A modernized and updated edition of the third edition of Walter Rudin's "Principles of Mathematical Analysis". The book has been retyped from scratch. The numbering for all definitions/theorems/propositions/etc. has remained the same, and an exhaustive list of the changes made can be found in a newly-added Forward. Originally created as a personal project by an anonymous group of typesetters.
Author(s): Walter Rudin, Anonymous Authors
Series: International series in pure and applied mathematics
Edition: 3
Publisher: McGraw-Hill, Inc
Year: 2024
Language: English
Pages: 331
Foreword
Preface
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
Basic Topology
Finite, Countable, and Uncountable Sets
Metric Spaces
Compact Sets
Perfect Sets
Connected Sets
Exercises
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
Summation by Parts
Absolute Convergence
Addition and Multiplication of Series
Rearrangements
Exercises
Continuity
Limits of Functions
Continuous Functions
Continuity and Compactness
Continuity and Connectedness
Discontinuities
Monotonic Functions
Infinite Limits and Limits at Infinity
Exercises
Differentiation
The Derivative of a Real Function
Mean Value Theorems
The Continuity of Derivatives
L'Hôpital's Rule
Derivatives of Higher Order
Taylor's Theorem
Differentiation of Vector-valued Functions
Exercises
The Riemann-Stieltjes Integral
Definition and Existence of the Integral
Properties of the Integral
Integration and Differentiation
Rectifiable Curves
Exercises
Sequences and Series of Functions
Discussion of the 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
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
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
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
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
