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Real Analysis a Comprehensive Course in Analysis, Part 1 Real Analysis A Comprehensive Course in Analysis, Part 1 Barry Simon Real Analysis A Comprehensive Course in Analysis, Part 1 http://dx.doi.org/10.1090/simon/001 Real Analysis A Comprehensive Course in Analysis, Part 1 Barry Simon Providence, Rhode Island 2010 Mathematics Subject Classification. Primary 26-01, 28-01, 42-01, 46-01; Secondary 33-01, 35-01, 41-01, 52-01, 54-01, 60-01. For additional information and updates on this book, visit www.ams.org/bookpages/simon Library of Congress Cataloging-in-Publication Data Simon, Barry, 1946– Real analysis / Barry Simon. pages cm. — (A comprehensive course in analysis ; part 1) Includes bibliographical references and indexes. ISBN 978-1-4704-1099-5 (alk. paper) 1. Mathematical analysis—Textbooks. I. Title. QA300.S53 2015 515.8—dc23 2014047381 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. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. For more information, please visit: http://www.ams.org/rightslink. Send requests for translation rights and licensed reprints to [email protected]. Excluded from these provisions is material for which the author holds copyright. In such cases, requests for permission to reuse or reprint material should be addressed directly to the author(s). Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article within proceedings volumes. c 2015 by the American Mathematical Society. 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 201918171615 To the memory of Cherie Galvez extraordinary secretary, talented helper, caring person and to the memory of my mentors, Ed Nelson (1932-2014) and Arthur Wightman (1922-2013) who not only taught me Mathematics but taught me how to be a mathematician Contents Preface to the Series xi Preface to Part 1 xvii Chapter 1. Preliminaries 1 §1.1. Notation and Terminology 1 §1.2. Metric Spaces 3 §1.3. The Real Numbers 6 §1.4. Orders 9 §1.5. The Axiom of Choice and Zorn’s Lemma 11 §1.6. Countability 14 §1.7. Some Linear Algebra 18 §1.8. Some Calculus 30 Chapter 2. Topological Spaces 35 §2.1. Lots of Definitions 37 §2.2. Countability and Separation Properties 51 §2.3. Compact Spaces 63 §2.4. The Weierstrass Approximation Theorem and Bernstein Polynomials 76 §2.5. The Stone–Weierstrass Theorem 88 §2.6. Nets 93 §2.7. Product Topologies and Tychonoff’s Theorem 99 §2.8. Quotient Topologies 103 vii viii Contents Chapter 3. A First Look at Hilbert Spaces and Fourier Series 107 §3.1. Basic Inequalities 109 §3.2. Convex Sets, Minima, and Orthogonal Complements 119 §3.3. Dual Spaces and the Riesz Representation Theorem 122 §3.4. Orthonormal Bases, Abstract Fourier Expansions, and Gram–Schmidt 131 §3.5. Classical Fourier Series 137 §3.6. The Weak Topology 168 §3.7. A First Look at Operators 174 §3.8. Direct Sums and Tensor Products of Hilbert Spaces 176 Chapter 4. Measure Theory 185 §4.1. Riemann–Stieltjes Integrals 187 §4.2. The Cantor Set, Function, and Measure 198 §4.3. Bad Sets and Good Sets 205 §4.4. Positive Functionals and Measures via L1(X) 212 §4.5. The Riesz–Markov Theorem 233 §4.6. Convergence Theorems; Lp Spaces 240 §4.7. Comparison of Measures 252 §4.8. Duality for Banach Lattices; Hahn and Jordan Decomposition 259 §4.9. Duality for Lp 270 §4.10. Measures on Locally Compact and σ-Compact Spaces 275 §4.11. Product Measures and Fubini’s Theorem 281 §4.12. Infinite Product Measures and Gaussian Processes 292 §4.13. General Measure Theory 300 §4.14. Measures on Polish Spaces 306 §4.15. Another Look at Functions of Bounded Variation 314 §4.16. Bonus Section: Brownian Motion 319 §4.17. Bonus Section: The Hausdorff Moment Problem 329 §4.18. Bonus Section: Integration of Banach Space-Valued Functions 337 §4.19. Bonus Section: Haar Measure on σ-Compact Groups 342 Contents ix Chapter 5. Convexity and Banach Spaces 355 §5.1. Some Preliminaries 357 §5.2. H¨older’s and Minkowski’s Inequalities: A Lightning Look 367 §5.3. Convex Functions and Inequalities 373 §5.4. The Baire Category Theorem and Applications 394 §5.5. The Hahn–Banach Theorem 414 §5.6. Bonus Section: The Hamburger Moment Problem 428 §5.7. Weak Topologies and Locally Convex Spaces 436 §5.8. The Banach–Alaoglu Theorem 446 §5.9. Bonus Section: Minimizers in Potential Theory 447 §5.10. Separating Hyperplane Theorems 454 §5.11. The Krein–Milman Theorem 458 §5.12. Bonus