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基本信息
- 原书名：Real and Complex Analysis,Third Edition
- 原出版社： McGraw-Hill
编辑推荐
分析领域内的一部经典著作
体例优美，实用性很强，列举的实例简明精彩
内容简介
作译者
作者：（美国）鲁丁
Walter Rudin，1953年于杜克大学获得数学博士学位。曾行后执教于麻省理工学院、罗切斯特大学、威斯康星大学麦迪逊分校、耶鲁大学等。他的主要研究兴趣集中在调和分析和复变函数。除本书外，他还著有另外两本名著：《Functional Analysis》和《Principles of Mathematical Analysis》，这些教材已被翻译成13种语言，在世界各地广泛使用。
Walter Rudin，1953年于杜克大学获得数学博士学位。曾行后执教于麻省理工学院、罗切斯特大学、威斯康星大学麦迪逊分校、耶鲁大学等。他的主要研究兴趣集中在调和分析和复变函数。除本书外，他还著有另外两本名著：《Functional Analysis》和《Principles of Mathematical Analysis》，这些教材已被翻译成13种语言，在世界各地广泛使用。
目录
Prefac
Prologue：The Ezponential Function
Chapter 1 Abstract Integration
Chapter 2 Positive Borel Measures
Chapter 3 Lp-Spaces
Chapter 4 Elementary Hilbert Space Theory
Chapter 5 Ezamples of Banach Space Techniques
Chapter 6 Complex Measures
Chapter 7 Differentiation
Chapter 8 Integration on Product Spaces
Chapter 9 Fourier Transforms
Chapter 10 Elementary Properties of Holomorphic Functions
Chapter 11 Harmonic Functions
Chapter 12 The Maximum Modulus Principle
Chapter 13 Approximation by Rational Functions
Chapter 14 Conformal Mapping
Chapter 15 Zeros of Holomorphic Functions
Chapter 16 Analytic Continuation
Chapter 17 Hp-Spaces
Chapter 18 Elementary Theory of Banach Algebras
Prologue：The Ezponential Function
Chapter 1 Abstract Integration
Chapter 2 Positive Borel Measures
Chapter 3 Lp-Spaces
Chapter 4 Elementary Hilbert Space Theory
Chapter 5 Ezamples of Banach Space Techniques
Chapter 6 Complex Measures
Chapter 7 Differentiation
Chapter 8 Integration on Product Spaces
Chapter 9 Fourier Transforms
Chapter 10 Elementary Properties of Holomorphic Functions
Chapter 11 Harmonic Functions
Chapter 12 The Maximum Modulus Principle
Chapter 13 Approximation by Rational Functions
Chapter 14 Conformal Mapping
Chapter 15 Zeros of Holomorphic Functions
Chapter 16 Analytic Continuation
Chapter 17 Hp-Spaces
Chapter 18 Elementary Theory of Banach Algebras
前言
This book contains a first-year graduate course in which 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; some of the basic ideas from functional analysis are also included.
Here are some examples of the way in which these connections are demon-strated and exploited. The Riesz representation theorem and the Hahn-Banach theorem allow one to "guess" the Poisson integral formula. They team up in the proof of Runge's theorem. They combine with Blaschke's theorem on the zeros of bounded holomorphic functions to give a proof of the Muntz-Szasz theorem,which concerns approximation on an interval. The fact that L2 is a Hilbert space is used in the proof of the Radon-Nikodym theorem, which leads to the theorem about differentiation of indefinite integrals, which in turn yields the existence of radial limits of bounded harmonic functions. The theorems of Plancherel and Cauchy combined give a theorem of Paley and Wiener which, in turn, is used in the Denjoy-Carleman theorem about infinitely differentiable functions on the real line. The maximum modulus theorem gives information about linear transform-ations on LP-spaces.
Since most of the results presented here are quite classical (the novelty lies in the arrangement, and some of the proofs are new), I have not attempted to docu-ment the source of every item. References are gathered at the end, in Notes and Comments. They are not always to the original sources, but more often to more recent works where further references can be found. In no case does the absenceof a reference imply any claim to originality on my part.
The prerequisite for this book is a good course in advanced calculus (settheoretic manipulations, metric spaces, uniform continuity, and uniform convergence). The first seven chapters of my earlier book "Principles of Mathe-matical Analysis" furnish sufficient preparation.
Experience with the first edition shows that first-year graduate students can study the first 15 chapters in two semesters, plus some topics from I or 2 of the remaining 5. These latter are quite independent of each other. The first 1S should be taken up in the order in which they are presented, except for Chapter 9, which can be postponed.
