Real Analysis and Foundations

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 71
2.1 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 Integral
7.1 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