h i s proves t h a t
k
f E
E Cm(n)
i s dense i n
cm(R)
such t h a t
1
y@l
, gl,.. . ,gl
E
M
M
over
such t h a t
Cm(n) ; t h e r e f o r e t h e r e e x i s t
Appendix I
1.7
Remark.
and
M
map Ta
I f the
i s a sub-module o f
f o r Cm(R;E)
217
i s d e f i n e d by
C m ( ~ ; E ) , t h e f o r m u l a t i o n o f W h i t n e y ' s theorem
should be " t h e c l o s u r e o f
M
i n C"(R;E)
coincides w i t h
n T i 1 ( T a ( M ) ) ' I . T h i s f o r m u l a t i o n i s t r u e s i n c e i t can be proven ( s e e acR , Tougeron [l]) t h a t 'ITa f E Ta (M) i f and o n l y i f f o r Malgrange [l] every
, T:
0
m
(M)".
f E T :
I t s p r o o f r e q u i r e s r e s u l t s and techniques
which go beyond t h e scope o f t h i s
note.
Previous r e s u l t s i n which t h e domain and t h e range had f i n i t e dimension a l l o w e d us t o assure t h a t i f
M
a c R
were a sub-module and
,
h
then
Ta(M)
was c l o s e d and t h u s
a l r e a d y seen i n c h a p t e r 10
M was c l o s e d t o o .
However, as we have
i n i n f i n i t e dimension t h i s does n o t g e n e r a l l y
o c c u r . T h e r e f o r e i t i s u s e f u l t h a t another more adequate f o r m u l a t i o n o f Whitney's theorem t o be extended t o t h e i n f i n i t e dimension be g i v e n . V
M
If
i s a sub-module o f
Cm(R;E)
we w i l l denote by
M
the
intersection.
8
n {f
=
E Cm(R;E)
F o r each
:
> 0
E
, there exists g
E M
such t h a t
aen k
- a kf ( a ) I 5
, f o r e v e r y k , / k l 5 in3 . I n a s i m i l a r way , i f M i s a submodule o f Cm(n;E) , M" = n { f 8 Cm(G;E): For each E > 0, t h e r e 13 g ( a )
E
aen
exists
1.8.
g
6
M
such t h a t
Theorem. L e t
t h e cZosure
M
meN
of
m c
M in
M
U
{m}
.
m
If M is a sub-moduZe of C (R;E) , coincides w i t h
Cm(n;E) A
Proof. x E R
I t i s obvious t h a t
and
p c
W
, for
V
M cM cM
every
n E
W
.
V
M. V
On t h e o t h e r hand
there e x i s t
gn E M
if
f c M
such t h a t
,
Appendix I
218
Hence
i s a covergent sequence i n
(TE 9,)
and s i n c e V
M
that
J! (M)
TE (Cm(Q;E))
i s closed i t f o l l o w s t h a t
T!
TE f
with l i m i t
T h i s proves
f E TE(M).
c M.
92. I n t h i s s e c t i o n we w i l l f o c u s o u r a t t e n t i o n on t h e s p e c i f i c case
E =
of
E
.
I t i s easy t o see t h a t
Cm(A;C)
i s a commutative Banach
a l g e b r a w i t h u n i t y , and w i t h a n a t u r a l i n v o l u t i o n , t h e manning Then
Cm(A;t) i s
A,
each maximal i d e a l i s t h e k e r n e l o f t h e e v a l u a t i o n a t a p o i n t i n
A .
I t i s n o t d i f f i c u l t t o prove t h a t t h e spectrum o f
An i d e a l
I
o f an a l g e b r a i s c a l l e d p r i m a r y i f t h e r e i s a unique
maximal i d e a l c o n t a i n i n g i t . ideal
J ( x ) = I f E Cm(A;k)
I n the algebra
Cm(Q;c) i t i s c l e a r t h a t t h e
vanishs i n a neighbourhood o f
: f
p r i m a r y i d e a l ; o t h e r examples o f p r i m a r y i d e a l s i n ideals 2.1
I ( x ) = { f E Cm(A;&) :
ma,.
For every
coincides w i t h Proof.
x E
I t i s clear that
f E I(x)
A
T!
x
1
is a
are the
C"(A;C)
I.
f = 0
J(x)
the closure
of
J(x)
in
Cm(A;C)
I(x).
o n l y have t o prove t h a t If
f .
i s a *-algebra.
