1、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: Finit
2、e 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 Di
3、fferential 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: Intro
4、duction 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 Theor
5、y 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
6、 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 Mode
7、rn 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 Professo
8、r 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 p
9、aper. 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 MA
10、THEMATICAL 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
11、, 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 Com
12、plex 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 Sequenc
13、es 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 F
14、unctions 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 LHospitals Rule De
15、rivatives of Higher Order Taylors 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 Proper
16、ties 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 Converge
17、nce 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 E
18、xercises 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 Dif
19、ferential Forms Integration CONTENTS vii 120 120 128 133 135 136 138 143 143 147 149 151 152 154 159 165 172 172 178 182 184 185 192 196 204 204 211 220 221 223 228 231 235 236 239 245 245 YID CONTENTS Primitive Mappings Partitions of Unity Change of Variables Differential Forms Simplexes and Chains
20、 Stokes Theorem Closed Forms and Exact Forms Vector Analysis Exercises Chapter 11 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
21、 Class !t2 Exercises Blbllograpby List of Special Symbols Index 248 251 252 253 266 273 275 280 288 300 300 302 310 310 313 314 322 325 325 332 335 337 339 PREFACE This book is intended to serve as a text for the course in analysis that is usually taken by advanced undergraduates or by first-year st
22、udents who study mathe-matics. The present edition covers essentially the same topics as the second one, with some additions, a few minor omissions, and considerable rearrangement. I hope that these changes will make the material more accessible amd more attrac-tive to the students who take such a c
23、ourse. Experience has convinced me that it is pedagogically unsound (though logically correct) to start off with the construction of the real numbers from the rational ones. At the beginning, most students simply fail to appreciate the need for doing this. Accordingly, the real number system is intr
24、oduced as an ordered field with the least-upper-bound property, and a few interesting applications of this property are quickly made. However, Dedekinds construction is not omit-ted. It is now in an Appendix to Chapter I, where it may be studied and enjoyed whenever the time seems ripe. The material
25、 on functions of several variables is almost completely re-written, with many details filled in, and with more examples and more motiva-tion. The proof of the inverse function theorem-the key item in Chapter 9-iS X PREFACE simplified by means of the fixed point theorem about contraction mappings. Di
26、fferential forms are discussed in much greater detail. Several applications of Stokes theorem are included. As regards other changes, the chapter on the Riemann-Stieltjes integral has been trimmed a bit, a short do-it-yourself section on the gamma function has been added to Chapter 8, and there is a
27、 large number of new exercises, most of them with fairly detailed hints. I have also included several references to articles appearing in the American Mathematical Monthly and in Mathematics Magazine, in the hope that students will develop the habit of looking into the journal literature. Most of th
28、ese references were kindly supplied by R. B. Burckel. Over the years, many people, students as well as teachers, have sent me corrections, criticisms, and other comments concerning the previous editions of this book. I have appreciated these, and I take this opportunity to express my sincere thanks
29、to all who have written me. WALTER RUDIN THE REAL AND COMPLEX NUMBER SYSTEMS INTRODUCTION A satisfactory discussion of the main concepts of analysis (such as convergence, continuity, differentiation, and integration) must be based on an accurately defined number concept. We shall not, however, enter
30、 into any discussion of the axioms that govern the arithmetic of the integers, but assume familiarity with the rational numbers (i.e., the numbers of the form m/n, where m and n are integers and n =fi 0). The rational number system is inadequate for many purposes, both as a field and as an ordered s
31、et. (These terms will be defined in Secs. 1.6 and 1.12.) For instance, there is no rational p such that p2 = 2. (We shall prove this presently.) This leads to the introduction of so-called irrational numbers which are often written as infinite decimal expansions and are considered to be approximated
32、 by the corresponding finite decimals. Thus the sequence 1, 1.4, 1.41, 1.414, 1.4142, . tends to J2. But unless the irrational number J2 has been clearly defined, the question must arise: Just what is it that this sequence tends to? 2 PRINCIPLES OF MATHEMATICAL ANALYSI This sort of question can be a
