Principles of Mathematical Analysis / Edition 3

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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.

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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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  • Anonymous

    Posted December 30, 2000

    Great Introduction to Graduate Analysis

    Great book. However, you need a little background in Topology to greatly appreciate this book.

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