Graduate Texts in Mathematics 280

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Graduate Texts in Mathematics bridge the gap between passive study andcreative understanding, offering graduate-level introductions to advanced topics inmathematics. The volumes are carefully written as teaching aids and highlightcharacteristic features of the theory. Although these books are frequently used astextbooks in graduate courses, they are also suitable for individual study.

More information about this series at http://www.springer.com/series/136

Graduate Texts in Mathematics

Christopher Heil

Introduction to Real Analysis

123

Christopher HeilSchool of MathematicsGeorgia Institute of TechnologyAtlanta, GA, USA

ISSN 0072-5285 ISSN 2197-5612 (electronic)Graduate Texts in MathematicsISBN 978-3-030-26901-2 ISBN 978-3-030-26903-6 (eBook)https://doi.org/10.1007/978-3-030-26903-6

Mathematics Subject Classification (2010): 28-XX, 26-XX, 42-XX, 46-XX, 47-XX, 54-XX

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Contents

Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1 Metric and Normed Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.1 Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.1.1 Convergence and Completeness . . . . . . . . . . . . . . . . . . . . 161.1.2 Topology in Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 171.1.3 Compact Sets in Metric Spaces . . . . . . . . . . . . . . . . . . . . . 181.1.4 Continuity for Functions on Metric Spaces . . . . . . . . . . . 20

1.2 Normed Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.2.1 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.2.2 Seminorms and Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.2.3 Infinite Series in Normed Spaces . . . . . . . . . . . . . . . . . . . . 241.2.4 Equivalent Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.3 The Uniform Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.3.1 Some Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.4 Holder and Lipschitz Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2 Lebesgue Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.1 Exterior Lebesgue Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.1.1 Boxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.1.2 Some Facts about Boxes . . . . . . . . . . . . . . . . . . . . . . . . . . 362.1.3 Exterior Lebesgue Measure . . . . . . . . . . . . . . . . . . . . . . . . 392.1.4 The Exterior Measure of a Box . . . . . . . . . . . . . . . . . . . . . 442.1.5 The Cantor Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.1.6 Regularity of Exterior Measure . . . . . . . . . . . . . . . . . . . . . 49

2.2 Lebesgue Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522.2.1 Definition and Basic Properties . . . . . . . . . . . . . . . . . . . . 532.2.2 Toward Countable Additivity and Closure under

Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552.2.3 Countable Additivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592.2.4 Equivalent Formulations of Measurability . . . . . . . . . . . . 62

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2.2.5 Caratheodory’s Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . 642.2.6 Almost Everywhere and the Essential Supremum . . . . . 66

2.3 More Properties of Lebesgue Measure . . . . . . . . . . . . . . . . . . . . . 712.3.1 Continuity from Above and Below . . . . . . . . . . . . . . . . . . 712.3.2 Cartesian Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742.3.3 Linear Changes of Variable . . . . . . . . . . . . . . . . . . . . . . . . 75

2.4 Nonmeasurable Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 812.4.1 The Axiom of Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 812.4.2 Existence of a Nonmeasurable Set . . . . . . . . . . . . . . . . . . 822.4.3 Further Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

3 Measurable Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873.1 Definition and Properties of Measurable Functions . . . . . . . . . . 87

3.1.1 Extended Real-Valued Functions . . . . . . . . . . . . . . . . . . . 883.1.2 Complex-Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . 92

3.2 Operations on Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 933.2.1 Sums and Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 933.2.2 Compositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 953.2.3 Suprema and Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 963.2.4 Simple Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

3.3 The Lebesgue Space L∞(E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1033.3.1 Convergence and Completeness in L∞(E) . . . . . . . . . . . 105

3.4 Egorov’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1073.5 Convergence in Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1113.6 Luzin’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

4 The Lebesgue Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1194.1 The Lebesgue Integral of Nonnegative Functions . . . . . . . . . . . . 119

4.1.1 Integration of Nonnegative Simple Functions . . . . . . . . . 1214.1.2 Integration of Nonnegative Functions . . . . . . . . . . . . . . . 123

4.2 The Monotone Convergence Theorem and Fatou’s Lemma . . . 1264.2.1 The Monotone Convergence Theorem . . . . . . . . . . . . . . . 1274.2.2 Fatou’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

4.3 The Lebesgue Integral of Measurable Functions . . . . . . . . . . . . 1324.3.1 Extended Real-Valued Functions . . . . . . . . . . . . . . . . . . . 1334.3.2 Complex-Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . 1354.3.3 Properties of the Integral . . . . . . . . . . . . . . . . . . . . . . . . . . 136

4.4 Integrable Functions and L1(E) . . . . . . . . . . . . . . . . . . . . . . . . . . 1384.4.1 The Lebesgue Space L1(E) . . . . . . . . . . . . . . . . . . . . . . . . 1384.4.2 Convergence in L1-Norm . . . . . . . . . . . . . . . . . . . . . . . . . . 1404.4.3 Linearity of the Integral for Integrable Functions . . . . . 1424.4.4 Inclusions between L1(E) and L∞(E) . . . . . . . . . . . . . . . 143

4.5 The Dominated Convergence Theorem . . . . . . . . . . . . . . . . . . . . 1464.5.1 The Dominated Convergence Theorem . . . . . . . . . . . . . . 1474.5.2 First Applications of the DCT . . . . . . . . . . . . . . . . . . . . . 149

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4.5.3 Approximation by Continuous Functions . . . . . . . . . . . . 1504.5.4 Approximation by Really Simple Functions . . . . . . . . . . 1534.5.5 Relation to the Riemann Integral . . . . . . . . . . . . . . . . . . . 154

4.6 Repeated Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1614.6.1 Fubini’s Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1614.6.2 Tonelli’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1684.6.3 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

5 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1775.1 The Cantor–Lebesgue Function . . . . . . . . . . . . . . . . . . . . . . . . . . 1785.2 Functions of Bounded Variation . . . . . . . . . . . . . . . . . . . . . . . . . . 182