Section: Fixed Point Theorems and Applications 468 Chapter 6. Tempered Distributions and the Fourier Transform 493 §6.1. Countably Normed and Fr´echet Spaces 496 §6.2. Schwartz Space and Tempered Distributions 502 §6.3. Periodic Distributions 520 §6.4. Hermite Expansions 523 §6.5. The Fourier Transform and Its Basic Properties 540 §6.6. More Properties of Fourier Transform 548 §6.7. Bonus Section: Riesz Products 576 §6.8. Fourier Transforms of Powers and Uniqueness of Minimizers in Potential Theory 583 §6.9. Constant Coefficient Partial Differential Equations 588 Chapter 7. Bonus Chapter: Probability Basics 615 §7.1. The Language of Probability 617 §7.2. Borel–Cantelli Lemmas and the Laws of Large Numbers and of the Iterated Logarithm 632 §7.3. Characteristic Functions and the Central Limit Theorem 648 §7.4. Poisson Limits and Processes 660 §7.5. Markov Chains 667 x Contents Chapter 8. Bonus Chapter: Hausdorff Measure and Dimension 679 §8.1. The Carath´eodory Construction 680 §8.2. Hausdorff Measure and Dimension 687 Chapter 9. Bonus Chapter: Inductive Limits and Ordinary Distributions 705 §9.1. Strict Inductive Limits 706 §9.2. Ordinary Distributions and Other Examples of Strict Inductive Limits 711 Bibliography 713 Symbol Index 765 Subject Index 769 Author Index 779 Index of Capsule Biographies 789 Preface to the Series Young men should prove theorems, old men should write books. —Freeman Dyson, quoting G. H. Hardy1 Reed–Simon2 starts with “Mathematics has its roots in numerology, ge- ometry, and physics.” This puts into context the division of mathematics into algebra, geometry/topology, and analysis. There are, of course, other areas of mathematics, and a division between parts of mathematics can be artificial. But almost universally, we require our graduate students to take courses in these three areas. This five-volume series began and, to some extent, remains a set of texts for a basic graduate analysis course. In part it reflects Caltech’s three-terms- per-year schedule and the actual courses I’ve taught in the past. Much of the contents of Parts 1 and 2 (Part 2 is in two volumes, Part 2A and Part 2B) are common to virtually all such courses: point set topology, measure spaces, Hilbert and Banach spaces, distribution theory, and the Fourier transform, complex analysis including the Riemann mapping and Hadamard product theorems. Parts 3 and 4 are made up of material that you’ll find in some, but not all, courses—on the one hand, Part 3 on maximal functions and Hp-spaces; on the other hand, Part 4 on the spectral theorem for bounded self-adjoint operators on a Hilbert space and det and trace, again for Hilbert space operators. Parts 3 and 4 reflect the two halves of the third term of Caltech’s course. 1Interview with D. J. Albers, The College Mathematics Journal, 25, no. 1, January 1994. 2M. Reed and B. Simon, Methods of Modern Mathematical Physics, I: Functional Analysis, Academic Press, New York, 1972. xi xii Preface to the Series While there is, of course, overlap between these books and other texts, there are some places where we differ, at least from many: (a) By having a unified approach to both real and complex analysis, we are able to use notions like contour integrals as Stietljes integrals that cross the barrier. (b) We include some topics that are not standard, although I am sur- prised they are not. For example, while discussing maximal functions, I present Garcia’s proof of the maximal (and so, Birkhoff) ergodic the- orem. (c) These books are written to be keepers—the idea is that, for many stu- dents, this may be the last analysis course they take, so I’ve tried to write in a way that these books will be useful as a reference. For this reason, I’ve included “bonus” chapters and sections—material that I do not expect to be included in the course. This has several advantages. First, in a slightly longer course, the instructor has an option of extra topics to include. Second, there is some flexibility—for an instructor who can’t imagine a complex analysis course without a proof of the prime number theorem, it is possible to replace all or part of the (non- bonus) chapter on elliptic functions with the last four sections of the bonus chapter on analytic number theory. Third, it is certainly possible to take all the material in, say, Part 2, to turn it into a two-term course.
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