Cm(A;t)
i.e.
f +
and
a E A
I(x)
i s closed, and t h a t
so we
G).
I(x) c a # x
@ = 1 i n a neighbourhood o f
J(x)cI(x)
a
E Cm(Rn) such t h a t
there exists @ = 0
and
i n a neighbourhood o f x; so we
have
and @ f = 0
i n a neighbourhood o f
t h e map f o F 0 plying 2.2 x
8
Lemma
A
belongs t o
Whitney's theorem
.
If
such t h a t
x , and thus r$f E J ( x )
J(x)
and
that
f E
T;
f o = T;
m).
I i s a closed primary ideal i n
G) c I.
f
.
.
If
a = x then
T h i s proves, ap-
Cm(A;t)
there exists
Appendix I
219
A such t h a t I c { f E Cm(A;C) : f ( x ) = 01 = M(x) . L e t f E J ( x ) , and A o = { y E A : f ( y ) = 01 . Since f E J ( x ) , x i s an i n t e r i o r p o i n t o f A"; so t h e r e e x i s t s @ e C"(R";t) such t h a t I#I(X) = 1 and @ ( y ) = 0 i f y does n o t belong t o A,, i n t e r i o r o f A. . T h e r e f o r e t h e i d e a l I 1 = I -+ I h E C m ( A ; t ) : h = 0 i n A \ i o l i s n o t c o n t a i n e d i n M(x) which i s t h e unique maximal i d e a l t h a t c o n t a i n s I , and t h e n I 1 = Cm ( A ; a ) . As a consequence t h e r e e x i s t g e I and h 8 Cm(A;E) , h = 0 i n A \ A o s u c h t h a t 1 = g t h and so f = f g t f h . Hence f = f g E I because f h = 0 . Proof.
x E
Let
From lemmas 2.1 and 2.2
,
i t follows
2 . 3 . Proposition. t o r every x E A , the i deal I ( x ) i s the l e a s t among a l l t h e c l o s e d primary ideaZs of Cm(A; E ) contained i n the maxima2 idea2
t f E Cm(A; C ) : f ( x ) Since obviously
I(x)
I(x)
= 0
I.
i s c o n t a i n e d i n a unique maximal i d e a l
# M(x) , n o t e t h a t
I(x)
M(x)
and
i s a c l o s e d i d e a l which i s n o t
an i n t e r s e c t i o n o f maximal i d e a l s . According t o t h e preceeding n o t a t i o n , i f
I
i s an i d e a l o f
?=
r l ( I t I ( a ) ) . So another f o r m u l a t i o n o f aEA W h i t n e y ' s s p e c t r a l theorem would be: Cm(A;c)
2.4.
i t results that
Theorem.
Every closed i deal i n
Cm(A;t)
i s the i n t e r s e c t i o n of
alZ closed primary i d e a l s t hat contain i t . A s i m i l a r t r e a t m e n t can be made t o t h e one j u s t mentioned changing
A
f o r an open subset o f Rn.
This Page Intentionally Left Blank
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Sobre l a densidad de subalgebras p o l i n o m i a l e s de funciones d e b i l mente d i f e r e n c i a b l e s
,
Congr6s du G.M.E.L.
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M a t h h a t i q u e s , Actes du 6"
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Ma1grange ,B.
[ll
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Margalef,J.
[ll
and Outerelo,E.
Un teorema de e x t e n s i d n de Whitney en dimensidn i n f i n i t a p.,
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235
INDEX
absolutely summing operator ................................... 15 algebra. topological .......................................... 3 weighted ............................................. 5 polynomial ...................................... 11 161 Nachbin polynomial ................................... 11 - m-admissible .........................................29 m-admissible of compact type ......................... 30 angel icy spaces ........................................... 14 94 approximate contacts .......................................... 45 approximation property ................................. 12.13. 155 115 120 Bernstein. theorem ...................................... bounded approximation property ................................ 14 weak approximation property ..................... 133 141 99 Cauchy. 0- ................................................... class Q' ..................................................... 70 collectionwise normal space .................................. 19 149 compact weak approximation property .......................... 108 composite subalgebras ........................................ 36 conditions (N) ................................................ (No) ............................................... 42 137 constant sign ................................................ 5 continuous linear maps ......................................... k-multilinear maps .................................. 5 symmetric k-multilinear maps ........................ 8 convergent. 0- ............................................... 99 derivative. H- ............................................... 71 HW- .............................................. 149 71 ,72 differentiable. H- ....................................... Hw- ......................................... 149
.. -. . .
