# Book:Walter Rudin/Principles of Mathematical Analysis/Second Edition

## Walter Rudin: Principles of Mathematical Analysis (2nd Edition)

Published $\text {1964}$, McGraw-Hill Inc.

### Contents

Preface

Chapter 1. The Real and Complex Number Systems
Introduction
Dedekind cuts
Real numbers
The extended real number system
Complex numbers
Euclidean spaces
Exercises

Chapter 2. Elements of Set Theory
Finite, countable and uncountable sets
Metric spaces
Compact sets
Perfect sets
Connected sets
Exercises

Chapter 3. Numerical Sequences and Series
Convergent sequences
Subsequences
Cauchy sequences
Upper and lower limits
Some special sequences
Series
Series of nonnegative terms
The number $e$
The root and ratio tests
Power series
Partial summation
Absolute convergence
Rearrangements
Exercises

Chapter 4. Continuity
The limit of a function
Continuous functions
Continuity and compactness
Continuity and connectedness
Discontinuities
Monotonic functions
Infinite limits and limits at infinity

Chapter 5. Differentiation
The derivative of a real function
Mean value theorems
The continuity of derivatives
L'Hospital's rule
Derivatives of a higher order
Taylor's theorem
Differentiation of vector-valued functions
Exercises

Chapter 6. The Riemann-Stieltjes Integral
Definition and existence of the integral
The integral as a limit of sums
Integration and differentiation
Integration of vector-valued functions
Functions of bounded variables
Further theorems on integration

Chapter 7. Sequences and Series of Functions
Discussion of main problem
Uniform convergence
Uniform convergence and continuity
Uniform convergence and integration
Uniform convergence and differentiation
Equicontinuous families of functions
The Stone-Weierstrass theorem
Exercises

Chapter 8. Further Topics in the Theory of Series
Power series
The exponential and logarithmic functions
The trigonometric functions
The algebraic completeness of the complex field
Fourier series
Exercises

Chapter 9. Functions of Several Variables
Linear transformations
Differentiation
The inverse function theorem
The implicit function theorem
The rank theorem
A decomposition theorem
Determinants
Integration
Differential forms
Simplexes and chains
Stokes' theorem
Exercises

Chapter 10. The Lebesgue Theory
Set functions
Construction of the Lebesgue measure
Measure spaces
Measurable functions
Simple functions
Integration
Comparison with the Riemann integral
Integration of complex functions
Functions of class $\LL^2$
Exercises

Bibliography
List of Frequently Occurring Symbols
Index

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