5.2.1 Definition and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 1835.2.2 Lipschitz and Holder Continuous Functions . . . . . . . . . . 1865.2.3 Indefinite Integrals and Antiderivatives . . . . . . . . . . . . . . 1875.2.4 The Jordan Decomposition . . . . . . . . . . . . . . . . . . . . . . . . 189

5.3 Covering Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1945.3.1 The Simple Vitali Lemma . . . . . . . . . . . . . . . . . . . . . . . . . 1955.3.2 The Vitali Covering Lemma . . . . . . . . . . . . . . . . . . . . . . . 196

5.4 Differentiability of Monotone Functions . . . . . . . . . . . . . . . . . . . 2005.5 The Lebesgue Differentiation Theorem . . . . . . . . . . . . . . . . . . . . 207

5.5.1 L1-Convergence of Averages . . . . . . . . . . . . . . . . . . . . . . . 2085.5.2 Locally Integrable Functions . . . . . . . . . . . . . . . . . . . . . . . 2105.5.3 The Maximal Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2105.5.4 The Lebesgue Differentiation Theorem . . . . . . . . . . . . . . 2135.5.5 Lebesgue Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

6 Absolute Continuity and the Fundamental Theorem ofCalculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2196.1 Absolutely Continuous Functions . . . . . . . . . . . . . . . . . . . . . . . . . 220

6.1.1 Differentiability of Absolutely Continuous Functions . . 2226.2 Growth Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2236.3 The Banach–Zaretsky Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 2296.4 The Fundamental Theorem of Calculus . . . . . . . . . . . . . . . . . . . . 234

6.4.1 Applications of the FTC . . . . . . . . . . . . . . . . . . . . . . . . . . 2356.4.2 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

6.5 The Chain Rule and Changes of Variable . . . . . . . . . . . . . . . . . . 2416.6 Convex Functions and Jensen’s Inequality . . . . . . . . . . . . . . . . . 245

7 The Lp Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2537.1 The ℓp Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

7.1.1 Holder’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2567.1.2 Minkowski’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2597.1.3 Convergence in the ℓp Spaces . . . . . . . . . . . . . . . . . . . . . . 2627.1.4 Completeness of the ℓp Spaces . . . . . . . . . . . . . . . . . . . . . 2647.1.5 ℓp for p < 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

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7.1.6 c0 and c00 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2667.2 The Lebesgue Space Lp(E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

7.2.1 Seminorm Properties of ‖ · ‖p . . . . . . . . . . . . . . . . . . . . . . 2717.2.2 Identifying Functions That Are Equal Almost

Everywhere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2727.2.3 Lp(E) for 0 < p < 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2737.2.4 The Converse of Holder’s Inequality . . . . . . . . . . . . . . . . 274

7.3 Convergence in Lp-norm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2777.3.1 Dense Subsets of Lp(E) . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

7.4 Separability of Lp(E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

8 Hilbert Spaces and L2(E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2898.1 Inner Products and Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . 289

8.1.1 The Definition of an Inner Product . . . . . . . . . . . . . . . . . 2908.1.2 Properties of an Inner Product . . . . . . . . . . . . . . . . . . . . . 2908.1.3 Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

8.2 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2958.2.1 Orthogonal Complements . . . . . . . . . . . . . . . . . . . . . . . . . 2968.2.2 Orthogonal Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . 2988.2.3 Characterizations of the Orthogonal Projection. . . . . . . 3028.2.4 The Closed Span . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3028.2.5 The Complement of the Complement . . . . . . . . . . . . . . . 3038.2.6 Complete Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

8.3 Orthonormal Sequences and Orthonormal Bases . . . . . . . . . . . . 3058.3.1 Orthonormal Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 3058.3.2 Unconditional Convergence . . . . . . . . . . . . . . . . . . . . . . . . 3078.3.3 Orthogonal Projections Revisited . . . . . . . . . . . . . . . . . . . 3088.3.4 Orthonormal Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3108.3.5 Existence of an Orthonormal Basis . . . . . . . . . . . . . . . . . 3128.3.6 The Legendre Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 3138.3.7 The Haar System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3148.3.8 Unitary Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316

8.4 The Trigonometric System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320

9 Convolution and the Fourier Transform . . . . . . . . . . . . . . . . . . . 3279.1 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

9.1.1 The Definition of Convolution . . . . . . . . . . . . . . . . . . . . . . 3289.1.2 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3299.1.3 Convolution as Averaging . . . . . . . . . . . . . . . . . . . . . . . . . 3319.1.4 Approximate Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3339.1.5 Young’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338

9.2 The Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3449.2.1 The Inversion Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3489.2.2 Smoothness and Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

9.3 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360

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9.3.1 Periodic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3619.3.2 Decay of Fourier Coefficients . . . . . . . . . . . . . . . . . . . . . . . 3629.3.3 Convolution of Periodic Functions . . . . . . . . . . . . . . . . . . 3659.3.4 Approximate Identities and the Inversion Formula . . . . 3659.3.5 Completeness of the Trigonometric System . . . . . . . . . . 3719.3.6 Convergence of Fourier Series for p 6= 2 . . . . . . . . . . . . . . 374

9.4 The Fourier Transform on L2(R) . . . . . . . . . . . . . . . . . . . . . . . . . 378

Hints for Selected Exercises and Problems . . . . . . . . . . . . . . . . . . . . 387

Index of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395

Preface

This text grew out of lecture notes that I developed over the years for the“Real Analysis” graduate sequence here at Georgia Tech. This two-semestersequence is taken by first-year mathematics graduate students, well-preparedundergraduate mathematics majors, and graduate students from a wide va-riety of engineering and scientific disciplines. Covered in this book are thetopics that are taught in the first semester: Lebesgue measure, the Lebesgueintegral, differentiation and absolute continuity, the Lebesgue spaces Lp(E),and Hilbert spaces and L2(E). This material not only forms the basis of a coresubject in pure mathematics, but also has wide applicability in science andengineering. A text covering the second semester topics in analysis, includingabstract measure theory, signed and complex measures, operator theory, andfunctional analysis, is in development.

