Fundamentals of Mathematical Analysis

378 Pages·2013·10.528 MB·English

by  Paul J. Sally Jr.

20
The UNDERPGRuAreD UanATdE A p TpEliXeTdS
SERIES
Sally
Fundamentals
of Mathematical
Analysis
Paul J. Sally, Jr.
American Mathematical Society
T h e UndERgRAdUATE TExTS • 20
SERIES
Pure and Applied
Sally
Fundamentals
of Mathematical
Analysis
Paul J. Sally, Jr.
American Mathematical Society
Providence, Rhode Island
EDITORIAL COMMITTEE
Paul J. Sally, Jr. (Chair) Joseph Silverman
Francis Su Susan Tolman
2010 Mathematics Subject Classification. Primary 15–01, 22B05,26–01, 28–01, 42–01,
43–01, 46–01.
For additional informationand updates on this book, visit
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Library of Congress Cataloging-in-Publication Data
Sally,PaulJ.,Jr.,1933–
Fundamentalsofmathematicalanalysis/PaulJ.Sally,Jr.
pagescm. —(Pureandappliedundergraduatetexts;volume20)
Includesbibliographicalreferencesandindex.
ISBN978-0-8218-9141-4(alk.paper)
1.Mathematicalanalysis. I.Title.
QA300.S23 2013
515—dc23
2012036735
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To my wife and fellow author, Judith D. Sally

Contents
Preface ix
Acknowledgments xiii
Chapter 1. The Construction of Real and Complex Numbers 1
1.1. The Least Upper Bound Property and the Real Numbers 2
1.2. Consequences of the Least Upper Bound Property 4
1.3. Rational Approximation 5
1.4. Intervals 8
1.5. The Construction of the Real Numbers 9
1.6. Convergence in R 13
1.7. Automorphisms of Fields 17
1.8. Complex Numbers 19
1.9. Convergence in C 20
1.10. Independent Projects 24
Chapter 2. Metric and Euclidean Spaces 33
2.1. Introduction 34
2.2. Definition and Basic Properties of Metric Spaces 34
2.3. Topology of Metric Spaces 36
2.4. Limits and Continuous Functions 44
2.5. Absolute Continuity and Bounded Variation in R 51
2.6. Compactness, Completeness, and Connectedness 56
2.7. Independent Projects 64
Chapter 3. Complete Metric Spaces 77
3.1. The Contraction Mapping Theorem and Its Applications to
Differential and Integral Equations 78
3.2. The BaireCategoryTheoremandtheUniformBoundedness
Principle 79
3.3. Stone-Weierstrass Theorem 82
3.4. The p-adic Completion of Q 85
3.5. Independent Projects 93
v
vi CONTENTS
Chapter 4. Normed Linear Spaces 101
4.1. Definitions and Basic Properties 102
4.2. Bounded Linear Operators 106
4.3. Fundamental Theorems about Linear Operators 109
4.4. Extending Linear Functionals 112
∞
4.5. Generalized Limits and the Dual of (cid:2) (F) 114
4.6. Adjoint Operators and Isometries of Normed Linear Spaces 116
4.7. Concrete Facts about Isometries of Normed Linear Spaces 119
4.8. Locally Compact Groups 123
4.9. Hilbert Spaces 126
4.10. Convergence and Selfadjoint Operators 134
4.11. Independent Projects 136
Chapter 5. Differentiation 141
5.1. Review of Differentiation in One Variable 142
5.2. Differential Calculus in Rn 149
5.3. The Derivative as a Matrix of Partial Derivatives 154
5.4. The Mean Value Theorem 158
5.5. Higher-Order Partial Derivatives and Taylor’s Theorem 160
5.6. Hypersurfaces and Tangent Hyperplanes in Rn 164
5.7. Max-Min Problems 166
5.8. Lagrange Multipliers 170
5.9. The Implicit and Inverse Function Theorems 175
5.10. Independent Projects 182
Chapter 6. Integration 191
6.1. Measures 192
6.2. Lebesgue Measure 194
6.3. Measurable Functions 205
6.4. The Integral 208
6.5. Lp Spaces 216
6.6. Fubini’s Theorem 220
6.7. Change of Variables in Integration 224
6.8. Independent Projects 227
Chapter 7. Fourier Analysis on Locally Compact Abelian Groups 237
7.1. Fourier Analysis on the Circle 238
7.2. Fourier Analysis on Locally Compact Abelian Groups 246
7.3. The Determination of Gˆ 249
7.4. The Fourier Transform on (R,+) 251
7.5. Fourier Inversion on (R,+) 254
7.6. Fourier Analysis on p-adic Fields 259
7.7. Independent Projects 263
CONTENTS vii
Appendix A. Sets, Functions, and Other Basic Ideas 271
A.1. Sets and Elements 271
A.2. Equality, Inclusion, and Notation 272
A.3. The Algebra of Sets 273
A.4. Cartesian Products, Counting, and Power Sets 277
A.5. Some Sets of Numbers 279
A.6. Equivalence Relations and the Construction of Q 284
A.7. Functions 290
A.8. Countability and Other Basic Ideas 297
A.9. The Axiom of Choice 305
A.10. Independent Projects 308
Appendix B. Linear Algebra 313
B.1. Fundamentals of Linear Algebra 313
B.2. Linear Transformations 319
B.3. Linear Transformations and Matrices 321
B.4. Determinants 324
B.5. Geometric Linear Algebra 332
B.6. Independent Projects 340
Bibliography 351
Index of Terminology 353
Index of Notation Definitions 361

Preface
Whathappenswhenastudentcompletesacoursein“APCalculus”,followed
by a routine course in multivariable calculus, a computational course in
linear algebra, and a formulaic presentation of differential equations? It is
time for some real mathematics. There is still a world of interconnected
labyrinths to explore, unscalable mountains to be climbed, and lands of
mystery to be discovered. The first step in this is a rigorous course in one-
variable analysis. This begins with a study of an ordered field in which
the least upper bound property holds (not to be confused with a complete
ordered field; see the project in Section 2.7.3). Here the student meets
Bolzano-Weierstrass, Heine-Borel, and a rigorous treatment of one-variable
differentiation and integration with careful attention paid to the pervasive
presence of the Mean Value Theorem. Then what? That is exactly the
reason behind this book.
Learning serious mathematics is about engaging with problems, from
kindergarten to graduate school and beyond. The preliminaries for reading
thisbookarealreadycontainedintheauthor’sbookTools of the Trade [27].
The first two chapters of Tools are Appendices A and B of this book. The
reader who is familiar with that material can jump right into Chapter 1.
The sequence of topics can be gleaned from the table of contents, so I will
not dwell on that.
There are three features here that should be discussed explicitly, espe-
cially since they are important for the use of this book as a text. The first,
andmostimportant,isthecollectionofexercises. Thesearespreadthrough-
out the chapters and should be regarded as an essential component of the
student’s learning. Some of these exercises comprise a routine follow-up to
the material, while others will challenge the student’s understanding more
deeply. The second feature is the set of independent projects presented at
the end of each chapter. These projects supplement the content studied in
their respective chapters. They can be used to expand the student’s knowl-
edgeandunderstandingorasanopportunitytoconductaseminarinInquiry
Based Learning (IBL) in which the students present the material to their
class. A brief glance will show that the independent projects cover a wide
range of interesting topics that hint at advanced areas of mathematics. The
ix