Real Analysis and Foundations: (Textbooks in Mathematics)

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

Table of Contents:
Number Systems The Real Numbers Appendix: Construction of the Real Numbers The Complex Numbers Sequences Convergence of Sequences Subsequences Limsup and Liminf Some Special Sequences Series of Numbers Convergence of Series Elementary Convergence Tests Advanced Convergence Tests Some Special Series Operations on Series Basic Topology Open and Closed Sets Further Properties of Open and Closed Sets Compact Sets The Cantor Set Connected and Disconnected Sets Perfect Sets Limits and Continuity of Functions Basic Properties of the Limit of a Function Continuous Functions Topological Properties and Continuity Classifying Discontinuities and Monotonicity Differentiation of Functions The Concept of Derivative The Mean Value Theorem and Applications More on the Theory of Differentiation The Integral Partitions and the Concept of Integral Properties of the Riemann Integral Change of Variable and Related Ideas Another Look at the Integral Advanced Results on Integration Theory Sequences and Series of Functions Partial Sums and Pointwise Convergence More on Uniform Convergence Series of Functions The Weierstrass Approximation Theorem Elementary Transcendental Functions Power Series . More on Power Series: Convergence Issues The Exponential and Trigonometric Functions Logarithms and Powers of Real Numbers Differential Equations Picard’s Existence and Uniqueness Theorem The Form of a Differential Equation Picard’s Iteration Technique Some Illustrative Examples Estimation of the Picard Iterates Power Series Methods Introduction to Harmonic Analysis The Idea of Harmonic Analysis The Elements of Fourier Series An Introduction to the Fourier Transform Appendix: Approximation by Smooth Functions Fourier Methods and Differential Equations Remarks on Different Fourier Notations The Dirichlet Problem on the Disc Introduction to the Heat and Wave Equations Boundary Value Problems Derivation of the Wave Equation Solution of the Wave Equation The Heat Equation Functions of Several Variables A New Look at the Basic Concepts of Analysis Properties of the Derivative The Inverse and Implicit Function Theorems Appendix I: Elementary Number Systems Appendix II: Logic and Set Theory Appendix III: Review of Linear Algebra

About the Author :
Steven G. Krantz is a professor of mathematics at Washington University in St. Louis. He has previously taught at UCLA, Princeton University, and Pennsylvania State University. He has written more than 65 books and more than 175 scholarly papers and is the founding editor of the Journal of Geometric Analysis. An AMS Fellow, Dr. Krantz has been a recipient of the Chauvenet Prize, Beckenbach Book Award, and Kemper Prize. He received a Ph.D. from Princeton University.

Review :
This is a “rigorous” introduction to real analysis for more advanced undergraduate students of mathematics. The text requires only that readers have a strong background in calculus. Krantz (Washington Univ., St. Louis) has removed several advanced chapters from earlier editions (CH, Apr'92, 29-4571) to strengthen the book's focus on basics. The only advanced topic that remains is harmonic analysis. The removal of advanced material enables the author to discuss the fundamental parts of the subject in more detail and to include more exercises (about ten per section, appearing in increasing order of difficulty). In the same spirit, there are multiple appendixes, as well as a very detailed glossary and a table of notation. The theorems, lemmas, and examples are always connected by some context, making it easy for students to see the big picture and how the discussed results fit within this picture. This reviewer’s only critical remark is that no exercises offer solutions, which makes self-study difficult. --M. Bona, University of Florida