This text is an introduction to real analysis. There are several classic anal-ysis texts that I keep close by on my bookshelf and refer to often. However, Ifind it difficult to use any of these as the textbook for teaching a first courseon analysis. They tend to be dense and, in the classic style of mathematicalelegance and conciseness, they develop the theory in the most general setting,with few examples and limited motivation. These texts are valuable resources,but I suggest that they should be the second set of books on analysis thatyou pick up.

I hope that this text will be the analysis text that you read first. The def-initions, theorems, and other results are motivated and explained; the whyand not just the what of the subject is discussed. Proofs are completely rigor-ous, yet difficult arguments are motivated and discussed. Extensive exercisesand problems complement the presentation in the text, and provide manyopportunities for enhancing the student’s understanding of the material.

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xiv Preface

Audience

This text is aimed at students who have taken a standard (proof-based)undergraduate mathematics course on the basics of analysis. A brief reviewof the needed background material is presented in the Preliminaries sectionof the text. This includes:

• sequences, series, limits, suprema and infima, and limsups and liminfs,• functions,• cardinality,• basic topology of Euclidean space (open, closed, and compact sets),• continuity and differentiability of real-valued functions,• the Riemann integral.

Online Resources

A variety of resources are available on the author’s website,

http://people.math.gatech.edu/∼heil/

These include the following.

• A Chapter 0, which contains a greatly expanded version of the mate-rial that appears in the Preliminaries section of this text, along withdiscussions and exercises.

• An Alternative Chapter 1, which is an expanded version of the materialpresented in Chapter 1, including detailed discussion, motivation, andexercises, focused on the setting of normed spaces.

• A Chapter 10, which provides an introduction to abstract measure the-ory.

• An Instructor’s Guide, with a detailed course outline, commentary, re-marks, and extra problems. The exposition and problems in this guidemay be useful for students and readers as well as instructors.

• Selected Solutions for Students, containing approximately one workedsolution of a problem or exercise from each section of the text.

• An Errata List that will be updated as I become aware of typographicalor other errors in the text.

Additionally, a Solutions Manual is available to instructors upon re-quest; instructions for obtaining a copy are given on the Birkhauser websitefor this text.

Preface xv

Outline

Chapter 1 presents a short review of metric and normed spaces. Studentswho have completed an undergraduate analysis course have likely encounteredmuch of this material, although possibly only in the context of the Euclideanspace Rd (or Cd) instead of abstract metric spaces. The instructor has theoption of beginning the course here or proceeding directly to Chapter 2. Theonline Alternative Chapter 1 presents a significantly expanded version ofthis chapter focused on normed spaces. (A detailed introduction to the moregeneral setting of metric spaces is available in the first chapters of the author’stext Metrics, Norms, Inner Products, and Operator Theory [Heil18].)

In Chapter 2 we begin the study of Lebesgue measure. The fundamentalquestion that motivates this chapter is: Can we assign a “volume” or “mea-sure” to every subset of Rd in such a way that all of the properties thatwe expect of a “volume” function are satisfied? For example, we want themeasure of a cube or a ball in Rd to coincide with the standard definition ofthe volume of a cube or ball, and if we translate an object rigidly in spacethen we want its measure to always remain the same. If we break an objectinto countably many disjoint pieces, then we want the measure of the originalobject to be the sum of the measures of the pieces. Surprisingly (at least tome!), this simply can’t be done (more precisely, the Axiom of Choice impliesthat it is impossible). However, if we relax this goal somewhat then we findthat we can define a measure that obeys the correct rules for a “large” classof sets (the Lebesgue measurable sets). Chapter 2 constructs and studiesthis measure, which we call the Lebesgue measure of subsets of Rd.

In Chapters 3 and 4 we define the integral of real-valued and complex-valued functions whose domain is a measurable subset of Rd. Unfortunately,we cannot define the Lebesgue integral of every function. Chapter 3 in-troduces the class of measurable functions and deals with issues related toconvergence of sequences of measurable functions, while Chapter 4 definesand studies the Lebesgue integral of a measurable function. The Lebesgueintegral extends the Riemann integral, but is far more general. We can de-fine the Lebesgue integral for functions whose domain is any measurable set.We prove powerful results that allow us, in a large family of cases, to makeconclusions about the convergence of a sequence of Lebesgue integrals, orto interchange the order of iterated integrals of functions of more than onevariable.

The Fundamental Theorem of Calculus (FTC) is, as its name suggests,central to analysis. Chapters 5 and 6 explore issues related to differen-tiation and the FTC in detail. We see that there are surprising examplesof nonconstant functions whose derivatives are zero at “almost every” point(and therefore fail the FTC). In our quest to fully understand the FTC we de-fine functions of bounded variation and study averaging operations in Chap-ter 5. Then in Chapter 6 we introduce the class of absolutely continuousfunctions, which turn out to be the functions for which the FTC holds. The

xvi Preface

Banach–Zaretsky Theorem plays a prominent role in Chapter 6, and it iscentral to our understanding of absolute continuity and its impact.

In Chapter 7 our focus turns from individual functions to spaces of func-tions. The Lebesgue spaces Lp(E) group functions by integrability proper-ties, giving us a family of spaces indexed by an extended real number p with0 < p ≤ ∞. For p ≥ 1 these are normed vector spaces of functions, whilefor 0 < p < 1 they are metric spaces whose metric is not induced from anorm. The case p = 2 is especially important, because we can define an innerproduct on L2(E), which makes it a Hilbert space. This topic is explored inChapter 8. In a metric space, all that we can do is define the distance be-tween points in the space. In a normed space we can additionally define thelength of each vector in the space. But in a Hilbert space, we furthermore havea notion of angles between vectors and hence can define orthogonality. Thisleads to many powerful results, including the existence of an orthonormalbasis for every separable Hilbert space. Even though a Hilbert space can beinfinite-dimensional, in many respects our intuitions from Euclidean spacehold when we deal with a Hilbert space.

