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An Introduction to Functional Analysis AN INTRODUCTION TO FUNCTIONAL ANALYSIS Charles Swartz New Mexico State University Las Cruces, New Mexico Marcel Dekker, Inc. New YorkBasel Hong Kong Library of Congress Cataloging-in-Publication Data Swartz, Charles. Functional analysis / Charles Swartz. p.cm. Includes bibliographical references and index. ISBN 0-8247-8643-2 1. Functional analysis.II. Title. QA320.S91992 515'.7--dc2O 91-44028 CIP This book is printed on acid-free paper. Copyright © 1992 by MARCEL DEKKER, INC. All Rights Reserved Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without per- mission in writing from the publisher. MARCEL DEKKER, INC. 270 Madison Avenue, New York, New York 10016 Current printing (last digit): 10 9 8 7 6 5 4 3 2 1 PRINTED IN THE UNITED STATES OF AMERICA TO NITA RUTH Preface These notes evolved from the introductory functional analysis course given at New Mexico State University.In the course, we attempted to cover enough of the topics in functional analysis in depth and to give a sufficient number of applications to give the students a feel for the subject. Of course, the choice of topics reflects the interests and prejudices of the author and many important and interesting topics are not covered at all. We have given references for further study at many points. The students in this course were assumed to have had an introductory course in point set topology and a course in measure and integration.The text assumes a knowledge of elementary point set topology including Tychonoffs Theorem and the theory of nets and a background in real analysis equivalent to an introductory course from a text such as [Ro] or [AB]; a few basic results from complex analysis are required, especially in the sections on spectral theory. The reader is also assumed to be familiar with Hilbert space; an V vi Preface appendix supplies the basic properties of Hilbert spaces which are required. Functional analysis is a subject which evolved from abstractions of situationswhichrepeatedlyoccurredinconcretefunctionspaces, differential and integral equations, calculus of variations,etc. The abstractions have repeatedly repaid their debts to its foundations with multitudes of applications. We have made an attempt to give at least enough of these applications to indicate the interplay between the abstract and the concrete and illustrate the beauty and usefulness of the subject. The first section of the book introduces the basic properties of topological vector spaces which will subsequently be used.In the second section the three basic principles of functional analysis -- the Hahn-Banach Theorem, the Uniform Boundedness Principle and the Open Mapping/ Closed Graph Theorems -- are established. Applications to such topics as Fourier series, measure theory, summability and Schauder basis are given. Locally convex spaces are studied in the third section.Topics such as duality and polar topologies are discussed and applications to topics such as Liapounoffs Theorem, the Stone-Weierstrass Theorem and the Orlicz-Pettis Theorem are given.We do not present an exhaustive study of the enormously large subject of locally convex spaces, but rather try to present the most important and useful topics and particularly stress the important special case of normed linear spaces.In the fourth section we study the basic properties of linear operators between topological vector spaces. The Uniform Boundedness Principle for barrelled spaces, the basic properties of the transpose or adjoint operator, and the special classes of projection, compact, weakly compact and absolutely summing operators are discussed. Preface Vii Applications to such topics as the integration of vector-valued functions, Schwartz distributions, the Fredholm Alternative and the Dvoretsky-Rogers Theorem are given.Finally, in section five we discuss spectral theory for continuous linear operators. We first establish the spectral theorem for compact symmetric operators and use this result to motivate the spectral theorem for continuous symmetric (Hermitian) operators. A simple straightforward proof of this version of the spectral theorem is given using the Gelfand mapping. We derive the spectral theorem for normal operators from basic results from Banach algebras; this affords the students an introduction to the subject of Banach algebras. Applications to such topics as Lomonosov's Theorem on the existence of invariant subspaces for compact operators, Sturm-Liouville differential equations, Hilbert-Schmidt operators and Wiener's Theorem are given. These notes cover many of the basic topics presented in the classic texts on functional analysis such as [TL], [Rul], [Cl], etc., although some of the topics covered in the applications such as the Nikodym Boundedness and Convergence Theorems, the Pettis and Bochner integrals, summability and the Orlicz-Pettis Theorem are often not treated in introductory texts. We do, however, include some material that is not found in the standard introductory