Approximation Problems in Analysis and Probability

APPROXI MATI 0 N PROBLEMS IN ANALYSIS AND PROBABILITY

NORTH-HOLLAND MATHEMATICS STUDIES 159 (Continuation of the Notas de Matematica)

Editor: Leopoldo NACHBIN Centro Brasileiro de Pesquisas Fisicas Rio de Janeiro, Brazil and University of Rochester New York, U.S.A.

NORTH-HOLLAND -AMSTERDAM

' NEW YORK OXFORD TOKYO

APPROXIMATION PROBLEMS IN ANALYSIS AND PROBABILITY

M.P. HEBLE Department of Mathematics University of Toronto Toronto, Canada

1989 NORTH-HOLLAND - AMSTERDAM ' NEW YORK

OXFORD TOKYO

ELSEVIER SCIENCE PUBLISHERS B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands Distributors for the U S A . and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 655 Avenue of the Americas NewYork, N.Y. 10010, U.S.A. Library o f Congress Cataloging-in-Publication Data

H e b l e . M. P. Approximation problems i n analysis and probability / M.P. Heble. p. c m . -- ( N o r t h - H o l l a n d m a t h e m a t i c s s t u d i e s ; 159) Includes bibliographical references. I S B N 0-444-88021-6 1 . A p p r o x i m a t i o n t h e o r y . 2. M a t h e m a t i c a l a n a l y s i s . 3. P r o b a b i l i t i e s . I. T i t l e . 11. S e r i e s . O A 2 2 1 .H375 1989 511'.4--dc20 89-16147 CIP

.ISBN: 0 444 88021 6 0 ELSEVIER SCIENCE PUBLISHERS B.V., 1989

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science Publishers B.V. / Physical Sciences and Engineering Division, P.O. Box 103, 1000 AC Amsterdam, The Netherlands. Special regulations for readers in the U.S.A. - This publication has been registered with the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A., should be referred to the publisher.

i

No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Printed in the Netherlands

To My mother Girijabai My uncle Rama Rao and Sushila, Ajay and Sucheta

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Vii

Table of Contents

ix

Introduction Chapter I. Weierstrass-Stone theorem and generalisations - a brief survey

8 1. Weierstrass-Stone theorem 2. 3. 4. 5.

Closure of a module - the weighted approximation problem Criteria of localisability A differentiable variant of the Stone-Weierstrass theorem Further differentiable variants of the Stone-Weierstrass theorem

Chapter II. Strong approximation in finitedimensional spaces 1. 2.

H. Whitney’s theorem on analytic approximation C” -approximation in a finitedimensional space

Chapter III. Strong approximation in infinitedimensional spaces § 1.

5 2.

§3. §4. 5 5. §6. § 7.

§8. §9. 0 10. fill.

Kurzweil’s theorems on analytic approximation Smoothness properties of norms in LP-spaces C”-partitions of unity in filbert space Theorem of Bonic and Frampton Smale’s Theorem Theorem of Eells and McAlpin Contribution of J. Wells and K. Sundaresan Theorems of Desolneux-Moulis Ck-approximation of Ck by Cw-a theorem of Heble Connection between strong approximation and earlier ideas of Bernstein-Nachbin Strong approximation - other directions

Chapter IV. Approximation problems in probability 1.

5 2. 53. $4. 5.

Bernstein’s proof of Weierstrass theorem Some recent Bernstein-type approximation results A theorem of H. Steinhaus The Wiener process or Brownian motion Jump processes - a theorem of Skorokhod

Appendix 1 : Appendix 2: Appendix 3: Appendix 4: Bibliography Index

Topological vector spaces Differential Calculus in Banach spaces Differentiable Banach manifolds Probability theory

1

2 6 19 31 34 41

41 61 77 77 95 99 101 103 107 111 121 127 153 154 169

170 172 178 183 189 20 1 215 223 229 237 243

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ix

Introduction The classical Weierstrass-Stone theorem and the Bernstein-type weighted approximation theorems were greatly extended by L. Nachbin. Another aspect of approximation theory, now called strong approximation and initiated by H. Whitney, had simultaneously developed, with contributions in finite-dimensional spaces as also in infinitedimensional spaces, by various individuals. At the same time, several approximation results in a probabilistic setting - from the elegant probabilistic proof of Weierstrass’ theorem by S. Bernstein to the later results on convergence of stochastic processes established by A.V. Skorokhod and other later authors - were being added to the literature. The material in this book covers some special aspects of the approximation theory of functions, viz. strong approximation in function spaces, as also certain approximation results concerning stochastic processes. The choice of topics reflects only the author’s taste. Within the narrow range of topics chosen, I have tried to do as thorough justice as I could, to the subject as also to the contribution of various individuals active in their respective areas; any possible omission of names is unintentional. This book is meant to be a monograph, of interest to research workers in the fields of analysis, probability, and stochastic processes. Graduate students, hopefully, will find it useful not merely as a source of information but also as an incentive to spur then on to do further work. The author has noted other monographs recently published, covering related topics. However, the contents of these books show that the overlap between these and my present monograph is negligible (e.g., cf. K. Sundaresan and S. Swami-

Introduction

x

nathan: “Geometry and non-linear analysis in Banach spaces”, Springer Verlag Lecture Notes in Math. No. 1131, 1986; and

J.G.Llavona: “Approximation of continuously dif-

ferentiable functions”, Notas de Matematica No. 130, North Holland 1986).

A quick description of the contents of this book appears to be in order. The material is divided into four chapters. The first chapter gives a quick survey of the classical Weierstrass-Stone theorem, Bernstein’s weighted approximation problem and Nachbin’s extension of the classical Bernstein approximation results. The material in this chapter excluding sections 4 and 5 , is mostly a summary of Professor Leopoldo Nachbin’s monograph: “Elements of approximation theory”. In Chapter I1 we present strong approximation results in a finite-dimensional space R” - first H. Whitney’s theorem on strong approximation by real analytic functions, and then some results on Coo-approximation (strong sense); the latter appear to have been commonly known and there are excellent expositions in several monographs, hence we have been content with only a summary in this book. Chapters I11 presents strong approximation results in finite- or infinitedimensional separable spaces, starting with Kurzweil’s extension of Whitney’s theorem (on analytic approximation), and ending with some recent results established by this author, as also an indication of possibilities in other directions. We also explain a connection between strong approximation results and the earlier Bernstein-Nachbin ideas. In Chapter IV we present some probabilistic approximation results, starting with a quick look at Bernstsein’s well-known proof of Weierstrass’ theorem with some recent developments, and ending with some results by A.V. Skorokhod on approximation of stochastic processes. Here, again, individual choice was the guiding factor. We thought it necessary

xi

Introduction

to leave out the enormous area of weak approximation - an area which has found excellent exposition in several monographs, e.g., M. Rosenblatt: “Markov processes, structure and asymptotic properties” (Springer Verlag 1971), and D. Pollard: “Convergence of stochastic processes”, (Springer Verlag 1984).

As for organisation of the the material, theorems, lemmas, etc., are numbered according to chapter and section; thus Theorem I1 2.1 means Theorem 1 in section 2 of Chapter 11. Equations and formulae are numbered consecutively, but the numbering is separate for each section and each chapter. We have used the common symbols: “3” for “there exists”, “V’ for “for all” or “for any”, “3” for “such that”,

“+” for “implies”,

and “H”for “if and only if”, C denotes the set of complex numbers, R the set of real numbers and R” the n-dimensional real Euclidean space. There are four appendices at the end of the book explaining basic background material without proofs, and with sufficient further references. The writing of this monograph was partially supported by an NSERC operating grant. Thanks are due to Shirley Chan and Pat Broughton for patient and expert typing of the manuscript. I am indebted to Professor Leopoldo Nachbin, first, for encouraging me to write this book for the Notas de Matematica series, and secondly for permission to summarise the material of his monograph: “Elements of approximation theory”.

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CHAPTER I Weierstrass-Stone theorem and generalisations

- a brief survey

In the first three sections of this chapter we shall review known results concerning the classical Weierstrass-Stone theorem (cf. [SO]),Bernsteiris approximation problem, (cf. [41])and further generalisations by L. Nachbin. The material in this chapter, excluding sections 4 and 5, is taken from L. Nachbin’s published lecture notes [39]and for this reason we shall often present only a summary, leaving it to the reader t o refer to his monograph for further details. Any missing proofs will be found in [43].For convenience in presentation, the chapter is divided into five subsections. For the concepts of functional analysis used we refer the reader to Appendix 1. Throughout this chapter we shall assume E to be a completely regular space, i.e., a Hausdorff topological space such that for any a E E , and any closed subset F c E not containing a, there is a continuous real-valued function f on E such that 0 5 f 5 1, f ( u ) = 1 and f ( z ) = 0 for any

2

E F.

We shall denote by C ( E ;C) the commutative algebra with unit of all continuous complex-valued functions on E ; if E = 4 we set C(E ;C) = ( 0 ) . For convenience we shall often write C ( E ) for C ( E ;C). Every compact set K norm

l l f l l ~ ‘kfmax{ If(z)l, z € K } on C ( E ) . Let I?

cE

determines an algebra semi-

be the family of such semi-norms (cf.

Appendix 1). We shall understand that C ( E ) is endowed with the compact-open topology viz. the topology rr determined by the family I? of semi-norms. Then C ( E )becomes a topological algebra, i.e., a topological vector-space which is also an algebra. If E is compact then C ( E ) becomes a Banach algebra.

Chapter I

2 We note that the topology

7,.

is the topology of uniform convergence on compact

sets. This fact will be utilised in certain proofs. In proving results for a C ( E )with E completely regular, it will be often convenient to prove any one such result on the assumption that E is a compact space, the result will then follow for a completely regular space. Sometimes it may be necessary to use the result: every element in C ( K ) ,where

I< c E is compact, has an extension to C ( E ) . The subalgebra of C ( E ) consisting of real-valued continuous functions on E will be denoted by C ( E ;R), and will be endowed with the subspace topology inherited from r r .

$1. Weierstrass-Stone theorem. The first non-trivial theorem that we should note is the following result, due to S. Kakutani and M.H. Stone. A subset L

c C ( E ;R) is called a lattice if f , g

E L

=+

sup(f, g) E L and inf ( f , g ) E L. Theorem 1.1.

(Kakutani-Stone)

Then f belongs to the closure of L

Proof.

Let L

c

C ( E ;R) be a lattice and f E C(E;R)

Vxl,3'2 E E , and VE > 0 , 3 g E L 3

Only the sufficiency part of the assertion meeds some attention. As noted ear-

lier, it is enough to prove the statement on the assumption that E is compact. So suppose E is compact. Let f € C ( E ;R) satisfy the condition (l),and let compactness, it follows that for any given t E E , 39, E L satisfying:

E

> 0. Then, using

Weierstrass-Stone theorem and generalisations

-

a brief survey

3

Then again using compactness, we find that 3g E L satisfying:

f(u)- E < g(u) < f ( u ) Thus 119 - f

+

E

for any u E E

.

l l ~< E , and hence g E z.This completes the proof.

The next step is to note a result concerning the closure of an ideal I

ideal in C ( E )is a nonempty subset I

c

c C ( E ) . An

C ( E )such that for any g E I, f g E I Vf E

C(E). Proposition 1.2.

Let I

c

C ( E ) be a n &:id, and let f E C ( E ) . Suppose N is the

closed subset of E consisting of all x E E 3 g(x) = 0 Vg E I. Then f E

Proof.

We note that N = nfEr f-l({O}),

f %f

= 0 on N .

hence N is closed. Also for each z E E the

delta function 6, is continuous (cf. Appendix). For each z E N , 6, vanishes on I, hence

6, = 0 on f. This proves

“j”.

Next suppose E is compact and suppose f E C ( E )vanishes on N . Let define X = {z E E 0

[If

I If(.)[

5 h 5 1, and h ( z ) = 1Vz

2

E}.

Then X is closed, and X

nN

6. Now let

c C ( E ;C) is said to be

+ f E X , where f is the complex conjugate of f .

(Weierstrass-Stone) Suppose A

c C ( E ) is a subalgebra,

assume to be self-adjoint in the complex case. Let f E C(E).Then f E A H (1)

Vz1,22

EE

h EI3

“+”. The proposition follows.

We turn now t o the Weientrass-Stone theorem. A subset X

Theorem 1.3.

> 0, and

E X (one such h does exist). Now set g = f h E I. Then

- 9113 < E , and hence f E f. This proves

self-adjoint if f E X

=

E

3 f(z1) # f ( ~ ) , 3 E g

A 3 d.1)

(2) Vx E E 3 f ( z ) # 0, 3g E A 3 g(x) # 0. For the proof the following lemmas are needed.

# 9(z2);

which we

4

Chapter I

Lemma 1.4. If k 1 0, and

I Ip(t) - It1 I
0, then 3 polynomial p

: R + R, 3 p ( 0 ) =

3 It1 5 k.

Since the statement is trivial for k = 0, we shall suppose k

Proof.

0 and

Lemma is true for k = 0 then it is true for any k

It1 5 1; then I k p ( i ) - It11 < k& for

It1

> 0. Also if the

> 0. For suppose I p ( t ) - It1 I
03w E W

V equivalence

3 lw(x) - f(x)l


0, and y E F .

Then

T I ({ y ) )

c

E is an equivalence class and is com-

pact. By assumption 3w, E W 3 Iw,(x) - f ( ~ ) l
0, 3gl,. . .,gm E C ( E ) ,and

Before stating the theorem of Choqukt-Deny, some definitions are in order. A subset

S c C ( E ;R) is called a sup-lattice if f , g E S if f , g E S

3 sup

( f , g) E S , and is called an inf-lattice

+ inf(f, g) E S. If 4, $ are continuous linear functionals on C ( E ;R), we write

If 4 2 0, we say that d is positive. Theorem 2.4.

(ChoquBt-Deny) Suppose S C C ( E ;R) is a sup-lattice and let f E

C ( E ;R). Then f E 3 (in (C(E; R))

@

(1) V positive 4 E C(E;R)' (the dual of C ( E ;R ) cf Appendix 1) and Vu E E

(2) V positive 4 E C ( E ;R)')

d(f) 2

inf(#(g) 1 g E S } .

At this point it is necessary to explain Bernstein's weighted approximation problem and for this purpose we should first explain the concept of a weighted locally convex space of continuous functions. The next step thereafter is to explain L. Nachbin's contribution towards extending the classical Bernstein approximation problem, viz. his work on the weighted approximation problem for modules.

We first turn to the concept of a weighted locally convex space. Let V be a set of upper-semicontinuous positive functions on E. We shall assume that V is directed, i.e., if

9

Weierstrass-Stone theorem and generalisations - a brief survey v1,v2

E V , then 3A > 0 and 3v E V 3

called weights. The vector subspace of

v1

5 Xv and v2 5

C(E ) consisting of

Xu. The elements of

V are

all f 3 v f is bounded on E ,

for each v E V , will be denoted by CVb(E). Then each v E V determines a semi-norm p,(f) = sup{v(z) If(x)l +

Iz

E E } on CVb(E). w e shall understand that CVb(E) is.

endowed with the natural topology i.e., the locally convex topology determined by the The vector subspace of C ( E ) consisting of all f 3 Vv E family of semi-norms {p,,(.)}uE~.

V and VE > 0 the closed subset {z E E

I v(z). If(z)l 2

E}

is compact, will be denoted

by CV,(E). It is clear that C V , ( E ) c C\’b(E), and the natural topology on CV,(E) is understood to be the topology induced by CVb(E).

A few observations are in order at this point. The family of semi-norms { p , , ( - > } , , ~ v in the preceding paragraph is directed because V itself is directed. If V consists of a single function v(-) then we shall denote CVb(E) and CV,(E) by C u b ( E )and C,,,(E), respectively, and if V consists of the constant function 1 then C,,,(E),Cuoo(E) will be denoted by Cb(E),and C,(E) if v x E E 3 v E

v

3 v(z)

>

respectively. CVb(E) and CV,(E) are Hausdorff spaces 0. C h ( E ) is a module over Cb(E),and CV,(E) is a

sub module over Cb(E). Furthermore, i f f E CVb(E),g E

C(E),and 191 5

If1

then

g E CVb(E);a similar remark holds for CV,(E).