Chapter 9 contains “extra” material that is usually not covered in ourreal analysis sequence here at Georgia Tech, but which has many striking ap-plications of the techniques developed in the earlier chapters. First we definethe operation of convolution. Then we introduce and study the Fourier trans-form and Fourier series. These results form the core of the field of harmonicanalysis, which has wide applicability throughout mathematics, physics, andengineering. Convolution is a generalization of the averaging operations thatwere used in Chapters 5 and 6 to characterize the class of functions forwhich the Fundamental Theorem of Calculus holds. The Fourier transformand Fourier series allow us to both construct and deconstruct a wide classof functions, signals, or operators in terms of much simpler building blocksbased on complex exponentials (or sines and cosines in the real case). Al-though Chapter 9 presents only a taste of the theorems of harmonic anal-ysis (which deserves another course, and a future text, to do it justice), wedo get to see many applications of all of the tools that we derived in earlierchapters, including convergence of sequences of integrals (via the DominatedConvergence Theorem), interchange of iterated integrals (via Fubini’s Theo-rem), and the Fundamental Theorem of Calculus (via the Banach–ZaretskyTheorem).

Many exercises and problems appear in each section of the text. The Ex-ercises are directly incorporated into the development of the theory in eachsection, while the additional Problems given at the end of each section providefurther practice and opportunities to develop understanding.

Preface xvii

Course Options

There are many options for building a course around this text. The coursethat I teach at Georgia Tech is fast-paced, but covers most of the text in onesemester. Here is a brief outline of such a one-semester course; a more detailedoutline with much additional information (and extra problems) is containedin the Instructor’s Guide that is available on the author’s website.

Chapter 1: Assign for student reading, not covered in lecture.Chapter 2: Sections 2.1–2.4.Chapter 3: Sections 3.1–3.5. Omit Section 3.6.Chapter 4: Sections 4.1–4.6.Chapter 5: Sections 5.1–5.2, and selected portions of Sections 5.3–5.5.Chapter 6: Sections 6.1–6.4. Omit Sections 6.5-6.6.Chapter 7: Sections 7.1–7.4.Chapter 8: Sections 8.1–8.4 (as time allows).Chapter 9: Bonus material, not covered in lecture.

Another option is to begin the course with Chapter 1 (or the online Al-ternative Chapter 1). A fast-paced course could cover most of Chapters1–8. A moderately paced course could cover the first half of the text in detailin one semester, while a moderately paced two-semester course could coverall of Chapters 1–9 in considerable detail.

Acknowledgments

Every text builds on those that have come before it, and this one is noexception. Many classic and recent volumes have influenced the writing, thechoice of topics, the proofs, and the selection of problems. Among those thathave had the most profound influence on my writing are Benedetto and Czaja[BC09], Bruckner, Bruckner, and Thomson [BBT97], Folland [Fol99], Rudin[Rud87], Stein and Shakarchi [SS05], and Wheeden and Zygmund [WZ77].I greatly appreciate all of these texts and encourage the reader to consultthem. Additional texts and papers are listed in the references.

Various versions of the material in this volume have been used over theyears in the real analysis courses that were taught at Georgia Tech, and Ithank all of the many students and colleagues who have provided feedback.Special thanks are due to Shahaf Nitzan, who taught the course out of earlierversions of the text and provided invaluable feedback.

Christopher HeilAtlanta, Georgia

April 27, 2019

Preliminaries

We use the symbol ⊓⊔ to denote the end of a proof, and the symbol ♦ todenote the end of a definition, remark, example, or exercise. We also use ♦to indicate the end of the statement of a theorem whose proof will be omitted.A few problems are marked with an asterisk *; this indicates that they maybe more challenging. A detailed index of symbols employed in the text canbe found at the end of the volume.

Numbers

The set of natural numbers is denoted by N = {1, 2, 3, . . . }. The set of integersis Z = {. . . ,−1, 0, 1, . . . }, Q denotes the set of rational numbers, R is the setof real numbers, and C is the set of complex numbers. We often refer to R asthe real line, and to C as the complex plane.

Complex Numbers. The real part of a complex number z = a+ib (wherea, b ∈ R) is Re(z) = a, and its imaginary part is Im(z) = b. We say that zis rational if both its real and imaginary parts are rational numbers. Thecomplex conjugate of z is z = a − ib. The modulus, or absolute value, of z is

|z| =√

zz =√

a2 + b2.

If z 6= 0 then its polar form is z = reiθ where r = |z| > 0 and θ ∈ [0, 2π). Inthis case the argument of z is arg(z) = θ. Given any z ∈ C, there is a complexnumber α such that |α| = 1 and αz = |z|. If z 6= 0 then α is uniquely givenby α = e−iθ = z/|z|, while if z = 0 then α can be any complex number thathas unit modulus.

Extended Real Numbers. The set of extended real numbers [−∞,∞] is

[−∞,∞] = R ∪ {−∞,∞}.

1

2 Preliminaries

We extend many of the normal arithmetic and order notations and oper-ations to [−∞,∞]. For example, if a ∈ [−∞,∞] then a is a real number ifand only if −∞ < a < ∞. If −∞ < a ≤ ∞ then we set a+∞ = ∞. However,∞ − ∞ and −∞ + ∞ are undefined, and are referred to as indeterminateforms. If 0 < a ≤ ∞, then we define

a · ∞ = ∞, (−a) · ∞ = −∞, a · (−∞) = −∞, (−a) · (−∞) = ∞.

We also adopt the following conventions:

0 · (±∞) = 0 and1

±∞ = 0.

The Dual Index. Let p be an extended real number in the range1 ≤ p ≤ ∞. The dual index to p is the unique extended real number p′ thatsatisfies

1

p+

1

p′= 1.

We have 1 ≤ p′ ≤ ∞, and (p′)′ = p. If 1 < p < ∞, then we can write p′

explicitly as

p′ =p

p − 1.

