Real Analysis and Foundations

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 130 books and more than 250 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.

• Preface 0 Background Material 0.1 Number Systems 0.1.1 The Natural Numbers 0.1.2 The Integers 0.1.3 The Rational Numbers 02 Logic and Set 0.2.1 And” and “Or” 0.2.2 “not” and “if then”0.2.3 Contrapositive, Converse, and “Iff” 0.2.4 Quantifiers 0.2.5 Set Theory and Venn Diagrams 0.2.6 Relations and Functions 0.2.7 Countable and Uncountable Sets 1 Real and Complex Numbers 1.1 The Real Numbers Appendix: Construction of the Real Numbers 1.2 The Complex Numbers 2 Sequences 712.1 Convergence of Sequences 2.2 Subsequences 2.3 Limsup and Liminf 2.4 Some Special Sequences 3 Series of Numbers 3.1 Convergence of Series 3.2 Elementary Convergence Tests 3.3 Advanced Convergence Tests 3.4 Some Special Series 3.5 Operations on Series 4 Basic Topology 4.1 Open and Closed Sets 4.2 Further Properties of Open and Closed Sets 4.3 Compact Sets 4.4 The Cantor Set 4.5 Connected and Disconnected Sets 4.6 Perfect Sets 5 Limits and Continuity of Functions 5.1 Basic Properties of the Limit of a Function 5.2 Continuous Functions 5.3 Topological Properties and Continuity 5.4 Classifying Discontinuities and Monotonicity 6 Differentiation of Functions 6.1 The Concept of Derivative 6.2 The Mean Value Theorem and Applications 6.3 More on the Theory of Differentiation 7 The Integral7.1 Partitions and the Concept of Integral 7.2 Properties of the Riemann Integral 7.3 Change of Variable and Related Ideas 7.4 Another Look at the Integral 7.5 Advanced Results on Integration Theory 8 Sequences and Series of Functions 8.1 Partial Sums and Pointwise Convergence 8.2 More on Uniform Convergence 8.3 Series of Functions 8.4 The Weierstrass Approximation Theorem 9 Elementary Transcendental Functions 9.1 Power Series 9.2 More on Power Series: Convergence Issues 9.3 The Exponential and Trigonometric Functions 9.4 Logarithms and Powers of Real Numbers 10 Functions of Several Variables 10.1 A New Look at the Basic Concepts of Analysis 10.2 Properties of the Derivative 10.3 The Inverse and Implicit Function Theorems 11 Advanced Topics 11.1 Metric Spaces 11.2 Topology in a Metric Space 11.3 The Baire Category Theorem 11.4 The Ascoli-Arzela Theorem 12 Differential Equations 12.1 Picard’s Existence and Uniqueness Theorem 12.1.1 The Form of a Differential Equation 12.1.2 Picard’s Iteration Technique 12.1.3 Some Illustrative Examples 12.1.4 Estimation of the Picard Iterates 12.2 Power Series Methods 13 Introduction to Harmonic Analysis 13.1 The Idea of Harmonic Analysis 13.2 The Elements of Fourier Series 13.3 An Introduction to the Fourier Transform Appendix: Approximation by Smooth Functions 13.4 Fourier Methods and Differential Equations 13.4.1 Remarks on Different Fourier Notations 13.4.2 The Dirichlet Problem on the Disc 13.4.3 Introduction to the Heat and Wave Equations 13.4.4 Boundary Value Problems 13.4.5 Derivation of the Wave Equation 13.4.6 Solution of the Wave Equation 13.5 The Heat Equation Appendix: Review of Linear Algebra Table of Notation Glossary Bibliography Index