We further note the following: (i) if 1’ is the set of characteristic functions of all compact subsets of E , then C ( E ) = C&( E ) = CV,(E) as locally convex spaces; (ii) if

V consists of just the constant function 1, then C&,(E) = Cb(E),and the topology on Cb(E) is defined by the single norm same norm

I l f l l ~ ; (iii) if E

I l f l l ~ ; also in this case CV,(E)

= C,(E), with the

= R”, and V consists of the Ipl for all C-valued polynomials

Chapter 1

10

#

C(R"); in this case if a norm 11x11 is fixed on R",

+ 11x11)"'

for m = 0,1,2,. . ., on R" then it is known that

p on R" then CVb(R") = CV,(R")

and W consists of functions (1

CWb(R") = CW,(R")

= CVb(R") = CV,(R") as locally convex spaces; in this case the

C(E)thus obtained are said to be rapidly decreasing at infinity.

elements of

We note the following result.

Proposition 2.5.

Ca(E)n CV,(E) is dense in CV,(E).

For stating the next theorem of Dieudonnb on dense subsets in tensor products

some terminology should be explained. For each i = 1,.. . ,n let E; be a completely regular space, Q a directed set of upper-semicontinuous positive functions on Ei; let

-

E = El x .. x En,and V = V1 x Vz x ... x V,. The following theorem of DieudonnC holds.

Theorem 2.6. fi

x

... x

(Dieudonnh)

fn, f; E

The set of all finite sums of tensor-products f = fi x

(CQ)=(E;),Z= I , . . . , n , is dense in CV,(E).

We turn now to the Bernstein approximation problem (cf. [43]). Let f be a C-valued function on R", and suppose f is locally bounded, i.e., bounded on every compact subset of R". Then f is said t o be rapidly decreasing at infinity if the following equivalent conditions hold: (1) pf is bounded on R" for any p E 'P(R") (the set of polynomials on W);

(2) pf

-+

0 at infinity for any p E P(R"). The implication (2) + (1) is clear. To see that

(1) + (2) define q on R" by q ( z ) = x:

+ xg + + xi, x =

(l),pqf is bounded for any p E P(R"), q

Bernstein problem (first form).

-+ 00

. . ,x,)

( ~ 1 , .

E R"; assuming

at infinity, hence pf -+ 0 at infinity.

Let w 2 0 be upper-semicontinuous on R" and

rapidly decreasing at infinity, i.e., P(R") C Cw,(R"),

or equivalently, P(Rn)C Cub@").

Weierstrass-Stone theorem and generalbations - a brief survey The weight w is said to be fundamental if P(R") is dense in Cw,(R");

11

and the Bern-

stein problem consists in finding necessary ;i.nd sufficient conditions for a given weight w to be fundamental. We remark here that the Weierstrass theorem means that every characteristic function of a compact subset of R" is a fundamental weight; and this implies that every w 2

0 which is upper-semicontinuous on R" and has compact support, is a fundamental weight.

Bernstein problem (second foTm). at infinity, i.e., P(R"). w

c C,(R")

Let C(R"), and suppose w is rapidly decreasing

or equivalently P(R") w

that the load w is fundamental if P ( R " ) .w is dense in C,(R").

c Cb(R").

We then say

The Bernstein problem

consists in finding necessary and sufficient conditions for a given load w to be fundamental.

For convenience in the sequel, we shall call these problems Bernstein's problem I and

Bernstein's problem 11, respectively. The next proposition follows.

Proposition 2.7.

Let w E C(R"), w 2 0. Then w is a fundamental load if and only if

w is a fundamental weight and w ( x ) > 0 for any x E R".

In order to explain the work of L. Nachbin in this area we have to explain the

"weighted approximation problem" for modules. Let A

c C(E)be a subalgebra containing the unit, and W c CV,(E)

subspace; we shall also assume 14' to he n inodiile over A i.e., AW

approximation problem consists in determining

be a vector

c W. The weighted

in CV,(E) under these circumstances.

In the special case in which A consists only of the constant functions, W is the most

Chapter I

12

general vector subspace of CV,(E). sists of all f

e CV,(E)

3 every.

In this case, all we can say about

+ E CV,(E)*

w is that

con-

vanishing on W must also vanish at f .

The general case of the weighted approximation problem is reduced to the special case just mentioned, by considering the subsets of E on which the functions belonging to A are constant, i.e., by introducing on E the equivalence relation E / A mentioned earlier. The following definition is formulated with this view in mind.

We say that W localisable under A in CV,(E)

Definition 2.8.

Vf E CV,(E), f

E

w (in CV,(E))

#

if the following holds:

Vv E V , VE > 0 and V equivalence class X modulo

EIA, 3w E W 3 .(.)a

lw(x) - f(x)l < E

vx E x .

The strict weighted approximation problem consists in finding necessary and sufficient conditions for W to be localisable under A in CV,(E). We note that if the following conditions are satisfied:

(1) A is separating on E ; (2) W is everywhere different from 0 in E i.e., Vx E E 3 w E W 3 ~ ( x #) 0; then W is localisable under A in CV,(E)

($

W is dense in CV,(E). Hence if the con-

ditions (1) and (2) are satisfied then corresponding to every sufficient condition for localisability to be established below there will be a corollary asserting density of W in

CV,(E). Furthermore the strict weighted approximation problem can be seen to be a generalisation of the Bernstein approximation problem, as follows. Consider the Bernstein problem I; let E = R", V = { w } , A = P(R"), W = P(R");or consider the Bernstein problem

Weierstrass-Stone theorem and generalisations - a brief survey

13

11; let E = R", V = {l},A = P(R"), W = P ( R " ) .w. Then condition (1) in the preced-

ing paragraph is satisfied; the condition (2) is always satisfied in the case of Bernstein's problem I; and as for Bernstein's problem 11, the condition (2) amounts t o saying that w(z)

# 0 for any z E

R", and in this case Proposition 2.7 justifies assuming the condi-

tion (2). Hence if these conditions (1) and ( 2 ) hold, then finding necessary and sufficient conditions for P(R") = Cwm(R") in Bernstein's problem I, or for P(R")w = Cm(R") in Bernstein's problem 11, is equivalent to finding necessary and sufficient conditions for localisabilit y. The next step is to consider how the weighted approximation problem can be reduced t o a finite-dimensional Bernstein problem. We shall denote by fl, the set of all upper-semicontinuous functions w 2 0 on R" which are fundamental weights in the sense of Bernstein. Let G ( A ) be a subset of A which topologically generates A as an algebra over C with unit i.e., 3 the subalgebra over C of A generated by G ( A ) and 1 is dense in A (in the topology of

C(E)); also let G ( W ) be a subset of W

3 G ( W ) generates W as a mod-

ule over A i.e., the submodule over A of W , generated by G ( W ) is dense in W for the topology of CVm(E). The following theorem now holds.

Theorem 2.9.

Suppose C(E)= C(E; R); if we let C(E)= C(E;C) then we shall as-

sume that G ( A ) consists of real-valued functions. Suppose further that Vv E V,Vul,. 3w

. . ,a,

E G ( A ) and Vw E G ( W ) , 3a,+l,.

E f l 3~ v(z) Iw(z)I 5

. . ,U N E

G ( A ) with N 2 n, and

w(al(z),. ..,nn(x),... ,aN(z)), foranyz E

E . Then W is

Chapter I

14

localisable under A in CVw(E). For the proof we shall need the following two lemmas.

L e m m a 2.10.

Let E = IIierEi be a Cartesian product of Hausdorffspaces and

K: a

collection of compact subsets E with an empty intersection. Then 3 finite subset J

I 3 if I I j denotes the natural projection E

4

c

IIiejEi then I I j ( K : ) has an empty inter-

section.

Lemma 2.11.

Let f E CVw(E),v E V , and

E

> 0.

Further suppose V equivalence

class X C E modulo E / A 3w E W 3 v(X)lw(x) - f(z)l

G ( A ) ,h i , . . .W , E G ( W ) and (~1,.. . ,Y(,

v(x).

12

cr;(al(z),

< EVX E X. Then g a l , . . . , a , E

E Cb(R") 3

. . . ,a,(r))wi(z) - f ( r )

IE

.

Vx E E

i=l

Proof.

Let F denote the space of all real-valued functions on G ( A ) ,and we shall as-

sume that F is endowed with the Cartesian product topology. Let

:

T

E

-t

F be the

continuous mapping which to x E E associates n ( z ) E F 3 the value of ~ ( x at) a E G ( A ) is a(.) E R. Let y E n ( E ) , then every a E G ( A ) is constant on n-'(y). topologically generates A, hence every a E A is constant on ~-'(y).

Now G ( A )

Therefore ~-l(y) is

-

contained in an equivalence class modulo E / A ~-l(y) is actually an equivalence class, for 7r is constant on every equivalence class.

Now by the assumption of the lemma, for each y E n ( E ) 3w, E f(z)l

< E Vx

E n-'(y).

W 3 v(r) Iwy(x)-

We shall assume wy is in the vector subspace of W generated by

G(W); for let 6 = SUP{+).

Iwy(.)

- f(x)l

I

5

E .-'(y>}

;

15

Weierstrass-Stone theorem and generalisations - a brief survey

then 6

< 8, for v(z) -

Iwu(z) - f(z)l attains a maximum on n-'(y).

generates W , hence 3 ~ 1 ,... ,a, E A and 3wl,.

Let

G ( W ) topologically

. . ,w, E G(W ) 3

X1, . . . , A, respectively, be the constant values of a l , . .. ,a, on the equivalence class

n-l(y). Then

I .(z).

+..*+Xr~r(~)-.f(z)I Ial(z)wl(z) + + ar(z)wr(z) - wy(z)I

+

Iwy(z) - f(z)l
0 and V equivalence class X

c E modulo E / A , 3w

EW3

a ) . ( .

~w(x) - f(i)l < E Vx E X. Then

by Lemma 2.11, gal,.. . , a , E G ( A ) , 3 ~ 1 ,... ,w, E G ( W ) and gal,.. . ,am E Ca(R") 3

By the preliminary remark in the last paragraph 3w:,. . . ,w k E W 3

for any x E E , and i = 1,.. . ,m. Putting together (7) and (8),and taking 6 =

2,we

find

v(x)* I W(X) - f(z) I< 2~ Hence f E

VX E E .

r.This proves localisability of W under A in CV,(E).

Corollary 2.12.

Suppose C ( E ) = C ( E ; R ) ;or suppose

C(E)= C(E;C) and that

G ( A ) consists of real functions. Suppose hrther that G ( A ) ,G ( W ) are both finite: G ( A ) = {q,. . . , a , , } , G ( W ) = {wl,.. .,to,,,};

and that

Vv E V,Vi= 1,. . . ,m 3w E 0,

3

v(z) I wi(z)15 w(al(z),.. . ,a,(z))V~ E E . Then W is localisable under A in CV,(E). Before stating the next theorem and its corollary, it is necessary to explain some notation. If x = (XI,.. .x,) E R", we shall denote by

151 the

point

. . , lznl). We

(1~11,.

shall also denote by 0: the set of all w E 0" which are decreasing in the sense that if

r,y E R" and IzI 5 IyI then Theorem 2.13.

W(X)

2 w(y);this implies W(X) = ~ ( 1 ~ 1 ) .

Suppose C(E) =

C(E;R); or suppose C(E)= C(E;C ) and that

A is self-adjoint. Also suppose that Vw E V,Val,.. . , a , E G ( A ) and Vw E G( W)

Chapter I

18

3a,+l,.

.. ,a N E G(A) with N

2 n and 3w E Rd, satisfying

Then W is localisable under A in CV,(E).

Remarks (1) We note that corresponding to Corollary 2.12, there is an analogous corollary of Theorem 2.13.

(3) Using the weighted Dieudonnk's Theor(im 2.6 on density in tensor products, the arguments of Theorems 2.9 or 2.13 are reduced to one-dimensional arguments. We shall denote by

rn the set of all upper semi-continuous y

a fundamental weight in the sense of Bernstein for any k that I?,

c R; however there are examples showing that

a certain k

> 0 then yf E R,

set of all y E ~ ( 5= )

for all f?

> k.

> 0.

I?,

2 0 on Rn 3 yk is

By taking k = 1 we see

# R,.

Also if yk E R, for

Furthermore we shall denote by :?I

I?, which are decreasing i.e., z,y E R", 1x1 5

IyI

+ $5)

the

2 y(y) hence

~(IzI).

Theorem 2.14.

Suppose A is self-adjoint and that Vv E V,Va E G ( A ) and Vw E G ( W )

37 E I'f 3 V(Z) lw(x)I 5 y( Ia(z)I)Vx E E. Then W is localisable under A in CV,(E).

19

Weierstrass-Stone theorem and generalbatwns - a brief survey

53. Criteria of localisability In this section several criteria of localisability due t o Nachbin, will be established; we find that each of these turns out to be a special case of the one immediately following.

Theorem 3.1.

Suppose C(E)=

C(E;R);

or suppose C ( E ) =

C(E;C) and that

A is self-adjoint. In either case, suppose further that Vv E V , Va E G ( A ) and Vw E

G(W)3C > 0 and 3c > 0 satisfying

Then W is localisable under A in CV,(E).

For the proof we need the following two lemmas.

Lemma 3.2.

Suppose V is a directed set of upper-semi continuous positive functions

of E . Suppose Cb(E) C CV,(E) and that A is a subalgebra of Cb(E) which is separating on E , contains the constant function 1 and is self-adjoint in the complex case. Then

A is dense in CVm(E).

Lemma 3.3.

Let y 2 0 upper-semicontinuous on R be 3 3C

y(t)

Then y E

Proof.

5

> 0,3c > 0 satisfying:

Ce'l*l V t E R.

rl. Clearly y(.) is rapidly decreasing at infinity, since trne-'ltl

any rn E R. Let t , x , y E R, z = z

+ i y E C, and define e ,

--f

0 aa t

--f

00,

for

E C(R;C) by e L ( t ) = eiZt for

Chapter I

20

t E R. We then note

provided (yI < c. Hence, e, E Cb(R,C)if JyI < c. Denote by S the open strip:

Now let

9 be a continuous linear functional on Cy,(R; C ) , and define f : S --+ C by:

f(z) = d ( e , ) , z E S.

Our claim now is: f(-)is analytic on S. To show this we write

for z E C, t E R. We shall show that if z E C and Iz1


0, then the function tme-ct

Therefore for m

>0

Now Stirling's formula

> 0, is an integer

on t >_ 0 attains its maximum at t =

$; and hence

Weierstrass-Stone theorem and generalisations - a brief survey

21

implies that

m -- e.

lim

(m!)llm

m-w

Hence using Cauchy's criterion for convergence of a series of positive terms, we see that

Thus from (2) and (3) we see that the series in (1) converges uniformly for t E R and z E C with IzI

< c.

Now define u, E C(R;C) by u,(t) CI(,(R;

C ) because tme-'l*l

+

0 as t

+

= (it)" 00.

for t E R, m = 0,1,. . .; then u, E

Because of the uniform convergence of (1)

for t E R we see that

where convergence is understood in the sense of Cy,(R;

C). Now let a E R: then we also

have m

where convergence is understood in the sense of C-y,(R;

C), hence eau, E C,, (R; C).

Thus we find

provided Iz - a1

< c.

Here a E R is arbitrary; thus f is seen to be analytic on S. Further-

more

Now suppose

4 vanishes on P(R;C).

Then f("'(0)

= 0 for m = 0,1,2,.

