Approximation of Vector Valued Functions PDF
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APPROXIMATION OF VECTOR VALUED FUNCTIONS This Page Intentionally Left Blank NORTH-HOLLAND MATHEMATICS STUDIES 25 Notas de Matemhtica (61) Editor: Leopoldo Nachbin Universidade Federal do Rio de Janeiro and University of Rochester Approximation of Vector Valued Functions JOaO B. PROLLA IMECC, Universidade Estadual de Campinas, Brazil 1977 NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM NEW YORK OXFORD @ North-Holland Publishing Company - 1977 AN rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner. North-Holland ISBN: 0 444 85030 9 PUBLISHERS : NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM NEW YORK OXFORD SOLE DISTRIBUTORS €OR THE U.S.A. AND CANADA: ELSEVIER NORTH-HOLLAND, INC. 52 VANDERBILT AVENUE, NEW YORK, N.Y. 10017 Library of Congress Cataloging in Publication Data Prolla, Joao B Approximation of vector valued functions. (Notas de matemitica ; 61) (North-Holland mathe- matics studies ; 25) Bibliography: p. Includes indexes. 1. .V ector valued functions. 2. Approximation theor I. Title. 11. Series. Q~l.Ni6 no. 61 [QA3201 5101.8s [ 515 .7l ISBN 0-444- 85 030-9 77-22095 PRINTED IN THE NETHERLANDS PREFACE This work deals with the many vari.ations of the Stone- ileierstrass Theorem for vector- valued functions and some of its a;)?lications. ?or a more detailed description of its contents sze the Iatroduction an5 the Tab1.e of Contents. The book is 1a:;re- ly se1i'- contained. The amount of Functional Inalvsis required is minimal, exce3t for Chapter 8. But the results oE this Chapter are not used e l ~ e ~ ~ h e'rThee. book can be used by rjraduate skudents who Iia-Je taken the usual first-year real and corrplex analvsis courses. Thc treatment of the subject has not anDeared in boo!< forn previously. Zven the proof of the Stone-Weierstrass Theo- ren is new, and due to S. 'lachado. 7Je also give results in non- archimedean approximation theory that are new and extend the Dieudonne - KaTlanskv Theorem to nonarchim.edean vector - valued function spaces. I thank Professor Silvio :lachado, from the Universich- cie Federal do 9io de Janeiro, for his valuable comments and re- narks on the suhject. Iilithout his he13 this vould be a different and ?oorer book. I thank also Pro-Fessor Lzo~oldo! Jaclibin, from the Unjversidade Federal do 710 de Janeiro and the Vniversity vi of Rochester, whose advice and encouragement was never failing. Finally, I wish to thank Angelica h4arquez and Elda Mortari for typing this monograph. JOE0 B. PROLLA Campinas, April 1977 CONTENTS ................................................... PREFACE v ............................................... INTRODUCTION ix . CHAPTER 1 THE COMPACT-OPEN TOPOLOGY 1 . ............................ § 1 Basic definitions 1 . ............................... § 2 Localizability 3 . ........................... § 3 Preliminary lemmas 4 . ........ § 4 Stone-Weierstrass Theorem for modules 7 . ................ § 5 The complex self-adjoint case 10 . ......................... § 6 Submodules of C(X;E) 13 5 7 . An example: a theorem of Rudin ............... 17 . ............................. § 8 Bishop's Theorem 19 . ............................ § 9 Vector fibrations 25 . .......................... 5 10 Extreme functionals 21 5 11 . Representation of vector fibrations .......... 35 . .................. p 12 The approxination property 40 . ................. Appendix Non-locally convex spaces 43 . ....................... CHAPTER 2 THE THEOREM OF DIEUDONNE 46 . ............................. CHAPTER 3 EXTENSION THEOREMS 52 . ............................ CHAPTER 4 POLYNOMIAL ALGEBRAS 57 . ................. 5 1 Basic definitions and lemmas 57 . .................. 2 Stone-Weierstrass subspaces 67 . ................................. 5 3 C(x)-modules 72 . ........... 5 4 Approximation of compact operators 74 . ......................... CHAPTER 5 WEIGHTED APPROXIMATION 79 . ................. 5 1 Definition of Nachbin spaces 79 . .. 5 2 The Bernstein-Nachbin approximation problem 80 . 3 3 Sufficient conditions for sharp localizability 88 . ............... 5 4 Completeness of Nachbin spaces 90 . ................ 5 5 Dual spaces of Nachbin spaces 96 . ....................... Appendix Fundamental weights 107 vii vii i CONTENTS . ... CHAPTER 6 THE SPACE Co(X;E) with the uniform topology 113 . .... CHAPTER 7 THE SPACE Cb(X;E) with the strict topology 127 . . .................. CHAPTER 8 THE c-PRODUCT OF L SCHWARTZ 138 5 1. General definitions ......................... 138 . .............. 2 Spaces of continuous functions 141 5 . .................. 3 The approximation property 144 . ......................... 5 4 Mergelyan's Theorem 146 5 S . Localization of the approximation property .. 149 . ........... CHAPTER 9 NONARCHIMEDEAN APPROXIMATION THEORY 153 . ............................... § 1 Valued fields 153 . ......................... § 2 Kaplansky's Theorem 156 . ............................... § 3 Normcd spaces 162 . ..................... § 4 Vector-valued functions 163 . ........................... § 5 Vector fibrations 171 . ........................... § 6 Some applications 181 § 7 . Bishop's Theorem ............................ 187 § 8 . Tietze Extension Theorem .................... 189 ................... § 9 . The compact-open topology 193 9 10 . The nonarchimedean strict topology .......... 198 .............................................. BIBLIOGRAPHY 206 .............................................. SYMBOL INDEX 213 ..................................................... INDEX 215 INTR ODUC T I ON The typical problem considered in this book is the following. One is given a vector subspace W of a locally convex space L of continuous vector-valued functions, which is a modu- le over an algebra A of continuous scalar-valued functions ,and the problem is to describe the closure of W i n the space L. In chapter 1 we start with the case in whichL=C(X;E) with the compact-open topoloay. Vhen the algebra A is self-ad- joint, the solution of the above problem is given by the Stone- -1Veierstrass theorem for modules. A very elegant and direct proof due to S. Machado (see [38!) is presentc.: here. As a co- rollary one gets the classical Stone-Weierstrass theorem for self-adjoint subalgebras of C(X;C). When the algebra A is not self-adjoint, a solution of the problem is given by Bishop's theorem. The proof that we include here is aqain due to S. Ma- chado (see [ 371). The main idea is to use a "strong" Stone- -!!eierstrass theorem for the real case plus a transfinite ar- gument. This is done in Machado's paper via Zorn's Lemma. Here we use the transfinite induction process found in the original paper of Bishop (see [ 8 ] ) . We prefer this new method over de Brange's technique, because it can be applied to other situa- tions in weighted approximation theory , namely where measure theoretic tools are either painful to apply or not available at all. In 5 9 of this Chapter we treat a special case of vec- tor fibrations, and prove in this context a "strong" Stone- -Weierstrass theorem due to Cunningham and Roy (see [ 153 ) . This result is used in the next section to characterize extrgoe fun* ix
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