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 .
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/
10 9 8 7 6 5 4 3 2 1 20 19 18 17 16 15
, 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. Hölder’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 Fréchet 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
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 .
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/
10 9 8 7 6 5 4 3 2 1 20 19 18 17 16 15
, 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. Hölder’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 Fréchet 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