. . by (4)

(taking a = 0). Since f is analytic on S, it follows that f is identically zero on S, hence

22

Chapter I

Vx

on R, i.e., q5(ez) = 0 all

ez,z E

E R. Denote by A the vector subspace of Cb(R; C) generated by

R. Then q5 = 0 on A .

It is then clear that A is a self-adjoint separating subalgebra of C @ ; C) containing By the last lemma A is dense in Cy,

1 and Cb(R;C) c Cy,(R;C). on Cy,(R;

(R;C), hence q5 = 0

C). Hence, we see that +=O

on

P ( R ; C ) + q5=0

Hence by the Hahn-Banach theorem we see h a t

on

Cy,(R;C).

P(R;C) is dense in Cy,(R; C).

7 E i l l . But now yk satisfies the same kind of assumption as y for integer

yk E il, for k

> 0 hence y

E

Proof of Theorem 3.1.

Hence

k > 0, hence

rl. This completes the proof of the Lemma. We now apply Theorem 2.14, and the last Lemma, taking

7 ( t )= Ce-'l*I for t E R , where we notice that y E I?!.

This proves the theorem.

We shall next turn to the quasi-analytical criterion of localisability. The following theorem will be established.

Theorem 3.4.

We shall suppose A is self-a.djoint, and also that Vv E V,Va E G ( A ) and

V w E G (W )we have

whereM,

= sup{v(z).

I

~ ( z ) ~ w ( Ix In: ) E

E} for rn

= 0,1,2, .... Then W is

localisable under A in CV,(E). Before turning to the proof of Theorem 3.4,we recall the following concepts from the area of infinitely differentiable functions. Suppose M = { M ,

1m

= 0, 1,2,

. . .} is a

sequence of strictly positive numbers. We shall denote by C(M) the set of all indefinitely

23

Weierstrass-Stone theorem and generalisations - a brief survey

differentiable complex-valued functions f, each defined on some open interval I C R (I depending on f), and satisfying the following estimates for its successive derivatives: for every compact subset K

c I, 3C > 0, and

3c

>0 3

Vx E I< and m = 0,1,2,. . .. We say thn.t C(M) is a quasi-analytic class if the following is true: i f f E C(M) and 3a E I such that f ( " ) ( a ) = 0

V m = O , l , 2 ,... , then

f E 0 on I.

Clearly this amounts to the requirement that every

f

E C(M) is determined within

C(M) by the knowledge of its Taylor series at a single point a E I (though the Taylor series o f f at a or at any other point in I are not assumed to be convergent

-

in fact may

fail to be convergent).

If M , = m!(m = 0,1,2,. . .) then by a theorem of Pringsheim, C(M) consists of complex valued functions which are analytic on open intervals of R , and hence can be called the analytic class. In this case C(M) is clearly quasi-analytic. On the other hand it is known that not every class is quasi-analytic. For instance, if

for m = 0, 1,2,. . . , then C(M) is not quasi-analytic, for the function f defined by: f ( x ) = e-i,O

< 2 < 1, and f(x)

= 0 for -1


(I/"'

= 1 , 2 , . . . and

1,2,. . . , then we shall have

Theorem 3.11.

E

I

E

E . If we choose c 3 0 < c
- c for m

= 1,2,

... .

l a ( ~ ) ~ w ( z5) lC for m = 0,1,2,

m!, and summing up, we obtain

hence

2 for m = 1,2, . . . . Let C = sup(1, Mo}. Then

. . . , and

I

E

E. Dividing by

30

Chapter I

Hence the sufficient condition of Theorem 10 holds. Conversely suppose 3C

> 0, and 3c > 0 3

We now note the elementary inequality

which has already been noted above in the course of the proof of Theorem 3.1. Then we find:

Choose c' 3 0

v ( z ) . la(z)mw(z)l

5 c

< c' 5 ceC-'/"' for m

= 1,2,

This proves the proposition.

(--)m m

for any

zE E

. . . and then -& M,

2

.

& for m = 1 , 2 , . . . .

31

Weierstrass-Stone theorem and generalisations - a brief survey $4. A differentiable variant of t h e Stone-Weierstrass t h e o r e m

In this section and the next we shall give an account of differentiable analogues of the Stone-Weierstrass theorem for certain algebras of r-times continuously differentiable functions. We shall explain a theorem due to L. Nachbin (cf. [44]), and in the next section mention some generalisations by Aaron and Prolla. Suppose M is a differentiable manifold of order r 2 1, and dimension n 2 1. Let

A be the algebra of Cr-functions i.e., r-times continuously differentiable functions on M endowed with the topology of uniform convergence of C ' functions up to order m on compact subsets of M . Nachbin established the following theorem ([44])p. 1550).

A necessary and sufficient condition for the algebra A ( B ) generated by

T h e o r e m 4.1.

a subset B c A to be dense in A is that the following conditions are satisfied:

(1) for each E E M 3 f E B 3 f ( E )

# 0;

(2) for each pair of points (,q E M , E (3) for each

Proof.

# 71, 3f E B

3

f([) # f (q);

< E M and for each tangent vector 0 # 0 at E, 3f E B 3 3 # 0. c M be compact, and W an

Only the sufficiency needs justification. Let K

open connected subset of M containing I< and such that

is compact. For each point

in M 3 a function which is not identically 0 in a neighbourhood of this point; hence 3 finite number of functions

Now let 3f1 f2

E B 3

E B3

E

f1,.

E M , and let

# #

. . ,fn

01

#

E B 3 { fl(z), . . . ,fn}

#

(0,. . . ,0} for z E

0 a tangent vector to A4 at

0. If n 2 2, then 3 tangent vector O2

0. Then if n 2 3, 3 tangcnt vector 0 3

Thus we obtain tangent vectors O1,.. . ,On at

#

v.

6. Then by hypothesis

0 at

# 0 at 6

E

3

3

= 0. Then let

$=

a

6 a.nd functions f1,. . . ,fn E B

3

= 0, etc.

$# 0

32

Chapter I

(i 5 i 5 n) and

= 0 (1 5 i

< j 5 n). with values in R",

Consider the linear mapping defined on the tangent space at which maps 6 +

- + cn&,

{ %,. . . %}. Each vector in R" is the image of a vector 6 = c161 +

i.e., this mapping is an isomorphism on R".

The implicit function theorem shows that the mapping z is a homeomorphism of order

T

2

. . ., gh E

B and 3 open subsets

w is compact, hence 3 functions

c M (1 5 i 5 b) covering

.

+ {gi(z),. . ,gk(z)} is a homeomorphism of order r of

Now set fa+(;-*),,+,

= gf. If [ , q E M , €

( z , y ) in a neighbourhood of

{ f l ( z > , . . . ,fn(z)}

(cf. DieudonnC: Foundations p. 272) of a neighbour-

hood of [ in M onto an open subset in R". The set gi,

+

([,r]).

#

r],

3 each mapping

onto an open subset of R".

then 3f E B 3 f(z)

#

f ( y ) for all

The space

is compact and disjoint from the diagonal of

w x w and thus 3 functions h l , .. . ,hc E

B 3 for ( z , y ) E Q we have

Write fa+bn+i = hi. Then we consider the mapping 9 : M

+RN

with

N = n+bn+c defhed by the mapping z

+

(fl(z), . . . ,f~(x)}. This mapping 9 is a homeomorphism

) order r# in of order r of W on the submanifold @ ( W of Consider the inverse @-'

RN.

: @ ( W )+ ( W ) . Let f E A. Then

tinuously differentiable of order

T

f@-l

is r-times con-

on 9 ( w ) . By a special case of a theorem of Whitney

Weierstrass-Stone theorem and generalisatwns - a brief survey

(cf. [63]),3 r-times continuously differentiable function on

@(K), hence f(z)

+ on RN

33

3 + ( z ) = f(@-'(z))

= @(fl(z),. . . , f N ( z ) ) on I 0 3

Definition.

PIU("E; F) is the subspace of P("'3; F ) consisting of those

m homoge-

neous continuous polynomials which are weakly uniformly continuous on bounded subsets of E (equivalently, on the unit ball in E ) .

37

Weierstrass-Stone theorem and generalisations - a brief survey

Deflnition. f :E

C,"(E; F ) is the space of m-times continuously differentiable mappings

F satisfying:

( a ) D j f ( z ) E Yw("E; F ) , for z E E , j 5 m.

(b) DJf : E

4

Y,JmE; F ) is weakly uniformly continuous on bounded subsets of

(i) If E is reflexive then f E CE(E;F ) iflfor each j 5 m, D; : E

4

Pw("E; F ) is

weakly continuous on bounded subsets of E (cf. Restrepo [49] for the case m = 1, E reflexive). (ii) C,"(E;F ) contains all functions of the form g o T where T is a continuous linear operator of finite rank and g E Cm( T ( E ) ;F ) (iii) C,"(E; F ) contains no non-zero function with bounded support except when F = 0 or dim E

Deflnition.

< 00.

f'

is the locally convex topology of uniform convergence of order m on

bounded subsets of E , endowed upon C,"(E;F ) , and is defined by all semi-norms of the form

where B is an arbitrary bounded subset of E. Each such semi-norm is well-defined. The topology 7s" on C,"(E; F ) is defined in an obvious manner. Deflnition.

A function f E Cm(E;F ) is said to t o be uniformly differentiable of order

38

Chapter I

m if V bounded B

cE

and VE > 0 36

> 0 3 if x E B , and y

E

E with llyll < 6 then

( N o t e : Restrepo [49]investigated uniform differentiability of order 1.) The first main result proved in [2] is the following theorem on uniform approximation of

C" mappings up to order m on bounded subsets of E. (cf Aron and Prolla 121p. 207) Suppose E , F are real Banach spaces

Theorem 5.1.

with E' having the bounded approximation property with constant a polynomial algebra A

c C,"(E; F ) is rr-dense

@

C;let m > 0. Then

the following conditions hold:

(a) A is a Nachbin polynomial algebra; (b) V continuous linear map

T

: E + E of finite rank, with

composite g o T belongs to the $'-closure

1 1 ~ 1 15

C, and Vg E A , the

of A.

Approximation up to order m in the compact-open topology. Definition.

Let U

CT(E;F ) (for m

c E be an open set, where E , F are real Banach spaces. Then

E N') is the space of functions

f E Crn(U; F ) 3 for each I

E

U and

v j 2 m, D j f ( z ) E Pw("E; F ) . C T ( E ;F ) is endowed with the locally convex topology of uniform convergence on compact sets of order m, defined by the family of semi-norms of the form

where j

5 m, and K is an arbitrary compact subset of U .

Remark.

39

Weierstrass-Stone theorem and generalisations - a brief survey

(i) When m = 0 or 1, C,"(U; F ) = C"(U; F ) , and for m

> 1, the two spaces are

generally different, though in certain cases, e.g. when E = co, F = R1,C,"(U;F ) =

C"(U; F ) Vm E N'. (ii) Cz(E;F ) is always a proper subset of C T ( E ;F ) if E is infinite dimensional. The second main result of [2] is the following theorem.

Theorem 5.2.

(cf 121 p. 210) Suppose E l F are real Banach spaces, E' having the

bounded approximation property with constant C. Let m E N', and U nonempty open set. A polynomial algebra A

c

E be a

c C,"(U; F ) is rF-dense in C,"(U; F ) e

the following conditions are satisfied: (a) A is a Nachbin polynomial algebra; (b) V continuous linear operator r : E open V

cU

C,"(U;F ) .

3 .(V)

+

E of finite rank and 11r11 5 C, Vg E A and V

c U , the composite g o ( T I v )

belongs to the closure of A

IV

in

This Page Intentionally Left Blank

CHAPTER I1 Strong approximation in finite dimensional spaces

The concept of strong approximation appears to have originated with H. Whitney (cf. [SS]), though he does not use the words "strong approximation" in his paper. In this chapter we shall present some results on strong approximation in a finite dimensional space R". The fist of these (Theorem 1.8 below) is Whitney's theorem on strong approximation by real analytic functions. We have presented the original proof (Lemma 6 in [66]), for we feel that this proof might suggest further possibilities (see also the Appendix by Stein in [l]).The second result in $2 of this Chapter, is a weaker result than Whitney's but is still interesting because it uses different techniques. This result on strong approximation by C" functions appears to be rather commonly known (cf. Munkres [42], Hirsch [21]); however, we have a.ttempted to be guided by the presentation

in [XI.

$1. Whitney's theorem on analytic approximation We shall first explain some notation. A point in R" shall be denoted either by a single variable, e.g., z,or by an ordered n.-tuple of real numbers e.g., ($1, we shall write z =

(21,

. . . , I,,) for a point in R". A

.. . , z"), and

multi-indez is a n ordered n-tuple

a = (all .. ., an)of non negative integers ai. With each multi-index a is associated the

differential operator

where Di =

&;so D pf(x) means

PI+...+..L

...

f(z1,

. . ., x,,).

The order IaJof D p is

Chapter II

42

defined by: la1 = a1 cy

+ .-

8

+a,; if la1 = 0, then Do f means f . Clearly

f p means (a1 f PI, . . . , a, f Pn),and cy 5 p means ai 5 pi, i

Ia+PI

= 1,

= IaI+

IpI,

... , n. We shall

write

where k =

(k1,

. . . , kn),

between two points z =

. .. , In)

1 =

(11,

(51,

. . . , z,)

d ( z , y ) = (x;!,Czi- yi)l)

112

=

and y = ( y l ,

. . ., y,)

in R" will be denoted by

- yII. However, a little further on we shall allow z

((2

and y to be complex: 2

are multi-indices with 1 5 k. The distance

= (z!

Y = (Y:

+ iz;, . . . , z:,

iz:),

+ iy;, . . . , y:, + i Y 3 ,

in which case d(z,y)' shall mean x;=l{(z> - y>)

+ i(y7 - y;)}',

where i =

in C.

d(z, E ) shall denote the distance from the point z to the set E , i.e.,

while d(A,B) shall denote the distance between the sets A and B, i.e.,

We shall have occasion to consider functions indexed by multi-indices, e.g. fo(z)= fo ,...,o(z),

OT

fa(.)

= fa,,...,a,,(z). We shall suppose A to be a closed set in R", bounded

or unbounded. Suppose f ( z ) is defined in A, and let m 2 0 be an integer. We shall say: f(2)

= fo(z) is of class

functions

fk(Z)

C" in A

in terms of the finclion~fk(Z) (with JLJ5 m ) if the

are defined in A for all k with lkl

5 m and satisfy, with z , d E A:

43

Strong approximation in finitedimensional spaces

meaning:

-k

Rk(z';x),

for each fk(z), with Ikl 5 m; here R k ( z ' ; z ) is assumed to satisfy: Vx" E A , VC 36

> 0 3 if 2,s' E A with

112 -

zoll < 6, llz'

>

0,

- zoll < 6 then

Note that if rn = 0, these conditions (1) and (2) mean that f ( z ) is continuous on the set

A , and also that these conditions are satisfied automatically at all isolated points of A, regardless of how the

fk(z)

It is clear that the

are defined there.

fk(Z)

are continuous and hence bounded in a neighbourhood of

each point in A. Thus i f f is of class C" in A in terms of the f k ( 2 ) with lkl

f is of class c"', rn'

< m in terms of the f k ( 5 )

5 rn, then

(with lkl 5 m'). w e shall say that any

arbitrary function f(s) is of class C-' in A , and that f(z) is of class

C" in A in terms

of the f r ( ( z )if the f k ( z ) are defined for all k and f is of class c" in A in terms of the fk(z)(lkl

5 rn) for each integer m 2 0.

Suppose f ( z ) is defined in a region R and is of class C" in terms of the

m). Let z = ( z 1 , . . . , z,),~' = (21, . . . , z

h

+ Azh, . . . ,

Zn);

fk(Z)

(lkl 5

then if Ik( < rn, we find:

Chapter ZZ

44

in R. Hence in this case f ( z ) is of class Cm in the ordinary sense, and the f&(z)are the partial derivatives of f(z). Also Taylor's theorem shows that the converse is true. We shall need a few lemmas, before turning to the proof of the approximation theorem of Whitney in question. L e m m a 1.1.