texts.We use the stronger notion of X convergence of sequences and X bounded sets developed by the members of the Katowice branch of the Mathematics Institute of the Polish Academy of Sciences under the direction of Piotr Antosik and Jan Mikusinski to establish versions of the Uniform Boundedness Principle which require no completeness or barrelledness assumptions on the domain space of the viii Preface operators.It is shown that the general version of the Uniform Boundedness Principle yields the classic version of the Uniform Boundedness Principle for complete metric linear spaces as a corollary and the general Uniform Boundedness Principle is employed to obtain a version of the Nikodym Boundedness Theorem from measure theory where the classic version of the Uniform Boundedness Principle is not applicable.These results are obtained from a beautiful result due to Antosik and Mikusinski on the convergence of the elements on the diagonal of an infinite matrix with entries in a topological vector space. The matrix theorem can be viewed as an abstract "sliding hump" technique and is used in many proofs in the text. The notion of X bounded sets is also used to obtain hypocontinuity results forbilinear maps which requireno completenessorbarrelledness assumptions. I would like to thank the many graduate students at New Mexico State University who took the introductory functional analysis course from me during the time that these notes were evolving into their final form and especially Chris Stuart, who read through the final manuscript.Special thanks go tomy colleague and friend Piotr Antosik,who introduced me to the matrix methods thatare employed throughout this text, and to Valerie Reed, who did sucha superb job of translating my handwriting from the original class notes. Charles Swartz Contents Preface v 1. Topological Vector Spaces (TVS) 1 1. Definition and Basic Properties 3 2. Quasi-normed and Normed Linear Spaces (NLS) 13 3. Metrizable TVS 31 4. Bounded Sets in a TVS 37 5. Linear Operators and Linear Functionals 43 6. Quotient Spaces 59 7. Finite Dimensional TVS 65 II. The Three Basic Principles 71 8. The Hahn-Banach Theorem 73 8.1 Applications of the Hahn-Banach Theorem in NLS 77 8.2 Banach Limits 86 8.3 The Moment Problem 89 9. The Uniform Boundedness Principle (UBP) 91 9.1 Bilinear Maps 105 9.2 The Nikodym Boundedness Theorem 112 9.3 Fourier Series 119 9.4 Vector-Valued Analytic Functions 123 9.5 Summability 125 10. The Open Mapping and Closed Graph Theorems 137 10.1 Schauder Basis 146 ix X Contents III. Locally Convex TVS (LCS) 155 11. Convex Sets 157 12. Separation of Convex Sets 165 13. Locally Convex TVS 169 13.1 Normability 182 13.2 Krein-Milman Theorem 184 14. Duality and Weak Topologies 189 15. The Bipolar and Banach-Alaoglu Theorems 199 16. Duality in NLS 209 17. Polar Topologies 231 18. The Mackey-Arens Theorem 237 19. The Strong Topology and the Bidual 243 20. Quasi-barrelled Spaces and the Topology j3*(E, E') 251 20.1 Perfect Sequence Spaces 256 21. Bornological Spaces 261 22. Inductive Limits 267 IV. Linear Operators 277 23. Topologies on Spaces of Linear Operators 279 23.1 The Krein-Smulian Theorem 290 24. Barrelled Spaces 295 25. The UBP and Equicontinuity 305 26. The Transpose of a Linear Operator 309 26.1 Banach's Closed Range Theorems 325 26.2The Closed Graph and Open Mapping Theorems for LCS 330 26.3 Vector Integration 334 26.4 The Space of Schwartz Distributions 347 26.5 The Lax-Milgram Theorem 366 26.6Distributions with Compact Support 371 26.7A Classical Theorem of Borel 379 27. Projections 381 28. Compact Operators 387 28.1 Continuity Properties of Compact Operators 397 28.2 Fredholm Alternative 401 28.3 Factoring Compact Operators 405 28.4Projecting the Bounded Operators onto the Compact Operators 416 29. Weakly Compact Operators 423 30. Absolutely Summing Operators 431 30.1 The Dvoretsky-Rogers Theorem 440 V. Spectral Theory 441 31. The Spectrum of an Operator 443 32. Subdivisions of the Spectrum and Examples 451 33. The Spectrum of a Compact Operator 457 33.1 Invariant Subspaces and Lomonosov's Theorem 461 34. Adjoints in filbert space 465 Contents xi 35. Symmetric, Hermitian and Normal Operators 469 36. The Spectral Theorem for Compact Symmetric Operators 485 36.1 Hilbert-Schmidt Operators 494 37. Symmetric Operators with Compact Resolvent 501 38. Orthogonal Projections and the Spectral Theorem for Compact Symmetric Operators 509 39. Sesquilinear Functionals 515 40. The Gelfand Map for Hermitian Operators 519 41. The Spectral Theorem for Hermitian Operators 523 42. Banach Algebras 533 43. Commutative Banach Algebras 545 44. Banach Algebras with Involutions 557 45. The Spectral Theorem for Normal Operators 567 Notation 573 Appendix: Hilbert Space 575 References 591 Index 597 AN INTRODUCTION TO FUNCTIONAL ANALYSIS Part I Topological Vector Spaces (TVS) 1 Definition and Basic Properties In this chapter we give the definition of a topological vector space and develop the basic properties of such spaces.
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