APPROXIMATION OF CONTINUOUSLY DIFFERENTIABLE FUNCTIONS
NORTH-HOLLAND MATHEMATICS STUDIES Notas de Matematica (112)
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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 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
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/
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
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 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
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)
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
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 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- 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 =
~ ~ , _ 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)
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
0. I f
dim(E)
(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
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 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 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 0 ( i = 1, x and y
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
0, such
"weak" ; thus
11
f(x)
-
: for a l l balls
B
in
E,
y 6 B
, I@(x-y)I
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$l,...,q~~
so we may choose p o i n t s
Qi(z.) = 6 , for 1 5 i , j 5 m J ij continuous on 28 , t h e r e i s some c o n s t a n t y, such t h a t
,I]
r > 0.
~ / 2 There i s c l e a r l y
and
no l o s s i n g e n e r a l i t y i n assuming t h a t t h e elements a r e l i n e a r l y independent
f o r any
E
E
i s uniformiy
0 < y
0 there i s a f i n i t e x, y E B,(E) s a t i s f y @ ( x - y ) = 0 (I$ 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
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}
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
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 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
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
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-
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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 ) / 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 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 . #
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
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
{$I,...
s a t i s f y i n g IOi(x-y)I
0
such
that
i s bounded i n
E",
and t h a t
Combining ( 2 ) and ( 3 ) sets
(4)
, we
conclude t h a t f o r a l l y 6 F " and a l l bounded
B c F", 1i m E+O
g(y
t
Ex)
- g ( y ) - g1 ( y ) ( e x ) ] = o
in
E;~*
uniformly
6
Chapter 11
184
for
x e B
and so
g1
i s the derivative o f
Assuming now t h a t derivatives o f rivative of
g
g.
(j - 1 ) th i s the j
are the f i r s t
gl,gzy...yg(j-l)
, where
g. L e t y
j < m , l e t us show t h a t g dg j e F" be f i x e d , and denote by C(y) t h e unique sym-
m e t r i c j - l i n e a r mapping a s s o c i a t e d t o
g.(y). J
u : FiW*+ P(J-lFiW*; E bw* " )
Let
be t h e l i n e a r mapping g i v e n by u ( x ) ( z ) = C(y)(x,z,.!j:!!.,z). t h a t g i v e n any bounded s e t
We must show
B c F",
As i n ( 2 ) , we f i r s t n o t e t h a t from t h e d e f i n i t i o n o f easy t o see t h a t f o r some 6 > 0,
dJ[A($)](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 ) 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
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
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 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=>13/4 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
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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This Page Intentionally Left Blank
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