Some examples are 1′ = ∞,(

32

)′= 3, 2′ = 2, 3′ = 3

2 , and ∞′ = 1.

The Notation F. In order to deal simultaneously with the complex planeand the extended real line, we let the symbol F denote a choice of either[−∞,∞] or C. Associated with this choice, we declare that:

• if F = [−∞,∞], then the word scalar means a finite real number c ∈ R;

• if F = C, then the word scalar means a complex number c ∈ C.

Note that a scalar cannot be ±∞; instead, a scalar is always a real or complexnumber.

Sets

The notation x ∈ X means that x is an element of the set X. We often referto an element of X as a point in X.

We write A ⊆ B to denote that A is a subset of a set B. If A ⊆ B andA 6= B then we say that A is a proper subset of B, and we write A ( B.

The empty set is denoted by ∅.A collection of sets {Xi}i∈I is disjoint if Xi∩Xj = ∅ whenever i 6= j. The

collection {Xi}i∈I is a partition of X if it is disjoint andS

i∈IXi = X.If X is a set, then the complement of S ⊆ X is X \S =

{x ∈ X : x /∈ S

}.

We sometimes abbreviate X \S as SC if the set X is understood. If A and B

Preliminaries 3

are subsets of X, then the relative complement of A in B is

B\A = B ∩ AC = {x ∈ B : x /∈ A}.

The power set of X is P(X) ={S : S ⊆ X

}, the set of all subsets of X.

The Cartesian product of sets X and Y is X×Y = {(x, y) : x ∈ X, y ∈ Y },the set of all ordered pairs of elements of X and Y. The Cartesian productof finitely many sets X1, . . . ,XN is

N∏

j=1

Xj = X1 × · · · × XN ={(x1, . . . , xN ) : xk ∈ Xk, k = 1, . . . , N

}.

Equivalence Relations

Informally, we say that ∼ is a relation on a set X if for each choice of x andy in X we have only one of the following two possibilities:

x ∼ y (x is related to y) or x 6∼ y (x is not related to y).

An equivalence relation on a set X is a relation ∼ that satisfies the followingconditions for all x, y, z ∈ X.

• Reflexivity: x ∼ x.

• Symmetry: If x ∼ y then y ∼ x.

• Transitivity: If x ∼ y and y ∼ z then x ∼ z.

For example, if we declare that x ∼ y if and only if x − y is rational, then ∼is an equivalence relation on R.

If ∼ is an equivalence relation on X, then the equivalence class of x ∈ Xis the set [x] that contains all elements that are related to x:

[x] = {y ∈ X : x ∼ y}.

Any two equivalence classes are either identical or disjoint. That is, if x and yare two elements of X, then either [x] = [y] or [x] ∩ [y] = ∅. The union ofall equivalence classes [x] is X. Consequently, the set of distinct equivalenceclasses forms a partition of X.

Intervals

An interval in the real line R is any one of the following sets:

• (a, b), [a, b), (a, b], [a, b] where a, b ∈ R and a < b, or

4 Preliminaries

• (a,∞), [a,∞), (−∞, a), (−∞, a] where a ∈ R, or

• R = (−∞,∞).

An open interval is an interval of the form (a, b), (a,∞), (−∞, a), or(−∞,∞). A closed interval is an interval of the form [a, b], [a,∞), (−∞, a],or (−∞,∞). We refer to [a, b] as a finite closed interval, a bounded closedinterval, or a compact interval.

The empty set ∅ and a singleton {a} are not intervals, but even so weadopt the notational conventions

[a, a] = {a} and (a, a) = [a, a) = (a, a] = ∅.

We also consider extended intervals, which are any of the following sets:

• (a,∞] = (a,∞) ∪ {∞} or [a,∞] = [a,∞) ∪ {∞}, where a ∈ R,

• [−∞, b) = (−∞, b) ∪ {−∞} or [−∞, b] = (−∞, b] ∪ {−∞}, where b ∈ R,or

• [−∞,∞] = R ∪ {−∞} ∪ {∞}.An extended interval is not an interval—whenever we refer to an “interval”without qualification we implicitly exclude the extended intervals.

Euclidean Space

We let Rd denote d-dimensional real Euclidean space, the set of all orderedd-tuples of real numbers. Similarly, Cd is d-dimensional complex Euclideanspace, the set of all ordered d-tuples of complex numbers.

The zero vector is 0 = (0, . . . , 0). We use the same symbol “0” to denotethe zero vector and the number zero; the intended meaning should be clearfrom context.

The dot product of vectors x = (x1, . . . , xd) and y = (y1, . . . , yd) in Rd orCd is

x · y = x1y1 + · · · + xdyd,

and the Euclidean norm of x is

‖x‖ = (x · x)1/2 =(|x1|2 + · · · + |xd|2

)1/2.

The translation of a set E ⊆ Rd by a vector h ∈ Rd (or a set E ⊆ Cd bya vector h ∈ Cd) is E + h = {x + h : x ∈ E}.

Preliminaries 5

Sequences

Let I be a fixed set. Given a set X and points xi ∈ X for i ∈ I, we write{xi}i∈I to denote the sequence of elements xi indexed by the set I. We call Ian index set in this context, and refer to xi as the ith component of the se-quence {xi}i∈I . If we know that the xi are scalars (real or complex numbers),then we often write (xi)i∈I instead of {xi}i∈I . Technically, a sequence {xi}i∈I

is shorthand for the mapping x : I → X given by x(i) = xi for i ∈ I, andtherefore the components xi of a sequence need not be distinct. If the indexset I is understood then we may write {xi} or {xi}i, or if the xi are scalarsthen we may write (xi) or (xi)i.

Often the index set I is countable. If I = {1, . . . , d} then we sometimeswrite a sequence in list form as

{xn}dn=1 = {x1, . . . , xd},

or if the xn are scalars then we often write

(xn)dn=1 = (x1, . . . , xd).