Let w ( z ) be a continuous function of one variable defined in an interval

I containing zo, let B be a closed set in I, and let

20;

be a fixed number. Suppose VE >

0 36>03

( I ) ifz E B and Iz - 201 < 6 then

-1

- wkI < e;

(2) i f l z - 201 < 6 and z fZ B then ~ ' ( z exists ) and J w ' ( z )- whI Then w ( z ) has a derivative at

20

< E.

and w ' ( z 0 ) = w i .

The proof of this lemma is omitted since it is elementary. We shall make use of the functions denoted below by $&(z'; 2). If z E A, and z' E E ( m < 00) then we define:

Thus $&(z'; z) is the value at z' of the polynomial of degree 5 m - lkl which approximates

fb(z)

to the (m- Ikl)th order at

2.

For fixed z, it is a polynomial in

Taylor's formula in terms of its value and derivatives at

I.

2'

given by

From (1) and (4) we then see

that

The P'' derivative of $b(z'; z) (as a function of

2')

at z', is $&+f(z';z). If we then ex-

45

Strong approximation in finitedimensional spaces

press

$k(Z";

x) in terms of its value and derivatives at z', we obtain $k+dz'

$k(Z'' - Z) =

e!

f

-

(,ti

(5)

This identity shows that +k(z";x') =

ce!

.fk+dz')(zft- ,r)f

f

Our next objective is to construct a suitable C" partition of unity on the set C ( A ) .

This will be done through several steps. We define a function, which shall be denoted by 0(z). Let R be the region defhed by:

1xhl


0 for some v, i.e., C ? T X> (X 0 in ) E - A. The function &(x)

# 0 in I,,- B,,

and only on this set. Further

C 4,,(~)= 1

if

Z E

E-A.

Y

It is clear that d,,(x) is C” at points x y’, X

#

v . The function ?~\(x) is C” in

ilarly c$,,= i G $ p -

#

y”.

Let Ux be a small neighbourhood of

U’, hence so is q5,, =

in

u’. Sim-

is C” in a small neighbourhood U,,of y”. Thus &(x) is C”

on E - A. We shall next derive convenient estimates for the derivatives of the functions d,,(z). Let C, C’ be two closed cubes of Ui>oI(i. The cubes C, C‘ are said to be of the same type if the sets in J’ can be brought into coincidence with the sets in J by a translation and by stretching of the axes. There are at most a finite number, say d, of possible types of cubes, and for some number

c,

3 at most c sets I,, with points in a given cube C.

Let C be a fixed cube of KO, and k 2 0 a fixed multi-index. Each &(x) is C”, and &(x)

# 0 only for a finite number of v, hence lDbq5,,(x)l

< Nk(C) V x E C and V v

= 1,2,.

..

49

Strong approximation in finitedimensional spaces

for some positive number

Nk(c).

Now let C E K,, and let C E KO of the same type as C'. If I q , .. . , I,y are the sets

Ix with points in C', let Ix,, . . ., Ix, be the corresponding sets with points in C which can be carried into the former by translation of the axes and stretching by a factor Each function

$.

4 x q corresponding to Ix, thereby is mapped into the function

corresponding to 1 x 4 . On differentiating lkl times w.r.t. s, we find

and therefore, as &(s) = 0 in c' for v

# A:, . . . , A:,

For a fixed k, there are at most d distinct values of

Nk(c);

let

N k

be the largest of

these. The following lemma has thus been proved

Lemma 1.5. For any multi-index k, 3 a number N k 3 if C is any cube of K, then

smooth extension of f(z).

The next result establishes a C"-

Lemma 1.6. Let A be a closed set in R" and let f(s)= fo(z) be of class C" (m finite or infide) in A in terms of the

fk(.)

(lkl 5 m). Then 3 function g(s) which is c" in

R" - A, and has the properties: (1) g ( z ) =

f(.)

(2)

= f k ( z ) in A, lkl

Dkg(z)

in A ,

5 m.

50

Chapter II

Proof.

Case I. First suppose m finite. Let

where the &(z) and $(x, c y ) = t,bo(x; 5") have been defined above ( v = 1,2,

. . .).

The

&(x) and $(x; 5") are Cooin R" - A , hence so is g(z). The function g(x) = f(x) is C" at all inner points of A. We only have to show that Dkg(Z) exists, equals

fk(Z),

and is

continuous, at all boundary points of A , for 1k1 5 m. Let xo be a fixed boundary point of -4, and let

9 where N = max{Nk, lkl

[

E

30

< E < 1. Let r] > 0 3

< min i 9 2 c { ( m+ 2)!}"(108fi)"N E

1

5 m } . Let M > 0 3

and let

'< min(l'

6(m

so small that lRk(x;

.")I

5 Ilt - z 11 rn-(kl 7 . '

0

Now let y* E R" - A 3 y* E B a l r ( t 0 ) .We now assert:

Contention. IDrg(y*) - jk(xoI
0.

Izt - z i l < 6 < 1; hence

Also &(z*,

< 9 < f , hence (using:

3')

Similarly, we find:

It)k(y*,z')

-fk(z*)I

Now suppose y* E C E K,; let

xxq

is at a distance

characteristic property of

&(2*,

= $k(z*,

2')

+ Rk(z*,

5')

< i.Hence

Ixl,. . . , Ixt be those sets Ix with points in C. Each

corresponding point yxq is at a distance sponding point

fk(Z*)

from z ' (cf. Lemma 1.4),hence each corre-




Then we

define

w e shall establish an inequality for Dkgoo(s) similar to one for Dkg(S) (case I), for any k. Let g("')(z) 1,2,

. . .).

be the extension of class C" obtained in the proof of case I (rn =

Let zo be a boundary pont of A , and let

zo.E S, and 3

$ < E.

Then choose 6

< 6,

E

> 0.

Then choose p

3 for any y* E R"

2 lkl + 2

3

- A with IJy*- zoll < 6, we

have IDkg(lk""(y*)- fk(zo)l

Let q 3

&,+I

I 6, < 6,,

<E .

where 6, = d ( f ; A ) ; then q 2 p . Define C, I p - 2 1 ILI. We now write

Using the definitions of g ( l k l ) ( z ) (from case I) and of gw(z) we find:

Note that DjEv(z) = qbyvi,(z;z”) - qbl,+j(x; z”). Replacing Ic by j in the expression for

5 rn - ( j l occur. Now replace rn by 7”

we see that only those terms with

&(z‘;z),

and Ikl successively; then on subtracting we obtain

Also

T,

>

4, hence

T,

>

bq+2,

number of terms in the sum is Ifj+t(T”)I




, therefore

55

Strong approximation in finitedimensional spaces

in particular at y*. Thus we find

for any point y' E R" - A within distance 6 of zo. Now again apply Lemma 1; and it follows that D k g ( " ) ( S ) exists and is continuous throughout R". This is true for every

Ic = (k1,. . . , kn). This proves the Lemma. We shall now turn to the following extension of the familiar Weierstrass approxima-

tion theorem.

Lemma 1.7. Suppose g ( z ) is of class Cm in R" (rn R". Then for each t > 0

Proof.

< co), and S be a compact

3G analytic in R" and satisfying:

Let Ra = Bk(0) ( b 2 0 ) , and consider the n-fold integral

where T is 3 @(m= 1. Then 0 5 @ ( b ) 5 1 Vb. Now replace y by tcy and b by

Kb,

obtain

Let v ( z ) of class

Drv(z)

set in

C"

be E 1 in

S1 E 0 outside some neighbourhood of S and 3

0 in S Vk (cf. Lemma 1.6). Put g'(z) = v ( z ) g ( z ) ,and let

and we

Chapter II

56

where n will be suitably chosen presently. Then G(s) is analytic in R". The function

111

- yll' is a function of z - y only and differentiating under the integral sign gives

where

Dp),Dp)denote differentiation with respect to s and

y, respectively. Then inte-

gration by parts llcl times yields

If we recall that @(m)= 1, we see that

Now let M

> 0 so large that

The functions Dkg'(s) are uniformly continuous on R", hence 3 6

Then let n

>0 3

> 0 so large that 1- q n 6 )

Let U = B&(s), and denote by

J1

and

J2

4M

the regions formed by replacing the domain of

integration on the right side of I D k G ( s ) - D k g ' ( s ) I we then obtain:

E

< -.

above, by U , and R"-U, respectively;

Strong approximation in finitedimensional spaces

hence

This completes the proof of Lemma 1.7. The preceding Lemma can be generalised to yield the next theorem which is the main theorem of Whitney that we are aiming at in this Chapter.

Theorem 1.8. Let R be an open set in R", and R1, R2, . . . bounded open sets 3

u

nzl

R,

= R, and 3

Rp c Pp+l for each p .

finite or finite), and suppose 61 2

~2

Suppose g is defined and of class C" in R (m

2 . . . are given positive number. Then 3 analytic

function G(z) in R, satisfying

where (YP =

Note that if R1,

. .. , R,

{

m p

if nz is finite i f m = 00, p = 1 , 2 , ...

are empty, then this statement means

= QLUQPUQ;.

In Lemma 1.6 we replace the closed set A by this set, and replace f(s)(of Lemma 1.6)

by a function which is function,

C"

in R", 3

3

1 in Qp, and

0 in

QL U Q:.

Then for each p , let up(.) be a

Chapter II

58

and D k u P ( z ) = 0 in QI,UQpU Q: (1k1 > 0). (If Rp+l = 0 we put up(.) but Rp-l = 0, then up(.) = 0 in Q:) and = 1 in

?zp+l.)

Now let 2,

Now define, successively, the analytic functions Gl(z), Gz(z),

= 0; if Rp+l # 0

2 1 be a number 3

. . . , by:

(If p = 1, the factor in brackets is g(y).) The constant tcp is so chosen that if we set

in Rp+i(IkI 5 ap+l;cf. Lemma 7; the constant

K~

will be further restricted a little later

on). From the definition of u p ( z )we see from the last two relations that

We then differentiate H p ( z ) ,replacing p by p - 1 in the preceding relations and we see that (cf. proof of Lemma 1.6, m

< 00):

The function up(.) and its derivatives are 0 in Rp-l, hence the preceding holds also in

Rp-l. Hence

2

I ~ k ~ ~ ( z ) l < in

R~ ( I ~I I ap>.

Strong approximation in finitedimensional spaces

59

We shall show that G(s) yields the required approximation to g(s). We note that

Dk{Gl(s)

+ + Gp(z)} converges uniformly in any compact subset of R (lkl 5 m);

hence G( x) is defined in R and


0 is so small that

the real points in the complex open ball 133p(xo)lie in some

&. Now if p

> q, x

E

U and y

E

R - Rp-l, then

cxi2


~ 6 2)4p2, ~

hence

Re (r&) > 3p2 Furthermore Hp(y) = 0 in R, and in R" - R,+z for p (recall: Hp(y) is determined before

K,),

> q.

Hence if Mk = max IHp(y)l

and if Vp = volume of Rp(p= 1,2,

~ G p ( z ' + i x f f l, - ~ih(x)ll

< q(x) v x E u] where h(.) E C k ( U , Y )and q(.) is an arbitrary positive continuous function on U , form a base. The Ck-fine topology on C k ( U ,Y ) is also denoted by C:(V, Y ) . We should point out that although the preceding definition is stated for Banach spaces we are emphasizing results in finite dimensional spaces in this chapter. So we shall now onward in this section deal with functions defined in an open set U in R" with values in R". Before proceeding to the theorem in question and its proof we shall need some preliminary lemmas. Let

and d(x) =

p(llz11'). The function b(.) is C" on R',P(.)

(b,m) while

4(.)

function

is C" on R",d(.)

1 on BJ;;(O)and

1 on (-m,a),P(.)

d(-)

d(.) also enables us to construct a Coo-function 0 : R"

0 on

0 outside B d ( 0 ) . This -t

R' 3 0 = 0 outside a

compact set and its Lebesgue integral in R"' equals 1. By the support of a continuous real-valued function f , denoted by Supp f, is meant the set f-l(R1 - {0}), so the complement of this set, C(Supp f ) is the largest open set on which f = 0. Let U =

{ U a } ,be ~ ~an open cover of U . By a Cj-partition of unity

subordinate to U we mean a family of Cj-maps A, : U -+ [O,l],aE A , 3

63

Strong approximation in finitedimensional spaces

(i) supp A,

c U,,

a E A;

(ii) {supp A a } , E ~ is a locally finite family; and (iii)

C

Aa(z) = 1, z E U .

uEA

The local finiteness property of {Supp X u } a E ~ ensures that each point of U has a neighbourhood on which all except a finite number of A, are 0, and the sum is locally a finite sum. We note that condition (iii) ensures that

where E" denotes the interior of E. We also note the following simple observation: If

V

= {V a } a E ~ is an open cover of U which refines U = { Up}pE~ , and if

V

has a

subordinate Coo-partition of unity, then so does U.

Lemma 2.1.

Every open cover of U has a subordinate CJ-partition of unity, for any

given j 3 0 5 j 5

00.

This result is well-known; for a sketch of the proof see 53 in Chapter 111. To approximate C'-maps by Cm-maps in the strong topology we need to approximate locally f on the

Ui,where { U i } ; E h is an open covering, by Coomaps whose deriva-

tives up to order r uniformly approximate those of

f. In a finite dimensional space this

is achieved by using the technique of convolutions, which we shall explain next. Let 6 : R" + R be a function with compact support. There is a smallest u 2 0 3 supp 6 is contained in the closed ball B,(O) C Rm. We call u the support radius of 6. Now suppose

U c Rm is an open set and f

:

U

+ R" a map. If 6 : R" -+ R has

compact support then we define the convohrtaon o f f by 6' to be the function denoted by

Chapter ZI

64

8 * f : U,

R", defined by

e*f(.)

=

J

@(Y)f(.

- YWY,

xE

uu

9

B..(O)

I Bu(z) c U } .

where U, = {x E U

Here we understand integration to be in the

Lebesgue sense, dy denoting Lebesgue mensure in Rm. We note that the integrand is 0 on the boundary of B,(O);we define the integralid to be 0 outside B,(O) and thus extend it to all of Rm. Then @

* f(.)

=

JRm

@(Y)f(.

- YWY,

5

E

UU .

Now for a fixed x E U, we make the change of variable: z = x - y. Then

e*f(.)

=

J

Be(=)

q x- Z)f(t)dZ

=

@(x - z)f(z)dz,

x E U,

,

where the integrand is defined to be 0 outside B,(x).

A function @ : R" port and

-+

R is called a convolution kernel if @ 2 0, @ has compact sup-

s,.,, @ = 1. Earlier we noted that such kernels which are further C",

do exist.

Since @ * f ( x )can be considered to be a weighted average of values of f near z, it is plausible to expect that @ * f might be an approsirnation to f in a neighbourhood of x, and the approximation might be smooth. We shall use the notation

i f f : U + R" is C', U

c Rm

is open,

K c U is any subset, and IlD'f(z)ll is the norm

of the kth derivative (as a continuous I:-linear symmetric operator) o f f at z. IlD"f(z)ll

We now need the following result.

Strong approximation in finitedimensional spaces

Lemma 2.2.

Let 8 : R"

--*

R have support radius u

65

> 0. Let U

set and f : U + R" be a continuous function. Then 8 * f : U,

+

C R" be an open

R" has the following

properties: (a) If 81,upp(0)0is C k ,1 5

kI 00, then

so is 8 * f , and for each finite

k

D k ( 8* f)(x) = Dk8 * f (z) . (b) I f f is C k Then

Dk(8*f ) = 8 * D k ( f ) . (c) Suppose f E

C', 0 5 r I 00. Let I< c U be compact. Given E > 03u > 0 3

K c U, and 8 is a C'-convolution kernel with support radius u , then 8 * f E C '

Proof.