The most important difference between this third edition and the previous ones is the entirely new chapter on differentiation. The basic facts about differen-tiation are now derived from the existence of Lebesgue points, which in turn is an easy consequence of the so-called "weak type" inequality that is satisfied by the maximal functions of measures on euclidean spaces. This approach yields strong theorems with minimal effort. Even more important is that it familiarizes stu-dents with maximal functions, since these have become increasingly useful in
several areas of analysis.
One of these is the study of the boundary behavior of Poisson integrals. A related one concerns HP-spaces. Accordingly, large parts of Chapters 11 and 17 were rewritten and, I hope, simplified in the process.
I have also made several smaller changes in order to improve certain details:For example, parts of Chapter 4 have been simplified; the notions of equi-continuity and weak convergence are presented in more detail; the boundary behavior of conformal maps is studied by means of Lindelof's theorem about asymptotic valued of bounded holomorphic functions in a disc.
Over the last 20 years, numerous students and colleagues have offered com-ments and criticisms concerning the content of this hook. I sincerely appreciated all of these, and have tried to follow some of them. As regards the present edition,my thanks go to Richard Rochberg for some useful last-minute suggestions, and I especially thank Robert Burckel for the meticulous care with which he examined the entire manuscript.
Walter Rudin
Here are some examples of the way in which these connections are demon-strated and exploited. The Riesz representation theorem and the Hahn-Banach theorem allow one to "guess" the Poisson integral formula. They team up in the proof of Runge's theorem. They combine with Blaschke's theorem on the zeros of bounded holomorphic functions to give a proof of the Muntz-Szasz theorem,which concerns approximation on an interval. The fact that L2 is a Hilbert space is used in the proof of the Radon-Nikodym theorem, which leads to the theorem about differentiation of indefinite integrals, which in turn yields the existence of radial limits of bounded harmonic functions. The theorems of Plancherel and Cauchy combined give a theorem of Paley and Wiener which, in turn, is used in the Denjoy-Carleman theorem about infinitely differentiable functions on the real line. The maximum modulus theorem gives information about linear transform-ations on LP-spaces.
Since most of the results presented here are quite classical (the novelty lies in the arrangement, and some of the proofs are new), I have not attempted to docu-ment the source of every item. References are gathered at the end, in Notes and Comments. They are not always to the original sources, but more often to more recent works where further references can be found. In no case does the absenceof a reference imply any claim to originality on my part.
The prerequisite for this book is a good course in advanced calculus (settheoretic manipulations, metric spaces, uniform continuity, and uniform convergence). The first seven chapters of my earlier book "Principles of Mathe-matical Analysis" furnish sufficient preparation.
Experience with the first edition shows that first-year graduate students can study the first 15 chapters in two semesters, plus some topics from I or 2 of the remaining 5. These latter are quite independent of each other. The first 1S should be taken up in the order in which they are presented, except for Chapter 9, which can be postponed.
The most important difference between this third edition and the previous ones is the entirely new chapter on differentiation. The basic facts about differen-tiation are now derived from the existence of Lebesgue points, which in turn is an easy consequence of the so-called "weak type" inequality that is satisfied by the maximal functions of measures on euclidean spaces. This approach yields strong theorems with minimal effort. Even more important is that it familiarizes stu-dents with maximal functions, since these have become increasingly useful in
several areas of analysis.
One of these is the study of the boundary behavior of Poisson integrals. A related one concerns HP-spaces. Accordingly, large parts of Chapters 11 and 17 were rewritten and, I hope, simplified in the process.
I have also made several smaller changes in order to improve certain details:For example, parts of Chapter 4 have been simplified; the notions of equi-continuity and weak convergence are presented in more detail; the boundary behavior of conformal maps is studied by means of Lindelof's theorem about asymptotic valued of bounded holomorphic functions in a disc.
Over the last 20 years, numerous students and colleagues have offered com-ments and criticisms concerning the content of this hook. I sincerely appreciated all of these, and have tried to follow some of them. As regards the present edition,my thanks go to Richard Rochberg for some useful last-minute suggestions, and I especially thank Robert Burckel for the meticulous care with which he examined the entire manuscript.
Walter Rudin
媒体评论
书评
书中还附有大量设计巧妙的习题——这些习题可以真实地检测出读者对课程的理解程序，有的还要求对正文中的原理进行论证。
书中还附有大量设计巧妙的习题——这些习题可以真实地检测出读者对课程的理解程序，有的还要求对正文中的原理进行论证。
