Real Analysis and Foundations, 3rd Edition

Back by popular demand, Real Analysis and Foundations, Third Edition bridges the gap between classic theoretical texts and less rigorous ones, providing a smooth transition from logic and proofs to real analysis. Along with the basic material, the text covers Riemann-Stieltjes integrals, Fourier analysis, metric spaces and applications, and differential equations. Offering a more streamlined presentation, this edition moves elementary number systems and set theory and logic to appendices and removes the material on wavelet theory, measure theory, differential forms, and the method of characteristics. It also adds a chapter on normed linear spaces and includes more examples and varying levels of exercises. Features • Presents a clear, thorough treatment of the theorems and concepts of real analysis • Includes a new chapter on normed linear spaces • Provides more examples throughout the text and additional exercises at the end of each section • Designates challenging exercises with an asterisk With extensive examples and thorough explanations, this best-selling book continues to give you a solid foundation in mathematical analysis and its applications. It prepares you for further exploration of measure theory, functional analysis, harmonic analysis, and beyond. Cover Page 1 Half Title Page 2 TEXTBOOKS in MATHEMATICS 3 Title Page 4 Copyright Page 5 Dedication 6 Preface to the Third Edition 8 Preface to the Second Edition 10 Preface to the First Edition 12 Table of Contents 16 Chapter 1 - Number Systems 20 1.1 The Real Numbers 20 EXERCISES 27 1.2 The Complex Numbers 28 EXERCISES 33 Chapter 2 - Sequences 34 2.1 Convergence of Sequences 34 EXERCISES 40 2.2 Subsequences 41 EXERCISES 44 2.3 Lim sup and Lim inf 45 EXERCISES 47 2.4 Some Special Sequences 48 EXERCISES 50 Chapter 3 - Series of Numbers 52 3.1 Convergence of Series 52 EXERCISES 56 3.2 Elementary Convergence Tests 58 EXERCISES 64 3.3 Advanced Convergence Tests 65 EXERCISES 70 3.4 Some Special Series 71 EXERCISES 76 3.5 Operations on Series 78 EXERCISES 81 Chapter 4 - Basic Topology 82 4.1 Open and Closed Sets 82 EXERCISES 87 4.2 Further Properties of Open and Closed Sets 88 EXERCISES 91 4.3 Compact Sets 92 EXERCISES 95 4.4 The Cantor Set 95 EXERCISES 98 4.5 Connected and Disconnected Sets 99 EXERCISES 101 4.6 Perfect Sets 102 EXERCISES 103 Chapter 5 - Limits and Continuity of Functions 104 5.1 Basic Properties of the Limit of a Function 104 EXERCISES 109 5.2 Continuous Functions 110 EXERCISES 115 5.3 Topological Properties and Continuity 115 EXERCISES 121 5.4 Classifying Discontinuities and Monotonicity 123 EXERCISES 126 Chapter 6 - Differentiation of Functions 130 6.1 The Concept of Derivative 130 EXERCISES 138 6.2 The Mean Value Theorem and Applications 139 EXERCISES 145 6.3 More on the Theory of Differentiation 146 EXERCISES 149 Chapter 7 - The Integral 152 7.1 Partitions and the Concept of Integral 152 EXERCISES 157 7.2 Properties of the Riemann Integral 159 EXERCISES 166 7.3 Another Look at the Integral 168 EXERCISES 172 7.4 Advanced Results on Integration Theory 172 EXERCISES 179 Chapter 8 - Sequences and Series of Functions 182 8.1 Partial Sums and Pointwise Convergence 182 EXERCISES 186 8.2 More on Uniform Convergence 187 EXERCISES 190 8.3 Series of Functions 191 EXERCISES 194 8.4 The Weierstrass Approximation Theorem 195 EXERCISES 199 Chapter 9 - Elementary Transcendental Functions 202 9.1 Power Series 202 EXERCISES 207 9.2 More on Power Series: Convergence Issues 208 EXERCISES 212 9.3 The Exponential and Trigonometric Functions 213 EXERCISES 218 9.4 Logarithms and Powers of Real Numbers 220 EXERCISES 222 Chapter 10 - Applications of Analysis to Differential Equations 224 10.1 Picard’s Existence and Uniqueness Theorem 224 10.1.1 The Form of a Differential Equation 224 10.1.2 Picard’s Iteration Technique 225 10.1.3 Some Illustrative Examples 226 10.1.4 Estimation of the Picard Iterates 228 EXERCISES 229 10.2 Power Series Methods 231 EXERCISES 239 Chapter 11 - Introduction to Harmonic Analysis 242 11.1 The Idea of Harmonic Analysis 242 EXERCISES 243 11.2 The Elements of Fourier Series 244 EXERCISES 250 11.3 An Introduction to the Fourier Transform 254 11.3.1 APPENDIX: Approximation by Smooth Functions 257 EXERCISES 259 11.4 Fourier Methods and Differential Equations 262 11.4.1 Remarks on Different Fourier Notations 262 11.4.2 The Dirichlet Problem on the Disc 263 EXERCISES 267 Chapter 12 - Functions of Several Variables 272 12.1 A New Look at the Basic Concepts of Analysis 272 EXERCISES 276 12.2 Properties of the Derivative 277 EXERCISES 282 12.3 The Inverse and Implicit Function Theorems 283 EXERCISES 288 Chapter 13 - Advanced Topics 290 13.1 Metric Spaces 290 EXERCISES 294 13.2 Topology in a Metric Space 295 EXERCISES 298 13.3 The Baire Category Theorem 299 EXERCISES 303 13.4 The Ascoli-Arzela Theorem 303 EXERCISES 306 Chapter 14 - Normed Linear Spaces 308 14.1 What Is This Subject About? 308 EXERCISES 309 14.2 What Is a Normed Linear Space? 309 EXERCISES 312 14.3 Finite-Dimensional Spaces 313 EXERCISES 314 14.4 Linear Operators 315 EXERCISES 317 14.5 The Three Big Results 318 EXERCISES 323 14.6 Applications of the Big Three 324 EXERCISES 334 Appendix I - Elementary Number Systems 336 Appendix II - Logic and Set Theory 354 Appendix III - Review of Linear Algebra 388 Table of Notation 396 Glossary 402 Bibliography 422 Back Cover 426 The first three editions of this popular textbook attracted a loyal readership and widespread use. Students find the book to be concise, accessible, and complete. Instructors find the book to be clear, authoritative, and dependable. The goal of this new edition is to make real analysis relevant and accessible to a broad audience of students with diverse backgrounds. Real analysis is a basic tool for all mathematical scientists, ranging from mathematicians to physicists to engineers to researchers in the medical profession. This text aims to be the generational touchstone for the subject and the go-to text for developing young scientists. In this new edition we endeavor to make the book accessible to a broader audience. This edition includes more explanation, more elementary examples, and the author stepladders the exercises. Figures are updated and clarified. We make the sections more concise, and omit overly technical details. We have updated and augmented the multivariable material in order to bring out the geometric nature of the topic. The figures are thus enhanced and fleshed out. Features A renewed enthusiasm for the topic comes through in a revised presentation A new organization removes some advanced topics and retains related ones Exercises are more tiered, offering a more accessible course Key sections are revised for more brevity