实分析与复分析（英文版·第3版）（独家销售）(09年度畅销榜NO.2)

　　 本书是分析领域内的一部经典著作。毫不夸张地说，掌握了本书，对数学的理解将会上一个新台阶。全书体例优美，实用性很强，列举的实例简明精彩。无论实分析部分还是复分析部分，基本上对所有给出的命题都进行了论证。另外，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度，有的还要求对正文中的原理进行论证。在第3版中，作者对一些新的课题进行了讨论，并力求全书条理清晰。本书适合作为高等院校数学专业研究生教材。
　　
　　

preface
prologue: the exponential function
chapter 1 abstract integration
set-theoretic notations and terminology
the concept of measurability
simple functions
elementary properties of measures
arithmetic in [0, ]
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

　　 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