Real Analysis and Foundations, Fourth Edition: (Textbooks in Mathematics)

The new edition of this popular text is revised to meet the suggestions of users of the previous edition. A readable yet rigorous approach to an essential part of mathematical thinking, this text 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.

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.