The part (b) follows by differentiating under the integral sign. The part (a)

follows by a change of variable, viz. z = x - y. (c) It suffices to let r = 0. Since dist (K,R" - U ) > 0, we can choose u

> 0 sufficiently small so that I( c U,. Also let

sufficiently small so that, if z E K and Now use the fact that

,s,

112 -

y1I

u

5 u then If ( z ) - f(y)l < E .

8 = 1. On integrating over R", we obtain:

This proves the Lemma. Thus we see that a proximated by

C"

C ' map from an open subset U of R" to R" can be C ' ap-

maps in neighbourhoods of compact subsets of

Kurzweil (see(301) for the proof of the next lemma.

U. We shall follow

Chapter II

66

Suppose X is a metric space and G

Lemma 2.3.

c

X an open set. Suppose G =

m

u

B; where we have written Bi for B t i ( z i )= the open ball in the centre zi and radius

i=l

&, i = 1,2, . . .. Then 3 locally finite open covering Remark.

The

{K}rlof G 3 K c Bi, i = 1,2, . . ..

in this lemma will turn out to have further properties: viz. each V;

will be “scalloped” to use the terminology of Lang [32] p. 35. Proof. 1 > el

Choose a sequence of positive numbers

< E~ > ... with lim

having defined

Vj-1,

u j= W

Clearly G =

~i

t-+ w

w

=0

.

Then define

{&i)zl 3 3ci V1 =

< 6;, i

= 1,2, . .

. , and

B I , VZ= Bz fl CB61-c1(21); and

define

Vj, and Vj c B j , j = 1,2, . . . ,

Let y E G, and let k - inf[rnly E B m ] . There is an integer e

>k

3y E B~~-3~~(zk).

Then

Hence

This means that 3 neighbourhood viz. B6,,-3rr(zk)of the point y which intersects only a finite number of the sets Vj, j = 1 , 2 , . . .. This shows that the covering

{Vj}glis locally

finite. This proves the lemma.

Lemina 2.4.

Suppose G

c R”

{ C i ) z l of G by compact set C;.

is an open set. Then 3 countable locally finite covering

67

Strong approximation in finitedimensional spaces

The proof is a slight modification of the proof of the last lemma. For each

Proof.

G3 open ball Bza(z)c G. Then 3 countable subcollection {Bai(zi)}& 3 G

u

=

i= 1

where we set Bi = Bai(zi). Then G =

U

0 0 -

Bi, and each

5

E

W

Bi,

is compact.

i=l

Let lim

i+w

~i

{&i}g1 be a sequence of positive nunihers 3 3&i < 6i, 1 > ~1 > EZ > ... , and

= 0. Define

generally having defined Cj-1, define

u

m

Then G =

Cj, Cj c Bj and the Cj are compact, for j = 1,2,

. . ..

j=1

Let y E G, and let k = inf [mly E

B&,- 3 c r

(2k)

Barn].There is an integer L? > k

3 y E

Then

hence

This means that 3 neighbourhood via.

B 6 h - 3 c r ( z k )of

the point y which intersects only

a finite number of the C,, j = 1 , 2 , . . . , i.e., the covering { C j } z l of G is locally finite.

This proves the lemma.

Theorem 2.5.

Let

U c R"

be an open set. Then Cw(U,R") is dense in C'(U, R") for

0 5 r < 00, in the Cr-fine topology.

Chapter I1

68

Proof.

Let f E C(U, R"). Let I( =

{I 0,

and denote by T(e',q ) the set of numbers t E [0,1] with the property: U is an open interval containing the point t , the 3 z(.) E C[O,11 satisfying: 11z11 < 1,lMz)Il

'

Lemma 1.6.

The set T(e',q) is finite.

I ( . )

= 0 if

7

6

U,

E'.

Proof of Lemma.

Suppose T(c ' ,q) is infinite. Then 3 numbers t , E T ( c ' , q ) , 3 open

Chapter III

88

intervals U, c [0,1], and 3 functions z,(.)

E C[O,l] for n = 1,2,

. . . , and satisfying

llznll 51 .

and

We now invoke the following theorem of Orlicz:

Lemma 1.7. (Orlicz 1421 Theorem 3, p. 247).

Suppose

is a sequence of

elements in a weakly complete Banach space; and suppose 3 positive number K 3

+ ii, + . .. + ?ti,,11 5 K

Ilii,

gers, k = 1,2,

. . .; then

for every finite sequence 1 5

the series

il


llznll - $Y(zn). Here

P ( s , , z ) is the differential of the mapping f. Then 3 open interval U containing t' 3 llP(zn,y)ll 5 f&r(zn)provided y satisfies: Iy(t)l 5 1 Vt E [0,1] and y ( t ) = 0 if t $! U . Now consider a fixed function y(.) E C[O, 11, and suppose

Let xn+l =

2,

+ y.

Then the above 3 required conditions on the sequence

(2,)

are

satisfied because 3

11zn+111

2 Izn+l(t')l = Izn(t')l+ q d z n ) 2 llznll+

1 ~ ~ ( z n ) ,

and then by the properties assumed regarding the mapping f ,

There must be an element zn 3 r 5 llznll

n = 1,2, . . . . Then the series

r

+ E ; for otherwise there would

satisfying the above 3 conditions and also llznll

be an infinite sequence of points z, T,


2 . When

0 on U - C1,and = 0 on

I by 4(1)

=

CO,q5-'(1)

C1.

Finally de-

= C1; further

for suitable Coo-smooth functions

kj.

Next let {Wp}gl be a countable open covering of U - (Co U C,) = 4-'(0, 1) by sets which meet only finitely many Vj. Suppwe x E W, is a critical point of

0= z

e k j ( z ) p j k ( z - ak).

Now

kj(z)

#

0 in W,. Since p j j ( z ) = 0 only when z = a j or

$ vj, we conclude that either z = a j or a non-trivial h e a r combination

(c

-fk)z-c-fkC&

that

c

-fk

4, so that

c

-fk(I-ak)

=

must be = 0. The centres { a & }are linearly independent; hence we find

# 0 , i.e., x

belongs to the linear span of

Let M p be the intersection of W, with this linear span; then M, is a finite-dimensional manifold containing all the critical points of theorem to

4

$1

4

R'. Now apply the Morse-Sard

WP

-+ R1; the set of critical values of this function has p1-measure 0. The IMP

set of critical values of q5 : U

-+

R' is a subset of a countable union of such null sets,

hence itself is a null set of R'. This completes the proof.

Proof of Theorem 6.1. and

E(.)

1c, : U

--t

Let U c 7-i be an open set,

4 :U

--t

F be a continuous map,

a continuous positive function on U . The object is to show that 3 residual C"

F3

I I ~ ( I ) - $(.)I1

< E ( X ) VX E U .

110

Chapter III

First let { U i } z l be a countable cover of U by open balls in U with centres at points { a i } E 1 such that II$(x) - $(ai)II

< c ( x ) Vx

E Uj. Let

{K}El

be a locally finite

scalloped refinement as in Lemma 6.3, and a subordinate partition of unity ing the residual functions in Lemma 6.4. Now set

Vx

bi = $ ( a i ) , i

{

us-

2 I, and define the C"-

E F.

Now use the proof of Lemma 6.3; this proof yields a countable open cover { W,},:, of U , each element of which meets only finitely many

< dim F , then $(W,)

K. If dim span

{bj

I Vj n W, # 8)

lies in a proper linear subspace of F , hence is meager. Otherwise

the set of critical points (resp. critical values) of

$1

-+

F lies in a finite dimensional

WP

manifold (resp. linear subspace). The Morse-Sard theorem ensures that this map has a meager set of critical values in F (more precisely, has Lebesgue mp-measure zero where mp is the dimension of the linear span of the relevant bi associated with W,). Hence the

set of critical values of $ : U

--.)

F is contained in a countable union of meager sets, and

is therefore meager, by Baire's theorem. This completes the proof of Theorem 6.1.

111

Strong approximation in infinitedimensionalspaces

57. Contributions of J. Wells and K . Sundaresan John Wells in 1968 (cf. [63]) showed that the techniques used in preceding results could not be used in certain spaces. Specifically he showed that in the space Q one cannot have a real valued function with bounded support and with a uniformly continuous derivative.

K. Sundaresan (cf. [61]) carried these ideas further, and showed that if a Banach space admits a non-trivial function with bounded support and having a uniformly continuous derivative then the space must be super-reflexive. The result of Wells ([63]) turns out to be a special case of Sundaresan's theorem. However we shall first present the theorem of Wells, for it has a simple proof which does not require the machinery used by Sundaresan, and also because it showed, for the first time, that in the space co, Ca-fine approximation of a C2-function by a

C" function is

not possible.

Theorem 7.1. (cf [63]). Let f E C'(c0, R) with a uniformly continuous derivative

D f (.). Then the support o f f is unbounded. Proof. Suppose the statement is not true. Then 3 f E C'(c0, R) with the properties:

f(0) = l,f(x) = 0 for

llzll 2

IlDf(z + h ) - D f ( z ) l l 5

1, and D f is uniformly continuous. Let N 3 llhll 5

k =+

i. Because of the mean value theorem, we then assert ,

Let E be the subset of co consisting of x such that the 2N - 1 of the first 2N components of x have absolute value

k,the remaining component has absolute value less than or

Chapter III

112

equal to

k,and all components after the first 2 N components are zero. The set E is

connected and even, hence we can choose, inductively, hl, h2, . . . , h N 6 E 3 D f ( h l

. . + hk-1)

hb = 0

and hl

It follows that llhl

+

+ + . + h k has at least 2 N - k components equal to k.

+ + h ~ l =l 1, and

c N

I

k= 1

1

1

-((hkll = 2 2

c N

k=l

1

1

N

2'

-= -

which is a contradiction. This proves the theorem of Wells. We shall next turn to the work of Sundaresan ([Sl]) mentioned above. The main result of his paper requires a good deal of machinery which we shall proceed t o explain. First some definitions and terminology are in order. A Banach space E is said t o be

smooth if for all x # 0, I E E ,

exists Vy E E. If this limit exists at a point x

# 0, Vy E E

then it is known that G. E

E' the dual of E , and also llGzII = 1 (see Kothe [29]). A smooth Banach space E is said to be uniformly smooth if the preceding limit is uniform Vx, y with ((z((= 1 = /(y((. The homogeneity of the norm implies the following lemmas. If E , F are Banach spaces, a function

f

:

E -+ F , having a uniformly continuous derivative on a set A

be said to be U.C.D. on A . A Banach space E is said to be U'-smooth if 3 function on E with bounded support.

c

E will

U.C.D. real

113

Strong approximation in infinitedimensional spaces

Lemma 7.2.

A Banach space E is uniformly smooth iff the norm is U.C.D. on regions

R(X,p) = { x E E

Lemma 7.3.

I X < 11x11 < p where 0 < X < p } .

The norm in a Banach space E is U.C.D. on regions R(X,p ) i$ the norm

is uniformly differentiable on bounded sets away from the origin. We shall also need the following lemmas, which are either results on differential calculus in Banach spaces (see Dieudonnk [9], Lang [32]), or consequences of the preceding definitions.

Lemma 7.4.

Let E be a Banach space, f : E

+

R a U.C.D. function, and D f the

derivative o f f . Then (a) if U is a bounded subset in E l then f

IU

is Lipschitzian: E M

>0

3 Vx,y E U ,

Ilf(x)- f(Y)II 5 MIIX - YII; (b) if the support o f f is bounded then f is Lipschitz everywhere in E , in particular f is uniformly continuous.

Lemma 7.5.

Suppose E is a U'-smooth Banach space, and X

function f on E 3 f(0) = 1, and f ( x ) = 0 for

Lemma 7.6.

11211

> 0,

then 3U.C.D. real

2 A.

I f f and g are two U.C.D. real functions on a Banach space E and the

support o f f (or the support of g) is bounded, then f g is U.C.D. with bounded support.

Lemma 7.7.

If E , F, G are three Banach spaces, f : E

+

F, g : F

+

G are U.C.D.

functions such that the derivatives D f , Dg are bounded on E + L ( E ,F ) , and on F -+

L(F, G ) , respectively, then the composite g o f i U.C.D. For the next lemma we refer ther reader to Nemirovski and Semenov ([45]).

Chapter III

114

Lemma 7.8. Suppose E is a uniformly convex and uniformly smooth Banach space. Then the restrictions of the uniformly continuously differentiable functions on E to any

-

-

closed ball U,(O) is dense in the space of uniformly continuous functions on Up(0) with the uniform topology. We now have to turn t o the concepts of super-reflexive Banach spaces and ultrapowers of normed linear spaces. If E, F are Banach space, then E is said to be finitely

represented in F , in symbols E 03

-t

F a C'-mapping. Suppose Bo

sup llD'f(x)ll

< 7. Let c > 0.

c B1 is an concentric open

Then 3 constants

X0,Xl

and 3g : B1

ball and

+F 3

g is

zEBo

C" and satisfies: SUP Ilg(x) - f(.)ll

ZEBO

< Xos;

and

SUP zEBo

I l ~ g ( ~ > 0.

Chapter III

130

(ii) The same property holds for D j { l - ‘ b p ( r ) }D, j 4 J z - c) and D j { l - d p ( x - c)) where p

> 0, and c is any fixed point in H , with the same constants as in (i).

Proof of Lemma 9.2.

Each derivative D j p ( z ) , of p(.),( j 2 1 ) , is continuous and

vanishes outside a compact set viz. [

5 x 5 11 in R1 , hence is bounded on R’

.

Next, D(11z11&) = F. where F, is the functional on ‘H defined by F.(h) = 2 ( x , h ) ,

z , h E ‘H, with llF.11~ = 211~11.Further Dz((llxll&)= G where G is the (constant) bilinear functional on ‘H x ‘H defined by G(h1, hz) = 2 ( h l , h 2 ) ,h1,hz E ‘H, with llGll2 = 2 . The further derivatives of

Ilzil& are all zero.

Then D2q5(z) = Djp(llxll&) is the: sum of a. finite number of terms each of which is bounded on ‘H, hence each derivative is bounded on ‘H. Then by the Chain Rule it follows that ((Dq5p(x)lll5

9where ( ( D $ ( z ) l (5 &Il; likewise IlD2dp(x)ll25 3 where

11D2q5(z)1125 M2Vx E ‘H. This proves the lemma. We shall make the convention.

Convention.

The constants

Mj

is chosen to be = 1, and for j = 1 ,

in Lemma 1 will be henceforth chosen as follows. MO

.. ., k ,

is chosen to be =l.u.b

llDjd(z)llj.

Z€?i

Next let @,,(z) = 4(z - z)d(z - 22). . . C$(X - z,,),

and define

Further define Mj,i(l) =

where

(;)Mi,

(I) is the binomial coefficient

0 L: j 5 k , 0 5 i 5 j

,

for integers s,t 3 0 5 t

5 s. Then define

Mj,i(2) = ( : ) [ M i , i ( l ) . M o + A [ i , i - 1 ( 1 ) . M l + . * . + M i , o ( l ) . M ; ]

131

Strong approximation in infinitedimensional spaces

Then by Leibnitz's Rule, for 0 5 j

I 2,O 5 i 5 ,j;

Now suppose (finite) Mj,i(p) for all integers p 3 0 have been defined, satisfying: for integers 0

I p I n - 1 (with 0 5 j I k,O I i I j)

5 q 5 p:

and

Then define

This shows

Now define: C ( n ) = 1

+ max{Mj,i(n) I 0 I j

I k,O 5 z I j }

for

n = 1,2,

. . .. We

also note that this preceding inequality concerning ~ ~ D i @ , , ( still z ) ~remains ~i true if some or all of the

4's

are replaced by (1 - 4)'s.

