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by Walter Rudin

Edition: 3RD 86Copyright: 1986

Publisher: McGraw-Hill Publishing Company

Published: 1986

International: Yes

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This is an advanced text for the one- or two-semester course in analysis taught primarily to math, science, computer science, and electrical engineering majors at the junior, senior or graduate level. The basic techniques and theorems of analysis are presented in such a way that the intimate connections between its various branches are strongly emphasized. The traditionally separate subjects of 'real analysis' and 'complex analysis' are thus united in one volume. Some of the basic ideas from functional analysis are also included. This is the only book to take this unique approach. The third edition includes a new chapter on differentiation. Proofs of theorems presented in the book are concise and complete and many challenging exercises appear at the end of each chapter. The book is arranged so that each chapter builds upon the other, giving students a gradual understanding of the subject.

**Chapter 1. Abstract Integration**

Set-theoretic notations and terminology

The concept of measurability

Simple functions

Elementary properties of measures

Arithmetic in [0, infinity]

Integration of positive functions

Integration of complex functions

The role played by sets of measure zero

Exercises

**Chapter 2. Positive Borel Measures**

Vector spaces

Topological preliminaries

The Riesz representation theorem

Regularity properties of Borel measures

Lebesgue measure

Continuity properties of measurable functions

Exercises

**Chapter 3. L^p-Spaces**

Convex functions and inequalities

The L^p-spaces

Approximation by continuous functions

Exercises

**Chapter 4. Elementary Hilbert Space Theory**

Inner products and linear functionals

Orthonormal sets

Trigonometric series

Exercises

**Chapter 5. Examples of Banach Space Techniques**

Banach spaces

Consequences of Baire's theorem

Fourier series of continuous functions

Fourier coefficients of L¹-functions

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Summary

This is an advanced text for the one- or two-semester course in analysis taught primarily to math, science, computer science, and electrical engineering majors at the junior, senior or graduate level. The basic techniques and theorems of analysis are presented in such a way that the intimate connections between its various branches are strongly emphasized. The traditionally separate subjects of 'real analysis' and 'complex analysis' are thus united in one volume. Some of the basic ideas from functional analysis are also included. This is the only book to take this unique approach. The third edition includes a new chapter on differentiation. Proofs of theorems presented in the book are concise and complete and many challenging exercises appear at the end of each chapter. The book is arranged so that each chapter builds upon the other, giving students a gradual understanding of the subject.

Table of Contents

**Chapter 1. Abstract Integration**

Set-theoretic notations and terminology

The concept of measurability

Simple functions

Elementary properties of measures

Arithmetic in [0, infinity]

Integration of positive functions

Integration of complex functions

The role played by sets of measure zero

Exercises

**Chapter 2. Positive Borel Measures**

Vector spaces

Topological preliminaries

The Riesz representation theorem

Regularity properties of Borel measures

Lebesgue measure

Continuity properties of measurable functions

Exercises

**Chapter 3. L^p-Spaces**

Convex functions and inequalities

The L^p-spaces

Approximation by continuous functions

Exercises

**Chapter 4. Elementary Hilbert Space Theory**

Inner products and linear functionals

Orthonormal sets

Trigonometric series

Exercises

**Chapter 5. Examples of Banach Space Techniques**

Banach spaces

Consequences of Baire's theorem

Fourier series of continuous functions

Fourier coefficients of L¹-functions

Publisher Info

Publisher: McGraw-Hill Publishing Company

Published: 1986

International: Yes

Published: 1986

International: Yes

**Chapter 1. Abstract Integration**

Set-theoretic notations and terminology

The concept of measurability

Simple functions

Elementary properties of measures

Arithmetic in [0, infinity]

Integration of positive functions

Integration of complex functions

The role played by sets of measure zero

Exercises

**Chapter 2. Positive Borel Measures**

Vector spaces

Topological preliminaries

The Riesz representation theorem

Regularity properties of Borel measures

Lebesgue measure

Continuity properties of measurable functions

Exercises

**Chapter 3. L^p-Spaces**

Convex functions and inequalities

The L^p-spaces

Approximation by continuous functions

Exercises

**Chapter 4. Elementary Hilbert Space Theory**

Inner products and linear functionals

Orthonormal sets

Trigonometric series

Exercises

**Chapter 5. Examples of Banach Space Techniques**

Banach spaces

Consequences of Baire's theorem

Fourier series of continuous functions

Fourier coefficients of L¹-functions