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Functional Analysis Principles of Functional Analysis SECOND EDITION Martin Schechter Graduate Studies in Mathematics Volume 36 American Mathematical Society http://dx.doi.org/10.1090/gsm/036 Selected Titles in This Series 36 Martin Schechter, Principles of functional analysis, second edition, 2002 35 JamesF.DavisandPaulKirk, Lecture notes in algebraic topology, 2001 34 Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, 2001 33 Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, 2001 32 Robert G. Bartle, A modern theory of integration, 2001 31 Ralf Korn and Elke Korn, Option pricing and portfolio optimization: Modern methods of financial mathematics, 2001 30 J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, 2001 29 Javier Duoandikoetxea, Fourier analysis, 2001 28 Liviu I. Nicolaescu, Notes on Seiberg-Witten theory, 2000 27 Thierry Aubin, A course in differential geometry, 2001 26 Rolf Berndt, An introduction to symplectic geometry, 2001 25 Thomas Friedrich, Dirac operators in Riemannian geometry, 2000 24 Helmut Koch, Number theory: Algebraic numbers and functions, 2000 23 Alberto Candel and Lawrence Conlon, Foliations I, 2000 22 G¨unter R. Krause and Thomas H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension, 2000 21 John B. Conway, A course in operator theory, 2000 20 Robert E. Gompf and Andr´as I. Stipsicz, 4-manifolds and Kirby calculus, 1999 19 Lawrence C. Evans, Partial differential equations, 1998 18 Winfried Just and Martin Weese, Discovering modern set theory. II: Set-theoretic tools for every mathematician, 1997 17 Henryk Iwaniec, Topics in classical automorphic forms, 1997 16 Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Volume II: Advanced theory, 1997 15 Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Volume I: Elementary theory, 1997 14 Elliott H. Lieb and Michael Loss, Analysis, 1997 13 Paul C. Shields, The ergodic theory of discrete sample paths, 1996 12 N. V. Krylov, Lectures on elliptic and parabolic equations in H¨older spaces, 1996 11 Jacques Dixmier, Enveloping algebras, 1996 Printing 10 Barry Simon, Representations of finite and compact groups, 1996 9 Dino Lorenzini, An invitation to arithmetic geometry, 1996 8 Winfried Just and Martin Weese, Discovering modern set theory. I: The basics, 1996 7 Gerald J. Janusz, Algebraic number fields, second edition, 1996 6 Jens Carsten Jantzen, Lectures on quantum groups, 1996 5 Rick Miranda, Algebraic curves and Riemann surfaces, 1995 4 Russell A. Gordon, The integrals of Lebesgue, Denjoy, Perron, and Henstock, 1994 3 William W. Adams and Philippe Loustaunau, An introduction to Gr¨obner bases, 1994 2 Jack Graver, Brigitte Servatius, and Herman Servatius, Combinatorial rigidity, 1993 1 Ethan Akin, The general topology of dynamical systems, 1993 Principles of Functional Analysis Principles of Functional Analysis SECOND EDITION Martin Schechter Graduate Studies in Mathematics Volume 36 American Mathematical Society Providence, Rhode Island Editorial Board Steven G. Krantz David Saltman (Chair) David Sattinger Ronald Stern 2000 Mathematics Subject Classification. Primary 46–01, 47–01, 46B20, 46B25, 46C05, 47A05, 47A07, 47A12, 47A53, 47A55. Abstract. The book is intended for a one-year course for beginning graduate or senior under- graduate students. However, it can be used at any level where the students have the prerequisites mentioned below. Because of the crucial role played by functional analysis in the applied sciences as well as in mathematics, the author attempted to make this book accessible to as wide a spec- trum of beginning students as possible. Much of the book can be understood by a student having taken a course in advanced calculus. However, in several chapters an elementary knowledge of functions of a complex variable is required. These include Chapters 6, 9, and 11. Only rudimen- tary topological or algebraic concepts are used. They are introduced and proved as needed. No measure theory is employed or mentioned. Library of Congress Cataloging-in-Publication Data Schechter, Martin. Principles of functional analysis / Martin Schechter.