.
.
. .
.
.
Index
236
dispersed. compact ............................................ 10 Uowker. theorem ........................................ 18. 19 93 embedded. C- ................................................. 18 17 fixed. z-filter ............................................... Fourier transform .................................... 195.196. 197 function. uniformly continuously differentiable ............... 54 semiproper ......................................... 108 infinitely nuclearly differentiable ................ 197 of bounded type 197 of bounded-compact type ....... 198 infinitely differentiable cylindric ............ 197. 199 quasi-di fferentiabl e ............................. 69 70 uniformly differentiable ........................... 115 -~ ................ of order m 116 12 Grothendieck. approximation property Grothendieck-Pietsch. theorem ........................... 15.95. 96 Hadamard differentiability ................................... 71 148 weakly differentiability ............................ 17 holomorphic function .......................................... G- .............................................. 17 homomorphisms ............................................ 177. 182 injective. spaces 18 82 James-Klee. theorem ........................................... Kaplansky. theorem 19.94 localizable ................................................... 20 8 multilinear mappings Nachbin conditions 65 theorem .........................................26.162 m-algebras .......................................... 36 Nachbin-Shirota. theorems .................................. 19.88 not vanish ..................................................... 1 nuclear. norm 10 Paley-Wiener-Schwartz. theorem .................. 195.196.201. 203 partitions of unity ........................................... 32 9 polarization formula ........................................... polynomials .................................................... 8 continuous n-homogeneous 9 continuous ....................................... 9
.
..
...............
Y
.
Y
..
.
Y
..........................
.
............................................
.........................................
........................................... ..........................................
. . .
.................................................
. .
..........................
Index
polynomials product
.
..
E-
237
........................ 9 ................ 10 ............................................. 12. 151
continuous o f f i n i t e t y p e n-homogeneous n u c l e a r c o n t i n u a u s
............................................. 115. 203 ............................................. 19 quas i - d e r i v a t i ve .............................................. 70 realcompact spaces ....................................... 16. 17 Rosenthal theorem ......................................... 20. 94 1 separates p o i n t s ............................................... smooth ........................................................ 58 - u’- .................................................. 54 - u n i f o r m l y ............................................ 58 90 space. NS- .................................................... Stone-Weierstrass h u l l ....................................... 11 , suhspace ................................... 11 theorem .... ............................... 20 s t r o n g l y s e p a r a t i n g ........................................... 1 superreflexive space ........................................ 57 37 s u p o r t i n g f a m i l y .............................................. 5 symnietric k - m u l t i l i n e a r maps ................................... T a y l o r . theorem ................................................ 5 T i e t z e , theorem ............................................... 82 topology. i n d u c t i v e l i m i t ...................................... 3 compact-open o f o r d e r m ....................... 7.65. 127 , f i n e ............................................... 53 , compact-compact o f o r d e r m .......................... 66 bw .................................................. 82 property (6)
pseudocompact s e t
.
.
..
. .
.
. .. .. .
bw*
.................................................
................................................. rbw* ................................................
cbw
82 83 84
.............. 86 ............ 119. 179 u l t r a p o w e r ................................................... 57 weak approximate c o n t a c t s ..................................... 45 weakly compactly generated spaces ............................. 17 - continuous ......................................... 79. 93 - u n i f o r m l y continuous ............................... 79. 90 - s e q u e n t i a l l y continuous ............................ 93. 94 ~
.. .
u n i f o r m convergence on weakly compact .o f o r d e r m on bounded s e t s
Index
238
. -
........................ 115. 178 differentiable .................................. 133. 134 92 weakly* uniformly continuous .................................. Weierstrass. theorem ...................................... 23. 127 Weierstrass-Stone. theorem .................................... 11
weakly
.
uniformly differentiable
239
INDEX OF SYMBOLS
.