33、nswered as soon as the so-called real number system is constructed. 1.1 Example We now show that the equation (1) p2 = 2 is not satisfied by any rational p. If there were such a p, we could write p = m/n where m and n are integers that are not both even. Let us assume this is done. Then (1) implies
34、(2) m2 = 2n2, This shows that m2 is even. Hence m is even (if m were odd, m2 would be odd), and so m2 is divisible by 4. It follows that the right side of (2) is divisible by 4, so that n2 is even, which implies that n is even. The assumption that (1) holds thus leads to the conclusion that both m a
35、nd n are even, contrary to our choice of m and n. Hence (I) is impossible for rational p. We now examine this situation a little more closely. Let A be the set of all positive rationals p such that p2 2. We shall show that A contains no largest number and B con-tains no smallest. More explicitly, fo
36、r every pin A we can find a rational q in A such that p q, and for every p in B we can find a rational q in B such that q 0 the number p2 -2 2p + 2 q=p-= . p+2 p+2 2 2 -2(p2 -2) q -(p + 2)2 . If p is in A then p2 -2 p, and (4) shows that q2 0, (3) shows that O q 2. Thus q is in B. 1.2 Remark The pur
37、pose of the above discussion has been to show that the rational number system has certain gaps, in spite of the fact that between any two rationals there is another: If r s then r (r + s)/2 A. If, in addition, there is an element of B which is not in A, then A is said to be a proper subset of B. Not
38、e that A c A for every set A. If Ac Band B c A, we write A= B. Otherwise A#: B. 1.4 Definition Throughout Chap. l, the set of all rational numbers will be denoted by Q. ORDERED SETS 1.5 Definition Let S be a set. An order on S is a relation, denoted by , with the following two properties: (i) If x e
39、 S and ye S then one and only one of the statements xy, x=y, yx is true. (ii) If x, y, z e S, if x y and y z, then .x z. The statement x x in place of x y. The notation x Sy indicates that x y. 1.6 Definition An ordered set is a set Sin which an order is defined. For example, Q is an ordered set if
40、r sis defined to mean thats -r is a positive rational number. 1.7 Definition Suppose S is an ordered set, and E c S. If there exists a /J e S such that x S fJ for every x e E, we say that Eis bounded above, and call /J an upper bound of E. Lower bounds are defined in the same way (with in place of s
41、 ). 4 PRINCIPLES OF MATHEMATICAL ANALYSIS 1.8 Definition Suppose S is an ordered set, E c S, and E is bounded above. Suppose there exists an ex e S with the following properties: (i) ex is an upper bound of E. (ii) If y cx is a lower bound of E. 1.9 Examples (a) Consider the sets A and B of Example
42、1.1 as subsets of the ordered set Q. The set A is bounded above. In fact, the upper bounds of A are exactly the members of B. Since B contains no smallest member, A has no least upper bound in Q. Similarly, B is bounded below: The set of all lower bounds of B consists of A and of all re Q with r S 0
43、. Since A has no lasgest member, B has no greatest lower bound in Q. (b) If cx = sup E exists, then cx may or may not be a member of E. For instance, let E1 be the set of all r e Q with r 0. Let E2 be the set of all r e Q with r S 0. Then sup E1 = sup E2 = 0, and O E1, 0 e E2 (c) Let E consist of al
44、l numbers 1/n, where n = 1, 2, 3, . Then sup E = 1, which is in E, and inf E = 0, which is not in E. 1.10 Definition An ordered set Sis said to have the least-upper-bound property if the following is true: If E c S, Eis not empty, and Eis bounded above, then sup E exists in S. Example l .9(a) shows
45、that Q does not have the least-upper-bound property. We shall now show that there is a close relation between greatest lower bounds and least upper bounds, and that every ordered set with the least-upper-bound property also has the greatest-lower-bound property. , THE REAL AND COMPLEX NUMBER SYSTEMS
46、 S 1.11 Theorem Suppose Sis an ordered set with the /east-upper-bound property, B c S, B is not empty, and B is bounded below. Let L be the set of a/ lower bounds of B. Then ex= supL exists in S, and ot: = inf B. In particular, inf B exists in S. Proof Since B is bounded below, L is not empty. Since
47、 L consists of exactly those y e S which satisfy the inequality y x for every x e B, we see that every x e B is an upper bound of L. Thus L is bounded above. Our hypothesis about S implies the ref ore that L has a supremum in S; call it ex. If y ex then (see Definition 1.8) y is not an upper bound o
48、f L, hence y B. It follows that ex x for every x e B. Thus ot: e L. If ex ex. In other words, ot: is a lower bound of B, but f3 is not if /3 ex. This means that ex= inf B. FIELDS 1.12 Definition A field is a set F with two operations, called addition and multiplication, which satisfy the following s
49、o-called field axioms (A), (M), and (D): (A) Axioms for addition (Al) If x e F and ye F, then their sum x + y is in F. (A2) Addition is commutative: x + y = y + x for all x, ye F. (A3) Addition is associative: (x + y) + z = x + (y + z) for all x, y, z e F. (A4) F contains an element O such that O +
50、x = x for every x e F. (AS) To every x e F corresponds an element -x e F such that X +(-x) = 0. (M) Axioms for multiplication (Ml) If x e F and ye F, then their product xy is in F. (M2) Multiplication is commutative: xy = yx for all x, ye F. (M3) Multiplication is associative: (xy)z = x(yz) for all