Similarly, if I = N then we may write

{xn}n∈N = {x1, x2, . . . },

or if each xn is a scalar then we usually write

(xn)n∈N = (x1, x2, . . . ).

A subsequence of a countable sequence {xn}n∈N = {x1, x2, . . . } is a se-quence of the form {xnk

}k∈N = {xn1, xn2

, . . . } where n1 < n2 < · · · .We say that a countable sequence of real numbers (xn)n∈N is monotone

increasing if xn ≤ xn+1 for every n, and strictly increasing if xn < xn+1

for every n. We define monotone decreasing and strictly decreasing sequencessimilarly.

The Kronecker Delta and the Standard Basis Vectors

Given indices i and j in an index set I (typically I = N), the Kronecker deltaof i and j is the number δij defined by the rule

δij =

{1, if i = j,

0, if i 6= j.

For each integer n ∈ N, we let δn denote the sequence

6 Preliminaries

δn = (δnk)k∈N = (0, . . . , 0, 1, 0, 0, . . . ).

That is, the nth component of the sequence δn is 1, while all other componentsare zero. We call δn the nth standard basis vector, and we refer to the family{δn}n∈N as the sequence of standard basis vectors, or simply the standardbasis.

Functions

Let X and Y be sets. We write f : X → Y to mean that f is a functionwith domain X and codomain Y. We usually write f(x) to denote the imageof x under f, but if L : X → Y is a linear map from one vector space X toanother vector space Y then we may write Lx instead of L(x). We also usethe following notation.

• The direct image of a set A ⊆ X under f is f(A) = {f(x) : x ∈ A}.• The inverse image of a set B ⊆ Y under f is

f−1(B) = {x ∈ X : f(x) ∈ B}.

• The range of f is range(f) = f(X) = {f(x) : x ∈ X}.• f is injective, or one-to-one, if f(x) = f(y) implies x = y.

• f is surjective, or onto, if range(f) = Y.

• f is bijective if it is both injective and surjective. The inverse function ofa bijection f : X → Y is the function f−1 : Y → X defined by f−1(y) = xif f(x) = y.

• Given S ⊆ X, the restriction of a function f : X → Y to the domain S isthe function f |S : S → Y defined by (f |S)(x) = f(x) for x ∈ S.

• The zero function on X is the function 0: X → R defined by 0(x) = 0 forevery x ∈ X. We use the same symbol 0 to denote the zero function andthe number zero.

• The characteristic function of A ⊆ X is the function χA : X → R given by

χA(x) =

{1, if x ∈ A,

0, if x /∈ A.

• If the domain of a function f is Rd, then the translation of f by a vectora ∈ Rd is the function Taf defined by Taf(x) = f(x − a) for x ∈ Rd.

Preliminaries 7

Cardinality

A set X is finite if either X is empty or there exists a positive integer n and abijection f : {1, . . . , n} → X. In the latter case we say that X has n elements.

A set X is denumerable or countably infinite if there exists a bijectionf : N → X.

A set X is countable if it is either finite or denumerable. In particular, N,Z, and Q are all denumerable and hence are countable.

A set X is uncountable if it is not countable. In particular, R and C areuncountable.

Extended Real-Valued Functions

A function that maps a set X into the real line R is called a real-valuedfunction, and a function that maps X into the extended real line [−∞,∞]is an extended real-valued function. Every real-valued function is extendedreal-valued, but an extended real-valued function need not be real-valued.An extended real-valued function f is nonnegative if f(x) ≥ 0 for every x.

Let f : X → [−∞,∞] be an extended real-valued function. We associateto f the two extended real-valued functions f+ and f− defined by

f+(x) = max{f(x), 0} and f−(x) = max{−f(x), 0}.

We call f+ the positive part and f− the negative part of f. They are eachnonnegative extended real-valued functions, and for every x we have

f(x) = f+(x) − f−(x) and |f(x)| = f+(x) + f−(x).

Given f : X → [−∞,∞], to avoid multiplicities of parentheses, brackets,and braces, we often write f−1(a, b) = f−1((a, b)), f−1[a,∞) = f−1([a,∞)),and so forth. We also use shorthands such as

{f ≥ a} = {x ∈ X : f(x) ≥ a},{f = a} = {x ∈ X : f(x) = a},

{a < f < b} = {x ∈ X : a < f(x) < b},{f ≥ g} = {x ∈ X : f(x) ≥ g(x)},

and so forth.If f : S → [−∞,∞] is an extended real-valued function on a domain S ⊆ R,

then f is monotone increasing on S if for all x, y ∈ S we have

x ≤ y =⇒ f(x) ≤ f(y).

We say that f is strictly increasing on S if for all x, y ∈ S,

8 Preliminaries

x < y =⇒ f(x) < f(y).

Monotone decreasing and strictly decreasing functions are defined similarly.

Notation for Extended Real-Valued and

Complex-Valued Functions

A function of the form f : X → C is said to be complex-valued. We have theinclusions R ⊆ [−∞,∞] and R ⊆ C, so every real-valued function is bothan extended real-valued and a complex-valued function. However, neither[−∞,∞] nor C is a subset of the other, so an extended real-valued functionneed not be a complex-valued function, and a complex-valued function neednot be an extended real-valued function. Hence there are usually two separatecases that we need to consider:

• extended real-valued functions of the form f : X → [−∞,∞], and

• complex-valued functions of the form f : X → C.

To consider both cases together, we use the notation F introduced earlier,which stands for a choice of either the extended real line [−∞,∞] or thecomplex plane C. Thus, if we write f : X → F then we mean that f couldeither be an extended real-valued function or a complex-valued function onthe domain X. Both possibilities include real-valued functions as a specialcase. As we declared earlier that, the word scalar means a finite real number(if F = [−∞,∞]) or a complex number (if F = C). Thus, a scalar-valuedfunction cannot take the values ±∞.

