GRADUATE STUDIES IN MATHEMATICS 191 Functional Analysis Theo Bühler Dietmar A. Salamon 10.1090/gsm/191 Functional Analysis GRADUATE STUDIES IN MATHEMATICS 191 Functional Analysis Theo Bühler Dietmar A. Salamon EDITORIAL COMMITTEE Dan Abramovich Daniel S. Freed (Chair) Gigliola Staffilani Jeff A. Viaclovsky 2010 Mathematics Subject Classification. Primary 46-01, 47-01. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-191 Library of Congress Cataloging-in-Publication Data Names: B¨uhler, Theo, 1978– author. | Salamon, D. (Dietmar), author. Title: Functional analysis / Theo B¨uhler, Dietmar A. Salamon. Description: Providence, Rhode Island : American Mathematical Society, [2018] | Series: Gradu- ate studies in mathematics ; volume 191 | Includes bibliographical references and index. Identifiers: LCCN 2017057159 | ISBN 9781470441906 (alk. paper) Subjects: LCSH: Functional analysis. | Spectral theory (Mathematics) | Semigroups of operators. | AMS: Functional analysis – Instructional exposition (textbooks, tutorial papers, etc.). msc | Operator theory – Instructional exposition (textbooks, tutorial papers, etc.). msc Classification: LCC QA320 .B84 2018 | DDC 515/.7—dc23 LC record available at https://lccn.loc.gov/2017057159 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 select pages 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 permission to reuse portions of AMS publication content are handled by the Copyright Clearance Center. For more information, please visit www.ams.org/publications/pubpermissions. Send requests for translation rights and licensed reprints to [email protected]. c 2018 by the author. All rights reserved. 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 http://www.ams.org/ 10987654321 232221201918 Contents Preface ix Introduction xi Chapter 1. Foundations 1 §1.1. Metric Spaces and Compact Sets 2 §1.2. Finite-Dimensional Banach Spaces 17 §1.3. The Dual Space 25 §1.4. Hilbert Spaces 31 §1.5. Banach Algebras 35 §1.6. The Baire Category Theorem 40 §1.7. Problems 45 Chapter 2. Principles of Functional Analysis 49 §2.1. Uniform Boundedness 50 §2.2. Open Mappings and Closed Graphs 54 §2.3. Hahn–Banach and Convexity 65 §2.4. Reflexive Banach Spaces 80 §2.5. Problems 101 Chapter 3. The Weak and Weak* Topologies 109 §3.1. Topological Vector Spaces 110 §3.2. The Banach–Alaoglu Theorem 124 §3.3. The Banach–Dieudonn´e Theorem 130 §3.4. The Eberlein–Smulyanˇ Theorem 134 v vi Contents §3.5. The Kre˘ın–Milman Theorem 140 §3.6. Ergodic Theory 144 §3.7. Problems 153 Chapter 4. Fredholm Theory 163 §4.1. The Dual Operator 164 §4.2. Compact Operators 173 §4.3. Fredholm Operators 179 §4.4. Composition and Stability 184 §4.5. Problems 189 Chapter 5. Spectral Theory 197 §5.1. Complex Banach Spaces 198 §5.2. Spectrum 208 §5.3. Operators on Hilbert Spaces 222 §5.4. Functional Calculus for Self-Adjoint Operators 234 §5.5. Gelfand Spectrum and Normal Operators 246 §5.6. Spectral Measures 261 §5.7. Cyclic Vectors 281 §5.8. Problems 288 Chapter 6. Unbounded Operators 295 §6.1. Unbounded Operators on Banach Spaces 295 §6.2. The Dual of an Unbounded Operator 306 §6.3. Unbounded Operators on Hilbert Spaces 313 §6.4. Functional Calculus and Spectral Measures 326 §6.5. Problems 342 Chapter 7. Semigroups of Operators 349 §7.1. Strongly Continuous Semigroups 350 §7.2. The Hille–Yosida–Phillips Theorem 363 §7.3. The Dual Semigroup 377 §7.4. Analytic Semigroups 388 §7.5. Banach Space Valued Measurable Functions 404 §7.6. Inhomogeneous Equations 425 §7.7. Problems 439 Contents vii Appendix A. Zorn and Tychonoff 445 §A.1. The Lemma of Zorn 445 §A.2. Tychonoff’s Theorem 449 Bibliography 453 Notation 459 Index 461 Preface These are notes for the lecture course “Functional Analysis I” held by the second author at ETH Z¨urich in the fall semester 2015. Prerequisites are the first year courses on Analysis and Linear Algebra, and the second year courses on Complex Analysis, Topology, and Measure and Integration. The material of Section 1.4 on elementary Hilbert space theory, Sub- section 5.4.2 on the Stone–Weierstraß Theorem, and the appendix on the Lemma of Zorn and Tychonoff’s Theorem was not covered in the lectures. These topics were assumed to have been covered in previous lecture courses. They are included here for completeness of the exposition. The material of Subsection 2.4.4 on the James space, Section 5.5 on the functional calculus for bounded normal operators, and Chapter 6 on un- bounded operators was not part of the lecture course (with the exception of some of the basic definitions in Chapter 6 that are relevant for infinitesimal generators of strongly continuous semigroups). From Chapter 7 only the basic material on strongly continuous semigroups in Section 7.1, on their infinitesimal generators in Section 7.2, and on the dual semigroup in Sec- tion 7.3 were included in the lecture course. 