Principles of Mathematical Analysis by Walter Rudin

Book Summary of Principles of Mathematical Analysis

The third edition of this well known text continues to provide a solid foundation in mathematical analysis for undergraduate and first-year graduate students. The text begins with a discussion of the real number system as a complete ordered field. (Dedekind's construction is now treated in an appendix to Chapter I.) The topological background needed for the development of convergence, continuity, differentiation and integration is provided in Chapter 2. There is a new section on the gamma function, and many new and interesting exercises are included.

This text is part of the Walter Rudin Student Series in Advanced Mathematics.
Table of Contents

• Chapter 1: The Real and Complex Number Systems
• Introduction
• Ordered Sets
• Fields
• The Real Field
• The Extended Real Number System
• The Complex Field
• Euclidean Spaces
• Appendix
• Exercises
• Chapter 2: Basic Topology
• 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
• The Root and Ratio Tests
• Power Series
• Summation by Parts
• Absolute Convergence
• Addition and Multiplication of Series
• Rearrangements
• Exercises
• Chapter 4: Continuity
• Limits of Functions
• Continuous Functions
• Continuity and Compactness
• Continuity and Connectedness
• Discontinuities
• Monotonic Functions
• Infinite Limits and Limits at Infinity
• Exercises
• Chapter 5: Differentiation
• The Derivative of a Real Function
• Mean Value Theorems
• The Continuity of Derivatives
• L'Hospital's Rule
• Derivatives of Higher-Order
• Taylor's Theorem
• Differentiation of Vector-valued Functions
• Exercises
• Chapter 6: The Riemann-Stieltjes Integral
• Definition and Existence of the Integral
• Properties of the Integral
• Integration and Differentiation
• Integration of Vector-valued Functions
• Rectifiable Curves
• Exercises
• 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: Some Special Functions
• Power Series
• The Exponential and Logarithmic Functions
• The Trigonometric Functions
• The Algebraic Completeness of the Complex Field
• Fourier Series
• The Gamma Function
• Exercises
• Chapter 9: Functions of Several Variables
• Linear Transformations
• Differentiation
• The Contraction Principle
• The Inverse Function Theorem
• The Implicit Function Theorem
• The Rank Theorem
• Determinants
• Derivatives of Higher Order
• Differentiation of Integrals
• Exercises
• Chapter 10: Integration of Differential Forms
• Integration
• Primitive Mappings
• Partitions of Unity
• Change of Variables
• Differential Forms
• Simplexes and Chains
• Stokes' Theorem
• Closed Forms and Exact Forms
• Vector Analysis
• Exercises
• Chapter 11: 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 L²
• Exercises
• Bibliography
• List of Special Symbols
• Index