Suprema and Infima

A set of real numbers S is bounded above if there exists a real number Msuch that x ≤ M for every x ∈ S. Any such number M is called an upperbound for S. The definition of bounded below is similar, and we say that S isbounded if it is bounded both above and below.

A number x ∈ R is the supremum, or least upper bound, of S if

• x is an upper bound for S, and

• if y is any upper bound for S, then x ≤ y.

We denote the supremum of S, if one exists, by x = sup(S). The infimum, orgreatest lower bound, of S is defined in an entirely analogous manner, and isdenoted by inf(S).

It is not obvious that every set that is bounded above has a supremum.We take the existence of suprema as the following axiom.

Preliminaries 9

Axiom (Supremum Property of R). Let S be a nonempty subset of R.If S is bounded above, then there exists a real number x = sup(S) that isthe supremum of S. ♦

We extend the definition of supremum to sets that are not bounded aboveby declaring that sup(S) = ∞ if S is not bounded above. We also declare thatsup(∅) = −∞. Using these conventions, every set S ⊆ R has a supremum inthe extended real sense.

If S = (xn)n∈N is countable, then we often write supn xn or supxn todenote the supremum instead of sup(S), and similarly we may write infn xn

or inf xn instead of inf(S).If (xn)n∈N and (yn)n∈N are two sequences of real numbers, then

infn

xn + infn

yn ≤ infn

(xn + yn) ≤ supn

(xn + yn) ≤ supn

xn + supn

yn.

Any or all of the inequalities on the preceding line can be strict. If c > 0 then

supn

cxn = c supn

xn and supn

(−cxn) = −c infn

xn.

Convergent and Cauchy Sequences of Scalars

Convergence of sequences will be discussed in the more general setting ofmetric spaces in Section 1.1.1. Here we will only consider sequences (xn)n∈N ofreal or complex numbers. We say that a sequence of scalars (xn)n∈N convergesif there exists a scalar x such that for every ε > 0 there is an N > 0 suchthat

n ≥ N =⇒ |x − xn| < ε.

In this case we say that xn converges to x as n → ∞, and we write

xn → x or limn→∞

xn = x or lim xn = x.

We say that (xn)n∈N is a Cauchy sequence if for every ε > 0 there existsan integer N > 0 such that

m, n ≥ N =⇒ |xm − xn| < ε. ♦

An important consequence of the Supremum Property is that the followingequivalence holds for any sequence of scalars:

(xn)n∈N is convergent ⇐⇒ (xn)n∈N is Cauchy.

10 Preliminaries

Convergence in the Extended Real Sense

Let (xn)n∈N be a sequence of real numbers. We say that the sequence (xn)n∈N

diverges to ∞ as n → ∞ if for each real number R > 0 there is an integerN > 0 such that xn > R for all n ≥ N. In this case we write

limn→∞

xn = ∞.

We define divergence to −∞ similarly.We say that limn→∞ xn exists or that (xn)n∈N converges in the extended

real sense if

• xn converges to a real number x as n → ∞, or

• xn diverges to ∞ as n → ∞, or

• xn diverges to −∞ as n → ∞.

For example, every monotone increasing sequence of real numbers (xn)n∈N

converges in the extended real sense, and in this case lim xn = supxn. Sim-ilarly, a monotone decreasing sequence of real numbers converges in the ex-tended real sense and its limit equals its infimum.

Limsup and Liminf

The limit superior, or limsup, of a sequence of real numbers (xn)n∈N is

lim supn→∞

xn = infn∈N

supm≥n

xm = limn→∞

supm≥n

xm.

Likewise, the limit inferior, or liminf, of (xn)n∈N is

lim infn→∞

xn = supn∈N

infm≥n

xm = limn→∞

infm≥n

xm.

The liminf and limsup of every sequence of real numbers exists in the extendedreal sense. Further,

(xn)n∈N converges in

the extended real sense⇐⇒ lim inf

n→∞xn = lim sup

n→∞xn,

and in this case lim xn = lim infxn = lim supxn.If (xn)n∈N and (yn)n∈N are two sequences of real numbers, then

lim infn→∞

xn + lim infn→∞

yn ≤ lim infn→∞

(xn + yn)

≤ lim supn→∞

xn + lim infn→∞

yn

Preliminaries 11

≤ lim supn→∞

(xn + yn)

≤ lim supn→∞

xn + lim supn→∞

yn,

as long as none of the sums above takes an indeterminate form ∞ − ∞or −∞ + ∞. Strict inequality can hold on any line above. If the sequence(xn)n∈N converges, then

lim infn→∞

(xn + yn) = limn→∞

xn + lim infn→∞

yn,

and likewise

lim supn→∞

(xn + yn) = limn→∞

xn + lim supn→∞

yn.

If (xn)n∈N is a sequence of real numbers, then there exist subsequences(xnk

)k∈N and (xmj)j∈N such that

limk→∞

xnk= lim sup

n→∞xn and lim

j→∞xmj

= lim infn→∞

xn.

In fact, if (xn)n∈N is bounded above then lim supxn is the largest possiblelimit of a subsequence (xnk

)k∈N, and likewise if (xn)n∈N is bounded belowthen lim inf xn is the smallest possible limit of a subsequence. Consequently,

lim infn→∞

(−xn) = − lim supn→∞

xn.

On occasion we deal with real-parameter versions of liminf and limsup.Given a real-valued function f whose domain includes an interval centeredat a point x ∈ R, we define

lim supt→x

f(t) = infδ>0

sup|t−x|<δ

f(t) = limδ→0

sup|t−x|<δ

f(t),

and lim inft→x f(t) is defined analogously. The properties of these real-parameter versions of liminf and limsup are similar to those of the sequenceversions.

Infinite Series

Infinite series in the general setting of normed spaces will be discussed inSection 1.2.3; here we restrict our attention to infinite series of scalars. If(cn)n∈N is a sequence of real or complex numbers, then we say that the infiniteseries

∑∞n=1 cn converges if there exists a scalar s such that the partial sums

sN =∑N

n=1 cn converge to s as N → ∞. In this case∑∞

n=1 cn is assigned the

12 Preliminaries

value s:∞∑

n=1

cn = limN→∞

sN = limN→∞

N∑

n=1

cn = s.