23 October 2017 Theo B¨uhler Dietmar A. Salamon ix Introduction Classically, functional analysis is the study of function spaces and linear operators between them. The relevant function spaces are often equipped with the structure of a Banach space and many of the central results re- main valid in the more general setting of bounded linear operators between Banach spaces or normed vector spaces, where the specific properties of the concrete function space in question only play a minor role. Thus, in the modern guise, functional analysis is the study of Banach spaces and bounded linear operators between them, and this is the viewpoint taken in the present book. This area of mathematics has both an intrinsic beauty, which we hope to convey to the reader, and a vast number of applications in many fields of mathematics. These include the analysis of PDEs, differential topology and geometry, symplectic topology, quantum mechanics, probability theory, geo- metric group theory, dynamical systems, ergodic theory, and approximation theory, among many others. While we say little about specific applications, they do motivate the choice of topics covered in this book, and our goal is to give a self-contained exposition of the necessary background in abstract functional analysis for many of the relevant applications. The book is addressed primarily to third year students of mathematics or physics, and the reader is assumed to be familiar with first year analysis and linear algebra, as well as complex analysis and the basics of point set topology and measure and integration. For example, this book does not include a proof of completeness and duality for Lp spaces. There are naturally many topics that go beyond the scope of the present book, such as Sobolev spaces and PDEs, which would require a book on its own and, in fact, very many books have been written about this subject; here we just refer the interested reader to [19, 28, 30]. We also restrict the xi xii Introduction discussion to linear operators and say nothing about nonlinear functional analysis. Other topics not covered include the Fourier transform (see [2,48, 79]), maximal regularity for semigroups (see [76]), the space of Fredholm operators on an infinite-dimensional Hilbert space as a classifying space for K-theory (see [5–7,42]), Quillen’s determinant line bundle over the space of Fredholm operators (see [71,77]), and the work of Gowers [31] and Argyros– Haydon [4] on Banach spaces on which every bounded linear operator is the sum of a scalar multiple of the identity and a compact operator. Here is a description of the contents of the book, chapter by chapter. Chapter 1 discusses some basic concepts that play a central role in the subject. It begins with a section on metric spaces and compact sets which includes a proof of the Arzel`a–Ascoli Theorem. It then moves on to establish some basic properties of finite-dimensional normed vector spaces and shows, in particular, that a normed vector space is finite-dimensional if and only if the unit ball is compact. The first chapter also introduces the dual space of a normed vector space, explains several important examples, and contains an introduction to elementary Hilbert space theory. It then introduces Banach algebras and shows that the group of invertible elements is an open set. It closes with a proof of the Baire Category Theorem. Chapter 2 is devoted to the three fundamental principles of functional analysis.TheyaretheUniform Boundedness Principle (a pointwise bounded family of bounded linear operators on a Banach space is bounded), the Open Mapping Theorem (a surjective bounded linear operator between Banach spaces is open), and the Hahn–Banach Theorem (a bounded linear func- tional on a linear subspace of a normed vector space extends to a bounded linear functional on the entire normed vector space). An equivalent formu- lation of the Open Mapping Theorem is the Closed Graph Theorem (a linear operator between Banach spaces is bounded if and only if it has a closed graph) and a corollary is the Inverse Operator Theorem (a bijective bounded linear operator between Banach spaces has a bounded inverse). A slightly stronger version of the Hahn–Banach Theorem, with the norm replaced by a quasi-seminorm, can be reformulated as the geometric assertion that two convex subsets of a normed vector space can be separated by a hyperplane whenever one of them has nonempty interior.