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Principles of mathematical analysis (International series in pure and applied mathematics) Bibliography: p Includes index 1 Mathematical analysis I Title QA300 R8 1976 515 75-17903 ISBN 0-07-054235-X PRINCIPLES OF MATHEMATICAL ANALYSIS • Copyright © 1964 1976 by McGraw-Hill Inc Al] rights reserved Copyright 1953 by McGraw-Hill Inc



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What is numerical analysis?

    Introduction Numerical analysis is a branch of Mathematics that deals with devising e?cient methods for obtaining numerical solutions to di?cult Mathematical problems. Most of the Mathematical problems that arise in science and engineering are very hard and sometime impossible to solve exactly.

What are the three parts of numerical analysis?

    Numerical analysis include three parts. The ?rst part of the subject is about the development of a method to a problem. The second part deals with the analysis of the method, which includes the error analysis and the e?ciency analysis.

Who invented the equivalence class of rational numbers?

    in which each real number is defined to be an equivalence class of Cauchy sequences of rational numbers (see Chap. 3), is carried out in Sec. 5 of the book by Hewitt and Stromberg. The cuts in Q which we used here were invented by Dedekind. The construction of R from Q by means of Cauchy sequences is due to Cantor.
INTERNATIONAL SERIES IN PURE AND APPLIED MATHEMATICS

PRINCIPLES OF

MATHEMATICAL ANALYSIS

INTERNATIONAL SERIES IN PURE

AND APPLIED MATHEMATICS

William Ted Martin, E. H. Spanier, G. Springer and

P. J. Davis. Consulting Editors

AHLFORS: Complex Analysis

BucK: Advanced Calculus

BUSACKER AND SAATY: Finite Graphs and Networks

CHENEY: Introduction to Approximation Theory

CHESTER: Techniques in Partial Differential Equations CODDINGTON AND LEVINSON: Theory of Ordinary Differential Equations CONTE AND DE BooR: Elementary Numerical Analysis: An Algorithmic Approach DENNEMEYER: Introduction to Partial Differential Equations and Boundary Value

Problems

DETTMAN: Mathematical Methods in Physics and Engineering GOLOMB AND SHANKS: Elements of Ordinary Differential Equations GREENSPAN: Introduction to Partial Differential Equations HAMMING: Numerical Methods for Scientists and Engineers

HILDEBRAND: Introduction to Numerical Analysis

HousEHOLDER: The Numerical Treatment of a Single Nonlinear Equation KALMAN, FALB, AND ARBIB: Topics in Mathematical Systems Theory

LASS: Vector and Tensor Analysis

McCARTY: Topology: An Introduction with Applications to Topological Groups

MONK: Introduction to Set Theory

MOORE: Elements of Linear Algebra and Matrix Theory

MosTOW AND SAMPSON: Linear Algebra

MouRSUND AND DURIS: Elementary Theory and Application of Numerical Analysis

PEARL: Matrix Theory and Finite Mathematics

PIPES AND HARVILL: Applied Mathematics for Engineers and Physicists

RALSTON: A First Course in Numerical Analysis

RITGER AND RosE: Differential Equations with Applications

RITT: Fourier Series

RUDIN: Principles of Mathematical Analysis

SHAPIRO: Introduction to Abstract Algebra

SIMMONS: Differential Equations with Applications and Historical Notes SIMMONS: Introduction to Topology and Modern Analysis SNEDDON: Elements of Partial Differential Equations

STRUBLE: Nonlinear Differential Equations

McGraw-Hill, Inc.

New York St. Louis San Francisco Auckland Bogota

Caracas Lisbon London Madrid Mexico City Milan

Montreal New Delhi San Juan Singapore

Sydney Tokyo Toronto

WALTER RUDIN

Professor of Mathematics

University

of Wisconsin,-Madison

THIRD EDITION

This book was set in Times New Roman.

The editors were A. Anthony Arthur and Shelly Levine Langman; the production supervisor was Leroy A. Young. R. R. Donnelley & Sons Company was printer and binder.

This book is printed on acid-free paper.

Library of Congress Cataloging in Publication Data

Rudin, Walter, date

Principles of mathematical analysis.

(International series in pure and applied mathematics)

Bibliography: p.

Includes index.

1. Mathematical analysis. I. Title.

QA300.R8 1976 515 75-17903

ISBN 0-07-054235-X

PRINCIPLES OF MATHEMATICAL ANALYSIS

Copyright © 1964, 1976 by McGraw-Hill, Inc. Al] rights reserved.

Copyright

1953 by McGraw-Hill, Inc. All rights reserved.

Printed

in the United States of America. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher.

28 29 30 DOC/DOC O 9 8 7 6 5 4 3 2 1 0

Preface

Chapter 1 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

Chapter 2 Basic Topology

Finite, Countable, and Uncountable Sets

Metric Spaces

Compact Sets

Perfect Sets

CONTENTS

lX 1 1 3 5 8 11 12 16 17 21
24
24
30
36
41

Vi CONTENTS

Connected Sets

Exercises

Chapter 3 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

Chapter 4 Continuity

Limits of Functions

Continuous Fur1ctions

Continuity and Compactness

Continuity and Connectedness

Discontinuities

Monotonic Functions

Infinite Limits and Limits at Infinity

Exercises

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

Exercises

42
43
47
47
51
52
55
57
58
61
63
65
69
70
71
72
75
78
83
83
85
89
93
94
95
97
98
103
103
107
108
109
110
110
111
114

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

Exercises

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

Exercises

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

Exercises

Chapter 9 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

Chapter

10 Integration of Differential Forms

Integration

CONTENTS vii

120
120
128
133
135
136
138
143
143
147
149
151
152
154
159
165
172
172
178
182
184
185
192
196
204
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