Real Analysis and Foundations, Second Edition introduces topics in real analysis, point set topology, and the calculus of variations in a treatment that bridges the gap between the highly theoretical texts and the less rigorous introductions currently available. For this edition, the author builds on the basics of Fourier analysis to introduce modern ideas on wavelets and applications to signal processing. He augments Fourier analytic theory with applications to differential equations and significantly expands on the development of the real analysis of several variables. All this along with more figures, streamlined proofs, and revamped exercises make this an even more lively and vital text than the popular first edition.

Table of Contents:
PREFACE TO THE SECOND EDITION PREFACE TO THE FIRST EDITION LOGIC AND SET THEORY Introduction "And" and "Or" "Not" and "If-Then" Contrapositive, Converse, and "Iff" Quantifiers Set Theory and Venn Diagrams Relations and Functions Countable and Uncountable Sets Exercises NUMBER SYSTEMS The Natural Numbers Equivalence Relations and Equivalence Classes The Integers The Rational Numbers The Real Numbers The Complex Numbers Exercises SEQUENCES Convergence of Sequences Subsequences Limsup and Liminf Some Special Sequences Exercises SERIES OF NUMBERS Convergence of Series Elementary Convergence Tests Advanced Convergence Tests Some Special Series Operations on Series Exercises BASIC TOPOLOGY Open and Closed Sets Further Properties of Open and Closed Sets Compact Sets The Cantor Set Connected and Disconnected Sets Perfect Sets Exercises LIMITS AND CONTINUITY OF FUNCTIONS Definition and Basic Properties of the Limit of a Function Continuous Functions Topological Properties and Continuity Classifying Discontinuities and Monotonicity Exercises DIFFERENTIATION OF FUNCTIONS The Concept of Derivative The Mean Value Theorem and Applications More on the Theory of Differentiation Exercises THE INTEGRAL Partitions and The Concept of Integral Properties of the Riemann Integral Another Look at the Integral Advanced Results on Integration Theory Exercises SEQUENCES AND SERIES OF FUNCTIONS Partial Sums and Pointwise Convergence More on Uniform Convergence Series of Functions The Weierstrass Approximation Theorem Exercises ELEMENTARY TRANSCENDENTAL FUNCTIONS Power Series More on Power Series: Convergence Issues The Exponential and Trigonometric Functions Logarithms and Powers of Real Numbers The Gamma Function and Stirling's Formula Exercises APPLICATIONS OF ANALYSIS TO DIFFERENTIAL EQUATIONS Picard's Existence and Uniqueness Theorem The Method of Characteristics Power Series Methods Exercises INTRODUCTION TO HARMONIC ANALYSIS The Idea of Harmonic Analysis The Elements of Fourier Series An Introduction to the Fourier Transform Fourier Methods in the Theory of Differential Equations Exercises FUNCTIONS OF SEVERAL VARIABLES Review of Linear Algebra A New Look at the Basic Concepts of Analysis Properties of the Derivative The Inverse and Implicit Function Theorems Differential Forms Exercises ADVANCED TOPICS Metric Spaces Topology in a Metric Space The Baire Category Theorem The Ascoli-Arzela Theorem The Lebesgue Integral A Taste of Probability Theory Exercises A GLIMPSE OF WAVELET THEORY Localization in the Time and Space Variables A Custom Fourier Analysis The Haar Basis Some Illustrative Examples Closing Remarks Exercises BIBLIOGRAPHY INDEX