Real Analysis and Foundations

Number Systems 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 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 Power Series Methods Introduction to Harmonic Analysis The Idea of Harmonic Analysis The Elements of Fourier Series An Introduction to the Fourier Transform Fourier Methods and Differential Equations Functions of Several Variables A New Look at the Basic Concepts of Analysis Properties of the Derivative The Inverse and Implicit Function Theorems Advanced Topics Metric Spaces Topology in a Metric Space The Baire Category Theorem The Ascoli-Arzela Theorem Normed Linear Spaces What Is This Subject About? What Is a Normed Linear Space? Finite-Dimensional Spaces Linear Operators The Three Big Results Applications of the Big Three Appendix I: Elementary Number Systems Appendix II: Logic and Set Theory Appendix III: Review of Linear Algebra Table of Notation Glossary Bibliography Index Exercises are included at the end of each section.