111. The proof of Theorem A depends upon suitable local C"-approximations in the neighbourhoods of the points

ii,

i = 1,2, . . . , to the given Ck-mapping f . These lo-

cal Coo-approximationsthen have to be put together; however, the customary technique

Chapter III

132

of partitions of unity could not be used directly in our problem. In our next lemma we obtain such a local approximation in a neighbourhood of each point of 0; this approximation is even analytic, and is valid in an arbitrary (separable or non-separable) Hilbert space, or even in a Banach space. Let f E Ck(R,F ) l x E R and

Lemma 9.3.

f E C”(R, F ) and 3

v >

0. Then 3f =

fz

:

R

-+

F 3

in a suitable neighbourhood U of I,f satisfies:

Proof of Lemma 9.3.

Let z E R,II > 0. Then define

f = f.

: R -+

F by

I ! I.

Here, as is customary, “(w)(j)” denotes the j-vector iw, . . . , wj. This expression for

f

is

the usual Taylor polynomial of order b of f around the point x. Changing our notation slightly, write

where

A0

= f(x) E F,Al = E Lt(‘H,F ) . Here

Ak =

-

of(.) E

L(‘H,F),A2 = $ D 2 f ( s ) E Li(‘H,F) ...,

L(‘H,F ) is t,lie space of continuous linear mappings ‘H -+

F , and for any integer j 2 2, Li(’H,F ) is the space of j-linear continuous symmetric mappings ‘H x

. . . x ‘H -+ F . j

Here we use the following notation. Suppose Gm(xl, . . . , 2,) ping V x

..- x V m

-+

is an rn-linear map-

W , where V ,W are vector spaces. For fixed 5 1 , . . . , x j € V

-

Strong approximation in infinitedimensional spaces

(1

5

j

5

m ) , we shall write

G m ( z l , . . . , xj)(j) for the mapping V x

133

x V + W

m- j

defined by

-

Similarly taking z1 = x 2 =

Vx

...

= x j = I, we shall write Gm(x)(J)for the mapping

. . . x V 4 W defined by

m- j

and likewise: Gm(x)(j)(x)m-j

ef j

m- j

A little calculation then shows that (more details will be found in [ I):

provided y

i x. (Here “P”

means “approximately equal to”.) Thus 36 = 6 ( z , v , f ) > 0 3

Then it suffices to let U = U ( 6 ) be the neighbourbood B a ( Z ) of x in

a. This proves

Lemma 9.3.

As a corollary of Lemma 9.3 we obtain:

Corollary 9.4.

The function fz in the preceding Lemma further satisfies (preserving

the notation of the last lemma): 36

> o satisfying v integers j E [o,ICI,sup UEU

~ l ~ j f ~ -( y )

Chapter ZZZ

134

Proof of Corollary 9.4.

We compare the Taylor expansion (cf. [33], p. 110) of f ,

or D f , or D2f, . . . , Dkf , respectively with fz(y), or Dfz(y),

. . . , or Dkfz(y). For any

integer j E [0, k], ~ l ~ j f z ( y-) ~ j f ( y > ljl =

11

[

+

~ j f ( z ) ~ j + ' f ( z )* (y - z)

- [Djf(z)

Now suppose 17

Dkf(z>. (y - z)k+ +(k - j ) ! * *

+ Dj+'f(Z). ( y - z) +

*

j

1

*.

> 0; then let 6 > 0 3su'p IlD'f(z) - Dkf(y)llk < 7. Then for such y UEU

we find that

This proves the corollary. We recall that X = {z1,z2, . . .} is a countable dense set in R. We shall write ~(z,),n = 1 , 2 ,

Convention.

.. .. &(a)

E,,

=

We shall now agree on the is bounded by 1 on R.

For otherwise we can replace

&(a)

by min

{E(.),

1).

Lemma 9.5. (a) For each z E n,3 open ball B,(z) SUP u,u'EB.(z)

c R satisfying

I 4 Y ) - 4Y')I

4Y) . 1 3 open ball Bp,(zn) c R 3 (a) holds in Bp,(zn) as also the folfowing: 3jn E Cm(R,F) satisfying:

135

Strong approximation in infinitedimensional spaces

Proof of Lemma 9.5. (a) Follows because

is positive and continuous.

E(.)

(b) For each n = 1 , 2 , . . . , 3 open Brn(zn)3 (a) holds in Bvn(zn).Also by Cor. to Lemma 2, 3 open ball Bra(zn) and 3 function

jnE C"(R, F ) satisfying:

Now let pn = min(rn, rk). Then

This proves Lemma 9.5.

From now on, we shall make the following

Convention. for n = 1,2,

The constants I m , g n ( z ) = gm(x). Therefore we

Chapter IIZ

140

In the next lemma we shall show that lim gn(z) exists not only on R' but exists, in fact n+w

uniformly, for all z E R.

Lemma 9.14.

For each integer j E [0, Ic], the sequence {Djgn(z)} is uniformly conver-

gent Vz E R. Hence the function g defined by: g(z) =n-w lim gn(z) at each point z E R is Ck-smooth in R and has the approximation property:

V integers j E [0, Ic], IlDjg(z)- DJf(z)[lj< ~ ( z ) Proof of Lemma 9.14.

Let N , N ' be intcgers 3 N'

> N > 0.

Vx E R

.

Then for any integer

j E [0, Ic], and z E R, by Lemma 9.9:

Hence for such j the sequence {Djgn(z)} is uniformly convergent Vz E R. For j = 0 this means { g , , ( . ) } is uniformly convergent in R. Hence ij(z) = lim gn(z) exists not merely n+w

at points in R' but at all points in R. Furthermore by Theorem 12 in [33], p. 117, we are able to assert that for each integer j E [l,Ic], lim Djg, must be = Djij, and further ,,--too

since {Dig,,} is uniformly convergent and each Djg, is continuous in 52, therefore Djg is continuous in R. Thus ij is Ck-smooth in R. Next for such integers j,and each point

,:R

E X,

IlDjg(zrn) - Djf(zrn)IIj = IIDJgn.,,,(~rn) - DJf(zrn)llj = JJD'gm(zm) - Djf(zrn)Ilj

141

Strong approximation in infinitedimensional spaces

because n.,

Let y E

5 m; and hence

(an')n R if (an')n Q # 0. Then

3 subsequence {xn,}

3, ,z

+y

and because

Djg(.),D j f ( . ) , ~ ( .are ) all continuous on R, we find:

Hence g has the property:

This completes the proof of Lemma 9.14 and hence Theorem A is proved.

V. We shall now turn to the proof of Tlicwrcmim 13 stated at the beginning of this section. We shall need the following two lriiinias. Although these two lemmas are stated for the particular functions g(.) and a(.),they are clearly true more generally. Lemma 9.15.

The function y(.) defined for x E R by:

is continuous on R.

Proof of Lemma 9.15. converging to

5.

Let

3'

E R, and let {x"}:!~ be a sequence of points in R

We first show that 7 = linisup y(x,)

will follow that y = liminf y(xn) 2 y(z). n-w

n-

W

5

y(z); by similar arguments it

Chapter 111

142

Taking a subsequence of {zn}if necessary we shall suppose lim 7(zn) = v. Supn+m

pose v

> 7(z). Then a simple argument

shows that for n sufficiently large, B,(,,)(zn)

would properly contain B,(,)(z). However this would clearly violate the supremum property of 7(z). We therefore conclude that v must be 5 7(z). Hence 7 must be I7(s)*

Similarly we conclude that 7 2 7(z), because otherwise the supremum property of 7(zn)would be violated for many

71%.

This completes the proof of the lemma.

The proof of the next lemma is very siiiiilar to that of Lemma 9.15, and hence will be omitted.

Lemma 9.16. The function X ( i ) defined for .T E R by

[

X(z) = sup 0

),corresponding to the unique solution

a1

= 0 = . . . = a,-1, a , = 1 of the simultaneous equations a1 +a2

+ ... + a ,

= 1,

a1+2a2+*..+ja, = j .

1

(5)

The remaining terms in the summation for D J g ( u , ( z ) ) correspond to solutions in non negative integers ai of the pairs of simultaneous equations

with i = 2 , 3 ,

. .., j - 1 in turn.

In each of these cases, any solution a (for a given i with

2 5 i 5 j - 1) will contain a non-zero ai with

ai

> 1. The corresponding contribution to

the summation representing Dj?j(un(z)) is zero (if j

> 1).

Chapter ZZZ

148

Then for a given j (with 1 I j I k):

This estimate clearly also holds for j = 0.

VII. We now want to put together the composites g(un(.)), n = 1,2, . . . , in a convenient manner, to obtain a suitable C" mapping. We proceed as follows. We define a C" function h(.) : R

+

and generally for any integer n 2 1, b(1,...,

[0,1] in the following manner. Let b(l) = b l , n)

= min(b1,

. . . , bn).

Let x E R, and define

for m = 1,2, . . , ,

Here the constants C ( j )are the ones defined in subsection II(c). For each m = 1, 2, . . . ,

hm(.) E Cw(R,R1). Further the sequence {hm(z)}z=l is constant from m = n onward, where for x E R, n = n, = inf[rn

I x E B6,,~(zm)].

Hence h(x) = lim h,(z) exists m-w

for each x E R, and the function h(.) E C"'(R, R').

We want t o note further properties of the function h(.). As noted, if x E R, and

Strong approximation in infinitedimensional spaces

where ai =

.(";I. ..i' iiiIfThen a ,

2, c+2

This means that

& E C-(n,

149

> a,+l for any rn = 1 , 2 , . . .. Hence

R').

With the notation of thc prcrrtling pnragraphs, for j = 0 , 1 , . . . , k, and making use of Lemma 9.9 (subsection II(c)), we find

sk+ 1 2{ 1 5+ 1 F + .+.&} . + k+2 1 {; + . .. $} + . . . + lc+2 1 {:+..+}, there being no more than ( k and j = O , 1 ,

..., k,

+ 1) terms in the sum in the right side. Thus, for z E R,

Chapter III

150

In general, induction yields: if m is any integer 2 1, then

This suggests that such an estimate might be valid for rn = 0. We verify that this actually is the case, as follows. For j = 0,1,

. . . , I;,

and, on the other hand, the sum in the left sidr in (8) is

We now need the following lemma.

Lemma 9.17.

Let

W1

= {w = ( w j , wj-1,

integer E [0, k], with norm llwll =

.. ., W O )

I

UI;E

Lt(H,R')}, where j is a fixed

j

IIwilli. r=O

Define T linear:

W1 + Lf(H, F

) by

Strong approximation in infinitedimensional spaces

151

where c; E Lf(H, F ) are fixed, the products being tensor products. Then llTll = max{l)ci(l;, 0 5

5j).

Proof of Lemma 9.17.

Hence llTll Imax{llc;ll;, 0

5 z 5 j}.

(ii) Let Wa = { ( a o , a l , . . . , a j ) } , the a; being elements of Lf(H, F ) , the norm in Wz being defined by: llall = max{)la;((;, 0 I i I j } . Then c =

(CO,

fixed i E [ O l j ] define w; E Lf(H,R1) of unit norm, and w = (0,

. . . , c j ) E Wz. For

..., w;,Ol ... , 0)

E

Wll where the ( j - i)th component is w;,the remaining components being 0's. Then llTwll = llw; llcj-illj-i

. cj-;llj

=

5

~ ~ c j - ; ~ ~ j - llTllllwll ;

= llTll. Hence for each i = 0,1,

.. . , j ,

5 llTll. Hence Omax llcilli I IlTll i.e.7 llcll 5 IITII. 0,

. . . , 2,. A

Q2(z)

> 0 ...,

Qm(z)> 0 ,

CP-map from one C“-Nash manifold

to another is called a CP-Nashm a p if the graph is seini-algebraic. A CP-Nesh vectorfield is similarly defined. N “ ( M ) denotes tlie ring of all CP-Nasli functions on A/r where A/r is a C‘-Nasli manifold. It is convenient and meaningful to restrict

1’

to be

< 03,

since it is

known (see [55]) that a Cm-Nash manifold and a Cm-Nash map are already of class C”.

Strong approximation in infinitedimensional spaces

155

One objective of Shiota's paper is to approximate a C'-Nash map between C"Nash manifolds by a CW-Nashmap. In the compact case an earlier result is attributed to Nash as also to Palais (see [55] for the references). In the non-compact case such a result is obtained in [55].

A stronger topology than either the uniform CP-topology or the compact open topology, is introduced in N ' ( M ) , as follows. Let fine

"fk

-+

0 as k

-+

00"

to mean the following:

for any C'-Nash vectorfields v1,

. . . , up# where T'

fk

v1

5

E N ' ( M ) , k = 1,2, . . . , and de-

. . . vp, f k

-+

0 uniformly as k

-+

00,

r . When M = R" and r = co,

this topology coincides with the usual topology on the space S of rapidly decreasing C" functions (cf. Rudin [52]), i.e., f k

k

-+

00,

for any multi-indices

(Y

-+

and

0 as k

p.

-+

00

+ ."opfk(.)

-+

0 uniformly, as

The space N ' ( M ) with this topology is not a

vector space, because a f f , 0 as a E R converges to 0 unless supp

(f)is compact. The

strong topology defined above is called the C'-topology on N ' ( M ) . The following theorems proved in [55].

Theorem 11.1. Let M I and Ma be CW-Nashmanifolds, and f : M I

-+

M2

a C'-Nash

mapping. Then f can be approximated by a C"-Nash mapping in the C'-topology. Further suppose the restriction off to a given compact Cw-Nashsubmanifold M3 of MI to be of class C". Then f can be approximated, fixing on

M3.

The proof uses a C"-Nash function on R" which is an approximation of 0 outside a small semi-algebraic neighbourhood of X ( X being an algebraic set in R"), and of 1

in another one. This function is required to hold a useful well-known property of a C"partition of unity. Several preliminary lemmas are needed for the proof, and though the

Chapter III

156

complete details of the proof of Theorem 11.1 would be outside the scope of this book, these leinmas themselves are interesting in their own right, and we shall csplain these in some detail. Let f E N‘(R”), and e ( z ) = integer, and

1 ~ =1 z:~ + ... + z.:

and k. Let U c

Leiiiiiia 11.2.

&,

where C is a positive number, k is a positive

Write

e as e c , h

when it is necessary t o emphasize C

W” be an open semi-algebraic neighbourhood

of f - ’ ( O ) .

Put

Define 1

F Then F -+ 0 on Vz and -+

= -2( ( f Z + e ) i + f }

f on Vl in the Cr-topologyas C and b +

00

in such a way

that k2k 5 C . Proof of Lemina 11.2. e ) i -+

We shall suppose r
0 then,

as in proof of Leiiirna 11.3,

outside

where

Yl = {f = I}, Y2 = {f = 2}, and (3 = f +

((f - 1 ) 2 + e,)

By Arguineiit 1, for sufficiently s ~ n a l el, l bourhood of

Y2.

U2

Here Fl and

*

F3

z}

=

el {f = 2 - -}

4

.

is contained in a given semi-algebraic neigh-

of Ifl, Y2respectively, and Clo,klo such that for each a],. . . , at

+ .. + loll 5

la11

Y3

L

Therefore to ensure (1) we only have to find open semi-algebraic neigh-

bourhoods U1 and

o with

Y3

TI

and any

~1

>~

1

and 0 kl

> k l o with k:kl 5 ~

can be replaced by

F30

respectively, because

= (3 - f -

((f - 1)2+ e1)')/2

1

we , have

>

163

Strong approximation in infinitedimensional spaces

and furthermore,

F l o ( s )> 0 w

> O for

F30(~)

z E R" - U1

u Uzaiid small

el

.

We shall denote by (2)o, ( 3 ) 0 , and (4)o the respective replaced quantities.

(30)

is trivial for some small U1

Uz

and

because

p ( t ) = tr'-L+l ( t - l)v'-!+i$(t)

for some polynomial $. Then by Lemma 11.3, F30 + F 1 0 on as C1 and

k1

00

3 k:kl

f1

= (( f - 1)'

+ e)

in the C"-topology

5 C1 and this, together with (3)0, provcs ( 2 ) o on Uz and

Then as in Lemma 11.3, (2)o on

where

W" - U1

U1

(4)o.

is reduced to

'.

