Real Analysis and Foundations

Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Contents
Preface
Acknowledgments
Background Material
0.1 Number Systems
0.1.1 The Natural Numbers
0.1.2 The Integers
0.1.3 The Rational Numbers
Exercises
0.2 Logic and Set Theory
0.2.1 "And" and "Or"
0.2.2 "Not" and "If Then"
0.2.3 Contrapositive, Converse, and "If"
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
Exercises
Note
1. Real and Complex Numbers
1.1 The Real Numbers
Appendix: Construction of the Real Numbers
Exercises
1.2 The Complex Numbers
Exercises
Note
2. Sequences
2.1 Convergence of Sequences
Exercises
2.2 Subsequences
Exercises
2.3 Lim sup and Lim inf
Exercises
2.4 Some Special Sequences
Exercises
Note
3. Series of Numbers
3.1 Convergence of Series
Exercises
3.2 Elementary Convergence Tests
Exercises
3.3 Advanced Convergence Tests
Exercises
3.4 Some Special Series
Exercises
3.5 Operations on Series
Exercises
4. Basic Topology
4.1 Open and Closed Sets
Exercises
4.2 Further Properties of Open and Closed Sets
Exercises
4.3 Compact Sets
Exercises
4.4 The Cantor Set
Exercises
4.5 Connected and Disconnected Sets
Exercises
4.6 Perfect Sets
Exercises
5. Limits and Continuity of Functions
5.1 Definition and Basic Properties of the Limit of a Function
Exercises
5.2 Continuous Functions
Exercises
5.3 Topological Properties and Continuity
Exercises
5.4 Classifying Discontinuities and Monotonicity
Exercises
6. Differentiation of Functions
6.1 The Concept of Derivative
Exercises
6.2 The Mean Value Theorem and Applications
Exercises
6.3 More on the Theory of Differentiation
Exercises
7. The Integral
7.1 Partitions and the Concept of Integral
Exercises
7.2 Properties of the Riemann Integral
Exercises
7.3 Change of Variable and Related Ideas
Exercises
7.4 Another Look at the Integral
Exercises
7.5 Advanced Results on Integration Theory
Exercises
Note
8. Sequences and Series of Functions
8.1 Partial Sums and Pointwise Convergence
Exercises
8.2 More on Uniform Convergence
Exercises
8.3 Series of Functions
Exercises
8.4 The Weierstrass Approximation Theorem
Exercises
9. Elementary Transcendental Functions
9.1 Power Series
Exercises
9.2 More on Power Series: Convergence Issues
Exercises
9.3 The Exponential and Trigonometric Functions
Exercises
9.4 Logarithms and Powers of Real Numbers
Exercises
10. Functions of Several Variables
10.1 A New Look at the Basic Concepts of Analysis
Exercises
10.2 Properties of the Derivative
Exercises
10.3 The Inverse and Implicit Function Theorems
Exercises
11. Advanced Topics
11.1 Metric Spaces
Exercises
11.2 Topology in a Metric Space
Exercises
11.3 The Baire Category Theorem
Exercises
11.4 The Ascoli-Arzela Theorem
Exercises
Note
12. Applications of Analysis to Differential Equations
12.1 Picard's Existence and Uniqueness Theorem Picard's 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
Exercises
12.2 Power Series Methods
Exercises
Note
13. Introduction to Harmonic Analysis
13.1 The Idea of Harmonic Analysis
Exercises
13.2 The Elements of Fourier Series
Exercises
13.3 An Introduction to the Fourier Transform
Appendix: Approximation by Smooth Functions
Exercises
13.4 Fourier Methods in the Theory of 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
Exercises
13.5 The Heat Equation
Fourier's Point of View
Exercises
Notes
Appendix: Review of Linear Algebra
Section A1. Linear Algebra Basics
Table of Notation
Glossary
Bibliography
Index