.......................... Ac(X) ......................... ............................ A ............................. bw ........................... bw* .......................... b.w.a.p. .................... c ( X ) .......................... c s ( F ) ......................... C(X;F) ..................... Cm(Q;F) ....................... C"(X;F) ....................... P ( X ) ......................... CF( X;F) ....................... C;(X. K.F) ..................... C F ( X ) ......................... C?(X) ......................... A(X)
1
2 8
h
W
co.
1
82 82 142 1
1 1.11 1 1
2 3 3 4 4
.....................
4
.......................
4
(IR)
CrnVm(X)
9
....................... 5 trnVW(X) ..................... 30 ................... 53. 65 c"'(E;F) C( E;F) ....................... 65 m , ,C (E.X) ..................... 73 C, (E.F) ...................... 79 C"(U;F)
........................ Cwk(E;F) ........................ Cwbu(E;F) ....................... Cwb(E;F)
79 79 79
.......................... 83 ............................. 83 , ,C bu( E";F) ..................... 92 Cw*b(E";F) ...................... 92 cwsc( E ;F ) ....................... 94 105 Cwbu(E;X) ...................... c : ~( E~;F) ...................... 116 ........................ 129 c!(E;F) c:~ ( E ;F ) ....................... 135 148 Cwk( E;X) ....................... , :C w( E;X) ...................... 149 c.w.a.p. ...................... 149 178 Cp(X;Y) ........................ Cp(Egw, ) ....................... 178 CP(Egw* ; Fgw* ) ............... 178 ccylk (E;F) ...................... 199 Ccylk ( E ) ........................ 199 E Z ............................. 12 E E F ........................... 12 E ' .............................. 21 C(Ebw)
cbw
Index o f symbols
240
........................... 2 1 E<< F ........................ 57 ....................... E(S. r ) 58
E*
......................... 82 ELw* ........................ 82 Fbw .......................... 87 Eg ......................... 197 197 ( E C ) ' ....................... ....................... 197 197 I N ( E ; F ) .................... ENb(E;F) ................... 197 ENbcE;F) ................... 198 E(M) ....................... 200 €INbc( E) .................... 201 Ebw
.......................... 1; .......................... L(E;F) ...................... k L ( E;F) ..................... L s ( k E;F) .................... LE(E;;F) ..................... Lc(E;F) ...................... I f / k,
7 48 5, 8 5. 8
5. 8 12 13
...................... 73 L @ ~ ~ ( " E ; F................... ) 98 98 L @ ~ ~ ( ~ E ................... ;F) LQc(n E;F) .................... 98 M ............................. 7 M ............................ 8 N ............................. 1 ITi ............................. 1 PI; ............................ 1
............................ 2 pi., .............................. 2 P("E;F) .......................... 9 P ~ ( ;F) ~ E ......................... 9
py(f)
P(E;F)
9 9
........................ 10 ........................ 10 PC('E;X) ........................ 72 pObU(n~ ;F) ...................... 97 pOb( nE;F) ....................... 97 P@, (".F) ....................... 97 p O S c ( " ~ ; ~ ...................... ) 97 pwbu( 'E ;F) ...................... 97 98 Pwb(nE;F) ....................... , , ,P ("E.F) ...................... 98 PWk( 3E;F) ....................... 148 ............................. 4 'k. p ........................... 1 ........................... 1 ..........................
n Lc( E;X)
n
........................... ..........................
........ ..........
Pf(E;F)
.........................
...........................
.. ........................... T x ( X ) ............................ ......................... Top;(X) T!
f
4 47 7 7 26 29
V
IN,
............................
1
............................. 54 ............................. 17 Z ( X ) ............................ 16 r o ............................... 2
UCD WCG
Index of symbols
rm .............................
z
rm .............................
2
.......................... 11 ......................... 15 np(E;F) ....................... 15 T~ .......................... 1. 65 T p .......................... 1.19 m T~ ...................... 2.65, 1 2 7 A(M)
np(T)
m
T~ ............................2.
8
m Ti
3
.............................
m ......................4.119.
179
T~
............................ 30 m T~ ......................... 66. 73 T~~ ........................... 86 Tb ............................ 90 TA
m
T~~
.....................
140. 149
.......................... 11 . /IN ........................
Q(Gn)
1
10
241
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