Principles of Mathematical Analysis (International Series in Pure and Applied Mathematics)

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. 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

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.

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. -- Publisher description