Real Analysis and Foundations, Third Edition

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.