This is one of the inequalities in the proof of Lemma 11.3.

Thus the Lemma follows. Next let X

c W"

be an algebraic set, I the ideal of R[xl . . . , x,,] defined by X,

namely consisting of polynomials vanishing on X, and h the square suiii of finite generators of I . Define f = h r ' / e 3 , with

c3,k3

> l and e3

=

eCs,ks.

Define e l , e2, r ' , (6, F1 and

Fz as we did earlier (before Lemina 11.4). Lemma 11.5.

The functions 4 ( F l ) and d(F2) are C"', and C" Nash functions, respec-

tively. Suppose U is a semi-algebraic neighbourhood of X. Tlien for small outside U and = 1 in another. Fix e 3 . Then, for some mation ofd(F1) in the CP'-topology.

el,

and

e3,

e3,

$(F1) = 0

( ~ ( F zis) an a.pp1,ox.i-

Chapter IZI

164

Proof. The first statement follows because of the definition. Next suppose sen that U 2 { h

e3

is to cho-

< 2e3}; this choice is possible by Argument 1. Then the second state-

ment follows. Th e last statement is Lemma 11.4. We shall denote by S i n g X the set of all the singular points of X. T he next leninia follows. Leiiinia 11.6.

Let Y c

X be

a connected component of X - Sing A;' let V be a semi-

algebraic neighbourliood of X - Y in W". Let g be a C'-Nasli function on R n r'-Aat on

Y . Then gd(F1) -i0 on R" - V in the C"-topology as C3,l i g -+

00

in such a way that

k i k s 5 C,. Proof of Leimila 11.6. index with la1 5

T'.

for large C3 and

123

Suppose E E N"(Rb)be of the same form as e , and a a multi-

What requires to be shown now is:

3 ktka

5 Cs. This inequality can be reduced t o simpler inequalities.

Since d(F1) = 0 outside W =

{f 5 2)) it is enough t o consider the inequality on W - V .

Furthermore, since

the earlier inequality can be replaced by

We note that if IyI > 0 then

Strong approximation in infinitedimensional spaces

Then because 4'(2 - f ) ,

. . . , 4(r')(2- f ) are bounded

165

on W , therefore it is enough to

prove:

This inequality in turn can be reduced to (for

1/31+ lal I + . . . +lafl + 17' 1 + :. . + IYk( 5 ? - I ,

and H = h"):

IDpgD"' H on W - V . If

I

e

. . . D"'HDy1e3 . . . Dyhe3/et"

15

E

= 0, then li = -1. In the proof of Lemma 11.2, the inequality:

DYiek+' < ole for some constant a , was established. Hence it is enough to prove: 3

1

... DaLH/HI 5 E

(DpgD"lH

on

liV - V - Y ,

l D P g 1 5 a on W - V where

1/31 5 r'. Here the inequality: H 5 2e3 on W as also the hypothesis: Dpg = 0 on

Y,has been used. Now consider the sets

2 =

{ z E R"

2' =

{ z E R"

I I

IDPgDYIH . . . D Y f H ( z ) l5 IDPg(r)l 5

E(Z)

H(z)} ,

~(5)).

These sets are semi-algebraic and contain Y . Using Argument 1, it is enough to prove that 2 and 2' are neighbourhoods of Y . For 2'this is clear. As regards 2,let

20

E Y

Chapter III

166

and consider a small neighbourhood of

(y,z) = (yl, . . . , ymlzm+1,

. . . , z,.)

ZO.

We can obtain a C" local coordinate system

around 2

h(y, 2) = y: t * . * t ym

20

and

3 (y, z ) = 0 at

Y

ZO

and

= {yl=...=ym=O}

(cf. Lemma 4.11 in Shiota [55]). Then by hypothesis

in a neighbourhood of 0 for some constant d'. Hence

pgD;1H . .. D,"'fqy,

I).

5 r"~.ll~~+~-l~l+~~'-l~ll+""l-l~~l - d")yl2r'+l .
0. Then uniform continuity of f on [0,1] implies that 36(~)> 0

[0,1]with Iz - yI

< & ( E ) , we have

If($)

- f(y)

3

Vx,y

< E . Now we apply Chebyehev’s inequality

(cf. Appendix 4), and obtain

5

u2(;)/q2

u2 being the variance of the binomial variable

max{lf(x)I

This completes Bernstein’s proof.

=

x ( l - x)

-,

nq2

Y . Now set 7 = $6( f), and M =

I 0 5 x 5 l}. Then if > f i ,we find n

E

Approximation problems in probability

171

More recently, Bernstein’s idea of using probabilistic techniques in approximation problems in analysis has been pushed further. We shall present a few of these recent results. However before turning to these recent developments we wish to note that Bernstein’s polynomials have been further used, to obtain accurate estimates of the errors of approximation, and these results have proved useful in semigroup theory (cf. Butzer and Behrens [6]).

Chapter IV

172

$2. Some recent Bernstein-type approximation results In the preceding section we saw that for a function

f

E C[O, 11, the Bernstein poly-

nomials n

where Pn,k(Z) = ( i ) z k ( l - z),-'

converge to f(z)uniformly on

be regarded as the result of an operator B, operating on

[o, 11. Here Bnf(z) can

f E C[O,11, for n

= 1 , 2 , . . ..

In

the same context it is known that

where w(6)= sup{lf(z) - f(y)I

I

Iz - yI

5 6, 0 5 z,y 5 l}, 6 > 0 (Popoviciu, cf. [3S]

p. 20). Further suppose f(2k)(z)exists at z, then (Bernstein, cf. [3S] pp. 22-23)

A number of such operators have been introduced after Bernstein. Many of these are special cases of an operator introduced by Feller [13]. Feller's operator is defined as follows. Let {X,, n 2 1) be a sequence of r.v.'s with distribution function ( d

. f.) F:,z(t)

with expectation E X , = z, and variance a ; ( z ) , z

being a continuous real parameter. For a continuous function f on W' define

Approximation problems in probability

Leiiiiiia 2.1.

If CT;(Z)

-+ 0 as n -+

00,

173

then lim L,f(z) = f ( z ) for every continuous n-ca

bounded function f . If further f is uniformly continuous and u k ( z ) -+ 0 uniformly, w.r.t.

x E W’, then the convergence of Lnf(z) to f ( z ) is uniform. Now suppose the continuous parameter z takes values in an interval I (perhaps infinite); let G(z) be a d.f. on I. We then obtain:

Leiiiiiia 2.2.

Suppose ok(z) 5 g(z),where g(.) is G-integrable and that the conditions

of Lemma 1 are valid. Then

The preceding scheme is now modified as follows. Let {Y,, n

2 1) be a sequence of

i.i.d. (independent, identically distributed) r.v.’s with mean z E I and variance 0”s). n

Let S,

=C yi.

Then the above expression for L,f(z) is modified to the following:

i=l

where F,,Z(t) is the distribution function of S,. The following theorem has been obtained by R.A. Khan (cf. [26] p. 195), thus extending, the above-mentioned result of Popoviciu and Bernstein to Feller’s operator defined in (1).

Theoreill 2.3.

Suppose {Y,,n

2 1) is a sequence of i.i.d.

r.v.’s with mean z E I

c W’,

and variance uz(z).Let A =sup ~’(x). Then for x E I the Feller operator defined in ( 1 ) ZEI

above satisfies:

Chapter I V

Ix-yl

5 6,z,y E R1}. firtherrnore, for x

5 yi and X = X (%) = [$I % xl] where [TI Clearly )%(fI f(x)1 5 w(f;6)(1 + A), and hence

Let Sn =

Proof of Theorem 2.3.

E [a,P] c I ,

-

i= 1

denotes the greatest integer ILn(f;Z)

5 T.

- f(x)l

-

I w(f;G)E(l+ A ) I w ( f ; b ) ( l +

{ + -}w(f;

= 1

EX2)

6) .

Now let 6 = n-’lz, and the proof is completed.

As regard monotonic convergence, the following theorems are proved in [26] p. 199 for the Feller operator defined earlier, when the function f(x) is convex.

Theorem 2.4.

Let {Yn, n 2 1) be a sequence of i.i.d. r.v.’s with mean x E I and vari-

ance uz(x). For a continuous convex and bounded function f on R1 defined the Feller

For the proof, the following lemma is needed. n

Lemma 2.5. Then

Let Y1,Yz

. . . be i.i.d r.v.’s with finite expectation.

Let Sn =

C

i=1

yi.

175

Approximation problems in probability

Proof of Lemma 2.5.

We note that E(Y1 ISn+1) = E(Yz(Sn+l)=

I

E(Y,+l (Sn+l); hence E ( K S,+1)

:+: 1 *.

= E('=-

Sn+l) =

. ..

=

Thus

This proves the Lemma. Proof of Theorem 2.4.

We note that

The function f is assumed to be convex; hence using the conditional version of Jensen's inequality and the preceding Lemma, we find

Thus

Now using the very first lemma in this topic (Lemma 2.1), the proof of the theorem is completed. More such results concerning Bernstein type operators are due, e.g., to Katherine Balazs (cf [3]). Some of the results in [3] are as follows. However, these are not exactly probabilistic in nature! For a function f on the positive half axis, with an,b, positive numbers 3 b,

-+

co,

Chapter IV

176

and a, =

% -+ 0 as n

-+ 00,

define the Bernstein type rational functions

Rn(f ;x) by

This particular positive linear operator has been investigated (see the references in [3]). For a function f defined in

(--00,

co),define the nth Bernstein type rational func-

tion RG(f;z) by:

where n

> 0 is even, and

a,, b, satisfying: a,, b, are

> 0,

b,

-+

00,

a, =

-+

0, as

n -+ co. The theorem established is Theorem 2.6.

(cf Theorem 2 [3], p . 196). Suppose f is continuous in

satisfies: f ( x ) = O(ealZI)for some a

for - A 5 x 5 A ; here n o f f in [ - A - E ; A

> 0. Then

for arbitrary fixed A

(-00,

co) and

> 0 and a > 0,

> 0 is even, w [ - A - ~ , A + ~ I ( ~.); denotes the modulus of continuity

+ a ] , and c1 = cl(ar;A; E ) > 0 is a number independent

of n.

Furthermore, a necessary and sufficient condition for the uniform convergence of

R i ( f ) to f is:

f

C[-00, m] where where C[-00, m] denotes the class of continuous

functions f 3 lim f ( z ) exists (finite) and hence f is uniformly continuous on [-co,co]. 14'~ The precise theorem is:

Approximation problems in probability Theorem 2.7.

lim n+m

177

(cf. Theorem 4, [3], p. 197). Suppose b, = np with 0

sup -ca o even

A slightly more abstract situation is considered by R. Wittman (cf. [64]). Suppose E is a locally convex space, and A

cE

a convex set. A function f : A

--t

W 1 is uniforinly

continuous if 3 continuous seminorm p on E 3

The result established in the following: Theorem 2.8.

(cf. Theorem 1, [64], p. 463). Suppose f : A

continuous convex function. Then for every E g on E satisfying: sup Ig(z) - f(x)I I zEA

E.

> 0 3 Lipschitz

-+

W' is a uniformly

continuous convex function

Chapter IV

178

$3. A theorem of Steinhaus Probabilistic ideas have been used in a function theoretic context, specifically in the context of random Taylor series. E. Borel, in 1896, formulated the statement that, with probability one, the circle of convergence of a power series with arbitrary coefficients is its natural boundary, i.e. consists only of singular points. This statement is only about plausibility. The correct formulation was given by H. Steinhaus [59] in 1929. The result can be stated as follows. 00

Theorem 3.1.

rneigmzn, where the

The series

&’s

are mutually independent ran-

0

dom variables uniformly distributed on [0,27r], and limsup r:ln n+cn

< 00,

has almost surely

the circle of convergence as its natural boundary. We shall give here an account of a theorem of H. Steinhaus in this context, formulated for Steinhaus series. This account follows Kahane [26], where also one can find references to further related developments. At this point a brief explanation of the probabilistic ideas involved in this theorem, appears to be in order, and the explanation given in the next paragraph appears to be the bare minimum of “probability theory” needed for this theorem. A more general account will be found in Appendix 4.

Probabilistic ideas; random variables

.

3-1 = [0,1] x [0,1] x ...

of infinite dimensions. An event is a subset E

c 3-1 and its probability p(&) is its (infinite

A random point is a point w

= ( w I , w ~ , . .) in the hypercube

dimensional) product Lebesgue measure, if it exists. If p ( E ) = 1 we say that E occurs al-

m o s t always (a.a.), of that & is almost certain. A r a n d o m variable (r.v.) is a measurable function of w , and is always denoted by a capital letter. With any hypothetical property

Approximation problems in probability

179

of a random variable or of a family of random variables we speak of the event of the oc-

currence of this property; we thus speak of the property occurring almost always. The expected value (or expectation) of a r.v. T is E ( T ) =

s . T(w)dw. For any given sequence

{Tk} of r.v.’s if Tk depends only on the kth component

wk,

we have E(II Tk) = IT E(Tk). k

k

The characteristic f u n c t i o n of a r.v. T is E(eiUT)where u E W’; if E(eiUT)= e-”’I2, T is said to be a (real) normal The r.v.’s Tj(j = 1,2,

T.v..

Two r.v.’s TI and T2 are orthogonal if E(TlT2) = 0.

. . . , ) are said to be

independent if 3 a mapping w

-+

w’ of

‘Ft -+ ‘H, which is measure-preserving 3 under this mapping Tj = function of w i a.a. Two r.v.’s T,T’ are said to follow (or to be subject t o ) the same law if T’(w)= T(w’)a.a.

A complex r.v. T is invariant under

r o t a t i o n if €or each

t E R’, T , and Teit are subject

to the same law. If the r.v.’s Tj,j = 1 , 2 , are real normal and independent, then for each

t

E

W’, TI cost

+ T2 sint is normal, and TI + aT2 is called a complex n o r m a l

r.v. A

complex normal r.v. is invariant under rotation. If T is a normal r.v. (real or complex), then a

+ bT is a Laplacian (or Gaussian) r.v., a , b being constants.

We shall need the following lemmas; for the proofs of the first two cf. Loeve [37], and for the third cf. Zygmund [67].

Kolmogorov’s Lemma.

Given an infinite sequence { T j } of independent r.v.’s, and an

event E which for each n = 1,2, . . . , does not depend upon the realisation (i.e. on the values of) Ti, j = 1,2, . . . , n, then p ( E ) = 0 or 1.

Kliinchin’s Lemma. j = 1,2,

If the Tj are real independent r.v.’s, 3 E(Tj) = 0 , E ( T j ) = 1 for

. . . , then the series

cjTj converges a.a. provided

C

IcjI2

< cm.

j

Zygmuud’s Lemma.

If the r.v.’s T,,j = 1,2, . . . , are subject t o the same law,

Chapter IV

180

then the series

c

cjTj

is not summable under any regular process of summability if

j ICjI2

= 00.

j

Random trigonometric series

Here we are concerned with random functions w into Fw(t)mapping tion on the unit circle. Consider a local property of F, e.g. F ( t ) being

IFI into a func-

> 0; or F ( t )

being continuous in t etc. We say at almost all points t , the property holds almost certainly meaning: the property holds almost with probability 1 at all points t except perhaps for t lying in a set of measure 0 on the unit circle. If this happens we say the property holds almost certainly (surely) almost everywhere, (a.s.a.a.). If a property holds at each t almost certainly, it does not follow that it holds almost certainly for all t . It is often much more difficult to determine the probability that a local property should hold everywhere than it is to determine such a probability at any (fixed) point.

