Principles of Mathematical Analysis

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  • Edition: 3rd
  • Format: Hardcover
  • Copyright: 1/1/1976
  • Publisher: McGraw-Hill Education

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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


Ordered Sets


The Real Field

The Extended Real Number System

The Complex Field

Euclidean Spaces



Chapter 2: Basic Topology

Finite, Countable, and Uncountable Sets

Metric Spaces

Compact Sets

Perfect Sets

Connected Sets


Chapter 3: Numerical Sequences and Series

Convergent Sequences


Cauchy Sequences

Upper and Lower Limits

Some Special Sequences


Series of Nonnegative Terms

The Number e

The Root and Ratio Tests

Power Series

Summation by Parts

Absolute Convergence

Addition and Multiplication of Series



Chapter 4: Continuity

Limits of Functions

Continuous Functions

Continuity and Compactness

Continuity and Connectedness


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 Higher-Order

Taylor's Theorem

Differentiation of Vector-valued Functions


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


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


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


Chapter 9: Functions of Several Variables

Linear Transformations


The Contraction Principle

The Inverse Function Theorem

The Implicit Function Theorem

The Rank Theorem


Derivatives of Higher Order

Differentiation of Integrals


Chapter 10: Integration of Differential Forms


Primitive Mappings

Partitions of Unity

Change of Variables

Differential Forms

Simplexes and Chains

Stokes' Theorem

Closed Forms and Exact Forms

Vector Analysis


Chapter 11: The Lebesgue Theory

Set Functions

Construction of the Lebesgue Measure

Measure Spaces

Measurable Functions

Simple Functions


Comparison with the Riemann Integral

Integration of Complex Functions

Functions of Class L²



List of Special Symbols


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