这是一部近代的数学名著，一直受到数学界的推崇。作为Rudin的分析学经典著作之一，本书在西方各国乃至我国均有着广泛而深远的影响，被许多高校用做数学分析课的必选教材。本书涵盖了高等微积分学的丰富内容，最精彩的部分集中在基础拓扑结构，函数项序列与级数。多变量函数以及微分形式的积分等章节。第3版经过增删与修订，更加符合学生的阅读习惯与思考方式。
　　本书内容相当精练，结构简单明了，这也是Rudin著作的一大特色。与其说这是一部教科书，不如说这是一部字典。　　

This book is intended to serve as a text for the course in analysis that is usuallytaken by advanced undergraduates or by first-year students who study mathematics.
　　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 attractive to the students who take such a course.
　　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 introduced as an ordered field with the least-upper-bound property, and a few interesting applications of this property are quickly made. However, Dedekind's construction is not omitted. It is now in an Appendix to Chapter 1, where it may be studied and enjoyed whenever the time seems ripe.
　　The material 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 simplified by means of the fixed point theorem about contraction mappings.
Differential 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 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 these 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 to all who have written me.
　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　WALTER RUDIN

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
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 Functions
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 Functiond
Exercises
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 Logarithrmic 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
Primitive Mappings
Partitions of Unity
Change of Variables
Differential Forms
Simplexes and Chains
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 Class 2
Exercises
Bibliography
List of Special Symbols
Index

无