Series of Real Numbers. Assume that every cn is a real number. Thenwe say that the series

∑∞n=1 cn converges in the extended real sense, or simply

that the series exists, if

• sN converges to a real number s as N → ∞, or

• sN diverges to ∞ as N → ∞, or

• sN diverges to −∞ as N → ∞.

Nonnegative Series. If every cn is a nonnegative real number (that is,cn ≥ 0 for every n), then the series

∑∞n=1 cn converges in the extended real

sense. Moreover, there are only two possibilities: Either the series convergesto a nonnegative real number or it diverges to infinity. We indicate whichpossibility holds as follows:

∞∑

n=1

cn < ∞ means that the series converges (to a finite real number),

while∞∑

n=1

cn = ∞ means that the series diverges to infinity.

Pointwise Convergence of Functions

If X is a set and {fn}n∈N is a sequence of extended real-valued or complex-valued functions whose domain is X, then we say that fn converges pointwiseto a function f if

f(x) = limn→∞

fn(x) for all x ∈ X.

In this case we write fn(x) → f(x) for every x ∈ X or fn → f pointwise.Note that this convergence can be in the extended real sense.

If {fn}n∈N is a sequence of extended real-valued functions whose domainis a set X, then we say that {fn}n∈N is a monotone increasing sequence if{fn(x)}n∈N is monotone increasing for each x, i.e., if

f1(x) ≤ f2(x) ≤ · · · for all x ∈ X.

In this case f(x) = limn→∞ fn(x) exists for each x ∈ X in the extended realsense, and we say that fn increases pointwise to f . We denote this by writing

fn ր f on X.

Preliminaries 13

Continuity

Continuity for the general setting of functions on metric spaces will be dis-cussed in Section 1.1.4. Here we define continuity for scalar-valued functionswhose domain is a subset E of Rd. We say that f : E → C is continuous onthe set E if whenever we have points xn, x ∈ E such that xn → x, it followsthat f(xn) → f(x).

Derivatives and Everywhere Differentiability

Let f be a scalar-valued function whose domain includes an open intervalcentered at a point x ∈ R. We say that f is differentiable at x if the limit

f ′(x) = limy→x

f(y) − f(x)

y − x

exists and is a scalar.Let [a, b] be a closed interval in the real line. A function f is everywhere

differentiable or differentiable everywhere on [a, b] if it is differentiable at eachpoint in the interior (a, b) and if the appropriate one-sided derivatives existat the endpoints a and b. That is, f is everywhere differentiable on [a, b] if

f ′(x) = limy→x, y∈[a,b]

f(y) − f(x)

y − x

exists and is a scalar for each x ∈ [a, b].We use similar terminology if f is defined on other types of intervals in R.

For example, x3/2 is differentiable everywhere on [0, 1] and x1/2 is differen-tiable everywhere on (0, 1], but x1/2 is not differentiable everywhere on [0, 1].

The Riemann Integral

Let f : [a, b] → R be a bounded, real-valued function on a finite, closed in-terval [a, b]. A partition of [a, b] is a choice of finitely many points xk in [a, b]such that a = x0 < x1 < · · · < xn = b. If we wish to give this partition aname then we will write:

Let Γ ={a = x0 < · · · < xn = b

}be a partition of [a, b].

The mesh size of Γ is |Γ | = max{xj − xj−1 : j = 1, . . . , n

}.

14 Preliminaries

Given a partition Γ ={a = x0 < · · · < xn = b

}, for each j = 1, . . . , n let

mj and Mj denote the infimum and supremum of f on the interval [xj−1, xj ]:

mj = infx∈[xj−1,xj ]

f(x) and Mj = supx∈[xj−1,xj ]

f(x).

The numbers

LΓ =

n∑

j=1

mj (xj − xj−1) and UΓ =

n∑

j=1

Mj (xj − xj−1),

are called lower and upper Riemann sums for f , respectively. We say that fis Riemann integrable on [a, b] if there exists a real number I such that

supΓ

LΓ = infΓ

UΓ = I,

where the supremum and infimum are taken over all partitions Γ. In thiscase, the number I is the Riemann integral of f over [a, b], and we write∫ b

af(x) dx = I.Here is an equivalent definition of the Riemann integral. Given a partition

Γ = {a = x0 < · · · < xn = b}, choose any points ξj ∈ [xj−1, xj ]. We call

RΓ =

n∑

j=1

f(ξj) (xj − xj−1)

a Riemann sum for f (note that RΓ implicitly depends on both the partitionΓ and the choice of points ξj). Then f is Riemann integrable on [a, b] if andonly if there is a real number I such that I = lim|Γ |→0 RΓ , where this meansthat for every ε > 0, there is a δ > 0 such that for any partition Γ with|Γ | < δ and any choice of points ξj ∈ [xj−1, xj ] we have |I −RΓ | < ε. In this

case, I is the Riemann integral of f over [a, b], and we write∫ b

af(x) dx = I.

We declare that a complex-valued function f on [a, b] is Riemann integrableif its real and imaginary parts are both Riemann integrable.

Every continuous function f : [a, b] → C is Riemann integrable. However,there exist discontinuous functions that are Riemann integrable. We will char-acterize the Riemann integrable functions on [a, b] in Section 4.5.5.

If g : [a, b] → C is continuous, then g is Riemann integrable on the interval[a, x] for each a ≤ x ≤ b, so we can consider the indefinite integral of g,defined by

G(x) =

∫ x

a

g(t) dt, x ∈ [a, b].

The Fundamental Theorem of Calculus implies that G is differentiable onthe interval [a, b], and G′(x) = g(x) for each x ∈ [a, b]. We will prove a moregeneral form of the Fundamental Theorem of Calculus in Section 6.4.