A r.v. is ( s t r i c t l y ) s t a t i o n a r y if the probabilities associated with it are invariant under translations t

--+

t - to, that is to say, if to each to 3 transformation

w -+

w’

(see

earlier in this section) 3 F,t(t) = Fw(t - t o ) .To say that a local property should hold a.s., a.e. is saying that is should hold a s . at a fixed point t o . The convolution of a stationary r.v. with a function which is certain (i.e. not subject to any randomness) is a r.v. which is stationary. It is also quite natural to consider formal trigonometric series

C A,eint, with coef-

ficients A , which are complex r.v.’s. Such a series is said to be s t a t i o n a r y if its convolution with anay trigonometric polynomial is stationary. We shall consider here only S t e i n h a u s series F i n which the coefficients A , are in-

Approximation problems in probability

181

dependent r.v.'s which are invariant under rotations. We shall use the notation F

N

CA,eint, so as to be able to use this symbol to identify F with a function, or a distribution, as the case may be.

Steinhaus' theorem We consider properties P of trigonometric series f on intervals, which are subject to the following conditions:

(1") If f satisfies P on

( a ,b ) ,

any translate ft satisfies P on ( a -t t , b -t t ) .

(2") If f satisfies P on two abutting intervals, then f satisfies P on their union. (3") If f satisfies P on an interval, then so does f

+ p , where p is any trigonometric poly-

nomial.

(4")If F is a Steinhaus series, then the statement: "F satisfies P on ( a , b)" is an event for which the probability exists. The theorem of Steinhaus referred to above is as follows.

Theorem 3.1.

(cf (261). Consider a Steinhaus series F and a property P . Then al-

most surely, F satisfies P everywhere or F satisfies P nowhere.

Proof. Let {I,} be a finite collection of intervals, all of the same length the circle, and

xc,, = p[on I,, F satisfies PI,

n = 1,2,

and

x, = p [ F satisfies P on at least one In]

.. .,

E,

which cover

182

Chapter I V

The condition (4") in the definition of the property P implies that x,,,, and xc both exist. Suppose x,

> 0. Now x,

_ E

L ~ [ om) , H

satisfying:

t(S," %dt)2,

which means that L2[0,m)c Lh[O,m). Further

Ji w h< m.

0 for n = 0 , j = 0, and j = 0,1,

. . . , 2"-l

NOW

let z(.) E L ~ [ om) , 3 :J

- 1 for n

z(t)hnj(t)dt =

2 1. A little calculation shows that

Approximation problems in probability

The system {gnj} is complete in L1[O, 11 i.e. if

3:

E

L'[O,11 and

I87

si 3 : ( t ) g n j ( t ) d t = 0 Vg,j

then z = 0 a.e. This implies that { h n j } is complete in L ~ [ O , C Ohence ) , in L2[0,co).This proves the Lemma. Now let H n j ( t ) =

soh,j(u)du. t

The next lemma gives a convenient estimate con-

cerning the functions Hnj(t). Zn-l

-1

Leiiiina 4.2. j=0

+ i)for t E [O,a],a
0, 3 only a finite number of‘values oft

such that the discontinuity at t is

> E.

Proof. For suppose 3 infinite sequence

{tk}

with t k

+

t o , say, and

Iz(tk

+ 0) - S ( t k -

0)l > E , then at t o , z ( t ) would not have a limit either from the right or from the left. Leiiiina 5.2.

Suppose t l , t z , . . . , tk are all the points in I where z ( t ) E D[O,11 has

discontinuities 2

E,

for some E

> 0. Then 36 > 0 3 if It’ - t”I < S and if both t’,t“ belong

to the same one of the subintervals ( O , t l ) , ( t l ,t z ) , . . . , (tk, l), then Iz(t’)- z(t”)(< E.

Approximation problems in probability

191

Proof. For suppose 3 sequences {t:}, {t:} both converging so some point t o and belonging to the same one of the intervals (0, tl), . . . , ( t h , 1) and 3 lx(tL) - “ ( t i ) [ Then t:, t: must lie on opposite sides of t o (otherwise I z ( t k ) - z(t:)l 2 possible!), hence Iz(to+O)-z(t0 -0)l

>

E.

E

>

E.

would be im-

Then t o must be one of t l , t z , . . . , t k ; but this

would contradict the conclusion above that t;, t: belong to the same one of the intervals

Lemma 5.3.

If z ( t ) E D[O,11, then Vq > 0 36 > 0 3 every t E 1 satisfies one of the

inequalities:

This can be stated a little more conveniently as

Lemma 5.3‘.

If z ( t ) E D[O,11, then

The justification of either statement is in Lemma 2. The next lemma is almost the converse of the assertion of Lemma 5.3.

Lemma 5.4.

If a function z ( t ) E D[O,11 satisfies the statement of Lemma 5.3’ then

3?(t) E D[O,11 which coincides with z ( t ) at all its points of continuity. For suppose z ( t ) satisfies the statement of Lemma 5.3’. Then z ( t ) must have, at each t E I , a limit from the left and a limit from the right. Otherwise 37 points tl

< tz < t3

in I 3 Iz(t1) - z(tz)l > 7 and Iz(tz) - z(t3)I

> 0 and three

> 7 . But this contradicts

Chapter IV

192

(*). Hence z ( t ) must equal either z ( t - 0) or z ( t

+ 0).

Setting Z(1) =t-1lim z ( t ) and

Z ( t ) = lim z ( t ' ) . :'+I :'>I

Skorokhod's space D[O,11 and his J-topology on this space are meant to be a generalisation of the space C[O, 11 of continuous functions on I with the uniform topology. The following definition is meaningful for functions ~ ( tE )C[O, 11.

Definition.

The sequence of functions {zn(t)}r=l

converges uniformly t o z ( t ) at the

point to E I if VE > 036 > 0 3

lim

n-oo

sup lzn(t)- z(t)l I 0 write

2'(t) = z ( t ) -

c

[ 2 ( s ) - z(s - O)]

.st

where the understanding is that the sum is over those discontinuities for which z(s) 2(s - 0 ) > E .

Definition 5.6'.

A sequence of functions {z,(t)} in D[O,11 is said to be J-convergent

t o zo(t) E D[O,11 if (a) for every E

>0

3 z 0 ( t )does not have jumps in absolute value equal to

zk(t) + zO(t) - z:(t) for almost all

& sup Iz:(t) - x $ ( t ) l = 0. (b) clim -+On-+m t E ~

t E I;

E,

~ , ( t )-

Chapter IV

194

Now suppose z ( t ) E D[O,11. Then for every C > 0, write

Ac(z(t)) =

min((z(t') - z ( t ) ( ; Iz(t) - z ( t " ) ( )

sup o< 11'

+

1'< 1 " j l 8"J E - 4p > 3p. Further

p . Similarly I y ( t ) - y ( t ' ) l

< p for t E [t',t'

+ h]. The Lemma

follows. Leiiiiiia 5.9.

Suppose the jumps of z ( t ) and y ( t ) do not exceed E , where E > 0 is

given, and suppose for some h

< f, p h ( z ( t ) , y ( t ) ) < p , then sup t

Proof of Lemma 5.9. We shall first show that

Iz(t) - y(t)l

< 2s + 5 p .

197

Approximation problems in probability

Let tl

< t2

and t' E

(tl,t2)

3 if s

< t' and Iz(t1) - z(t')l >

p then Iz(t1) - z(s)l

5 p.

Then Iz(t1) - z(t')l I Iz(t1)

- z(t'

- 0)l + Iz(t' - 0)

- s(t')l

However rnin(lz(t1) - z(t')l; Iz(t') - z ( t 2 ) ( )< p , hence Iz(t') - z(t2)I Iz(t1) - z(t2)I

Similarly

+

I E p .

< p , and therefore

< E + 2P. It1

- t21 < h

+ Iy(t1) - y(t2)I < E + 2p.

If t E [0,1] then 3 t' 3 It - t'I


0

Suppose pc,(z,(t),

z o ( t ) ) -+ 0 and C,

3 zo(t) does not have jumps equal in absolute value to

E.




E

(tj-

%,t j + %) would have two points t'

- 6p

loo(t'') -

- 0)l

>

hence for some j , t" =

E,

Then the interval

tj.

and t y ) at which the jumps of zn(t) exceed e

in absolute value, which is impossible. Hence

c

I,(t) - 5 k ( t ) =

( z n ( t y ) - z,(t:n) - 0))

,

tpst and

By Lemma 5.8, and the condition that pc, (zn(t), z o ( t ) ) z n ( t ) + zo(t) at every point of continuity of

Iz,(tin) zn(t)

~ ( t )Hence . as

- 0) - zo(ti - 0)l -+ 0. Also because C,

- zk(t)

-+

zo(t) - z t ( t ) as t

#

ti.

-+

---t

0, it is clear that

---t 00,

0, therefore tf"'

Therefore z k ( t )

-+

Izn(t!n)) - z o ( t i ) / -+

ti. Hence

z t ( t ) for all points t of

continuity of z f ( t ) (as this is fulfilled for xn(t)). We conclude also that condition (a) of Definition 1 is fulfilled for zn(t). We shall now show that when A c ( z ( t ) )< e, &( z ' ( t ) ) t3

< 2.5. If tl < t 2 < t 3 , with

- tl < C, then

provided z ( t ) has no jump exceeding

E

in absolute value in ( t l l t 3 ) ; and if a jump with

199

Approximation problems in probability

absolute value exceeding E does exist at, say, t' E ( t l ,t z ) , then min ( I z c ( t l )- zc(tz)l, ) z c ( t 2 )- zc(t3)1) I ( z C ( t z )- zC(t3)1

+ min Iz(t' - 0) - @)I,

I(z(t')- z(t3)1

I 2Ac(z(t)) ; and a similar inequality holds if z ( t ) has a jump exceeding E in absolute value in Therefore n-m lim A, Now consider

(t2,tS).

( z k ( t ) ) = 0. inf

f0

3 Vt

> s,

Appendix 1

204

E

c tV. Suppose X is a TVS. To each a E X

Translation and multiplication operators.

we associate the translation operator Ta defined by: Tax = a

+ x, x

E

X ; and to each

scalar X we associate the multiplication operator M A : M A X= Xx,x E X. Then these two operators Ta and M A both homeomorphisms of X onto X . This last statement implies the following: every vector topology on X is translation

invariant i.e. a set E

cX

is open if and only if for each a E X , a

+ E is open.

In a T.V.S. X the term local base means a local base of neighbourhoods at 0. Thus a local base of a T.V.S. X is a collection

B of neighbourhoods of 0 such

that every

neighbourhood of 0 contains a member of B. The open sets of X are then precisely those that are unions of translates of members of

B.

A metric d on a vector space X is translation invariant if d(z vz,y,z E

+ z , y + z ) = d(x,y)

X.

The following definition explains some of the types of T.V.S.'s that we might encounter. X here denotes a T.V.S. with topology

7.

Definition. (a) X is locally convez if 3 local base

B

consisting of convex subsets.

(b) X is locally bounded if 0 has a bounded neighbourhood. ( c ) X is locally compact if 0 has a neighbourhood with compact closure.

(d) X is metrisable if (e) X is an F-space if

T

T

is induced by a metric d. is induced by a complete invariant metric.

( f ) X is a Fre'chet space if X is a locally convex F-space.

Topological vector spaces

205

(g) A n o r m on a vector space X is a non negative valued function denoted by

11211,

hav-

ing the properties: ))z))= 0

112

only if

+ YII Ilbll + IlYll

2

=0

VX,Y

EX

.

A vector space X with a norm on X is called a n o r m e d linear space. If a vector space X is normed then d ( z , y ) =

((3 -

yII defines a distance (or metric) on X. If

X is complete w.r.t. this metric, X is called a B a n a c h space. (h) A T.V.S. X is normable if 3 norm on X 3 the metric induced by the norm on X is compatible with the topology on X .

(k) A T.V.S. X has the Heine-Borel property if every closed and bounded subset of X is compact.

Theorem.

If B is a local base for a T.V.S. X then every member of t3 contains the

closure of some member of B. Hence:

Corollary.

Every T.V.S. is a HausdorfTspace.

Theorem.

In a T.V.S. X

(a) every neighbourhood of 0 contains a balanced neighbourhood of 0. (b) Every convex neighbourhood of 0 contains a convex balanced neighbourhood of 0.

Thus Theorem. (a) Every T.V.S. has a balanced local base.

Appendix 1

206

(b) Every locally convex space has a balanced convex local base. Suppose X and Y are vector spaces over the same field K. A mapping T : X + Y is called linear if

T ( a x + P y ) = aTx+PTy V x , y E X and Va, p E K. For a linear mapping we often write Tx instead of T ( x ) . A linear mapping T : X

Theorem.

+K

is called a linear functional.

Let X and Y be T.V.S.S. If T : X

+

Y is continuous at 0 then T is con-

tinuous, and in fact uniformly continuous, i.e., for each neighbourhood W of 0 in Y,3 neighbourhood V of 0 in X 3

y- x E

V

+Ty -Tx

EW

.

For a linear functional on a T.V.S., the following is true. Theorem.

Suppose F is a linear functional on a T.V.S. X , 3 Tx

# 0 for some x

E X.

Then the following four statements are equivalent: (a) F is continuous.

(b) The null-space N ( F ) is closed. (c) N ( F ) is not dense in X. (d) F is bounded in some neighbourhood of 0. The simplest models of Banach spaces are the standard real of complex n-dimensional Euclidean spaces R" or C" over R1 or 43, respectively, normed by means of the usual Euclidean metric.

Topological vector spaces

For example if z =

($

. . . , zn), z;

(21,

E

207

C is a point (i.e. vector) in C" then

llzll

=

112

(zil')

is a norm on C"; likewise if z = (zl,. . . , zn),zi E R' is a point (or 112

vector) in R", then

( ( ~ 1 1=

is a norm on R".

These are by n o means the only norms that can be introduced on W" or C", respectively.

Theorem.

Suppose X is a complex T.V.S., Y is a subspace of X, and dimY = n

where n is a positive integer. Then (a) every isomorphism of Y onto C" is a homeomorphism;

( b ) Y is closed. Theorem. (a) Every locally compact T.V.S. is finite dimensional. (b) If a T.V.S. X is locally bounded and has the Heine-Borel property then X is finite dimensional. Before turning on to some of the most useful type of T.V.S.s we shall mention the general characteristics of a bounded linear transformation (or linear mapping). A linear mapping T : X

4

Y ,where X, Y are T.V.S.'s, is

bounded

if T maps bounded sets into

bounded.

Theorem.

Suppose X , Y are T.V.S.s and T : X

among the following four properties of X ,

If further X is metrisable then

4

Y is a linear mapping. Then

Appendix I

208

so that for a metrisable T.V.S. X , all four statements are equivalent.

(a)

T is continuous;

(b)

T is bounded;

(c) if z,

-+

0 then {Tz,,n = 1, 2, 3, . . .} is bounded;

(d) if xn

-+

0 then T x n -+ 0.

Among the most useful kind of T.V.S.s occurring in analysis are the locally convex ones, for the topological structure of a locally convex space X can be specified by a special family of non negative (non linear) functions on X called semi-norms.

A s e m i - n o r m on a vector space X is a real-valued function p ( . ) on X with the properties:

(4

P(X

+ Y) I P(X) + dY),

(b) p ( a i ) = IaIp(z), Vx,y E

X

(c) p ( i ) # 0 if x

is a norm.

# 0 then p

and V a E K. If further p satisfies

A family P of semi-norms on X is called separating if to each 5

#

0

3 semi-norm p E

p 3 d X ) # 0. If the vector space X is also an algebra, an algebra semi-norm p ( . ) on X is a seminorm which further satisfies

(4

P(X -Y)I P(.)P(Y)

VX,Y E X ,and

(b) if X further has a unit e then p ( e ) is either 1 or 0.

A subset A c X is called absorbing if each z E X lies in tA for some t > 0. Suppose A c X is absorbing; then the Minkowski functional

~ A ( x )= inf[t

I

PA(.)

of A is defined by:

> 0 x E tA] .

Topological vector spaces

We note p ~ ( x )