—2nd ed. p. cm. — (Graduate studies in mathematics, ISSN 1065-7339 ; v. 36) Includes bibliographical references and index. ISBN 0-8218-2895-9 (alk. paper) 1. Functional analysis. I. Title. II. Series. QA320 .S32 2001 515.7—dc21 2001031601 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Assistant to the Publisher, American Mathematical Society, P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to [email protected]. c 2002 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at URL: http://www.ams.org/ 10 9 8 7 6 5 4 3 2 17 16 15 14 13 12 BSD To my wife, children, and grandchildren. May they enjoy many happy years. Contents PREFACE TO THE REVISED EDITION xv FROM THE PREFACE TO THE FIRST EDITION xix Chapter 1. BASIC NOTIONS 1 §1.1. A problem from differential equations 1 §1.2. An examination of the results 6 §1.3. Examples of Banach spaces 9 §1.4. Fourier series 17 §1.5. Problems 24 Chapter 2. DUALITY 29 §2.1. The Riesz representation theorem 29 §2.2. The Hahn-Banach theorem 33 §2.3. Consequences of the Hahn-Banach theorem 36 §2.4. Examples of dual spaces 39 §2.5. Problems 51 Chapter 3. LINEAR OPERATORS 55 §3.1. Basic properties 55 §3.2. The adjoint operator 57 §3.3. Annihilators 59 §3.4. The inverse operator 60 §3.5. Operators with closed ranges 66 §3.6. The uniform boundedness principle 71 ix x Contents §3.7. The open mapping theorem 71 §3.8. Problems 72 Chapter 4. THE RIESZ THEORY FOR COMPACT OPERATORS 77 §4.1. A type of integral equation 77 §4.2. Operators of finite rank 85 §4.3. Compact operators 88 §4.4. The adjoint of a compact operator 95 §4.5. Problems 98 Chapter 5. FREDHOLM OPERATORS 101 §5.1. Orientation 101 §5.2. Further properties 105 §5.3. Perturbation theory 109 §5.4. The adjoint operator 112 §5.5. A special case 114 §5.6. Semi-Fredholm operators 117 §5.7. Products of operators 123 §5.8. Problems 126 Chapter 6. SPECTRAL THEORY 129 §6.1. The spectrum and resolvent sets 129 §6.2. The spectral mapping theorem 133 §6.3. Operational calculus 134 §6.4. Spectral projections 141 §6.5. Complexification 147 §6.6. The complex Hahn-Banach theorem 148 §6.7. A geometric lemma 150 §6.8. Problems 151 Chapter 7. UNBOUNDED OPERATORS 155 §7.1. Unbounded Fredholm operators 155 §7.2. Further properties 161 §7.3. Operators with closed ranges 164 §7.4. Total subsets 169 §7.5. The essential spectrum 171 §7.6. Unbounded semi-Fredholm operators 173 §7.7. The adjoint of a product of operators 177 Contents xi §7.8. Problems 179 Chapter 8. REFLEXIVE BANACH SPACES 183 §8.1. Properties of reflexive spaces 183 §8.2. Saturated subspaces 185 §8.3. Separable spaces 188 §8.4. Weak convergence 190 §8.5. Examples 192 §8.6. Completing a normed vector space 196 §8.7. Problems 197 Chapter 9. BANACH ALGEBRAS 201 §9.1. Introduction 201 §9.2. An example 205 §9.3. Commutative algebras 206 §9.4. Properties of maximal ideals 209 §9.5. Partially ordered sets 211 §9.6. Riesz operators 213 §9.7. Fredholm perturbations 215 §9.8. Semi-Fredholm perturbations 216 §9.9. Remarks 222 §9.10. Problems 222 Chapter 10. SEMIGROUPS 225 §10.1. A differential equation 225 §10.2. Uniqueness 228 §10.3. Unbounded operators 229 §10.4. The infinitesimal generator 235 §10.5. An approximation theorem 238 §10.6. Problems 240 Chapter 11. HILBERT SPACE 243 §11.1. When is a Banach space a Hilbert space? 243 §11.2. Normal operators 246 §11.3. Approximation by operators of finite rank 252 §11.4. Integral operators 253 §11.5. Hyponormal operators 257 §11.6. Problems 262 xii Contents Chapter 12. BILINEAR FORMS 265 §12.1. The numerical range 265 §12.2. The associated operator 266 §12.3. Symmetric forms 268 §12.4. Closed forms 270 §12.5. Closed extensions 274 §12.6. Closable operators 278 §12.7. Some proofs 281 §12.8. Some representation theorems 284 §12.9. Dissipative operators 285 §12.10. The case of a line or a strip 290 §12.11. Selfadjoint extensions 294 §12.12. Problems 295 Chapter 13. SELFADJOINT OPERATORS 297 §13.1. Orthogonal projections 297 §13.2. Square roots of operators 299 §13.3. A decomposition of operators 304 §13.4. Spectral resolution 306 §13.5. Some consequences 311 §13.6. Unbounded selfadjoint operators 314 §13.7. Problems 322 Chapter 14. MEASURES OF OPERATORS 325 §14.1. A seminorm 325 §14.2. Perturbation classes 329 §14.3. Related measures 332 §14.4. Measures of noncompactness 339 §14.5. The quotient space 341 §14.6. Strictly singular operators 342 §14.7. Norm perturbations 345 §14.8. Perturbation functions 350 §14.9. Factored perturbation functions 354 §14.10. Problems 357 Chapter 15. EXAMPLES AND APPLICATIONS 359 §15.1. A few remarks 359 Contents xiii §15.2.
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