目录
5 60 本教材点评
5.5 实分析 4 本
《Fourier Analysis》
作者：Elias M. Stein; Rami Shakarchi
出版商：Princeton University Press
出版年：2003
ISBN：9780691113845
适用范围：高年级本科生，研究生
推荐强度：9
作者简介：Elias M. Stein （1931-2018），美国数学家，是调和分析的领军人物之一，自1963年至去世一直担任普林斯顿大学的数学教授。他于1949年毕业于史岱文森高中，之后进入芝加哥大学学习。1955年施泰因在Antoni Zygmund的指导下获得芝加哥大学博士学位并开始在麻省理工学院授课。1958年成为芝加哥大学助理教授，1963年成为普林斯顿大学终身教授。他在1984年和2002年获得斯蒂尔奖，于1993年获得肖克奖，于1999年获得沃尔夫奖，于2001年获得美国国家科学奖章。此外他还是美国国家科学院院士。2005年获得伯格曼奖以表彰他在实分析、复分析以及调和分析领域的贡献。2012年成为美国数学学会会士。
Rami Shakarchi，在耶鲁大学获学士学位，在普林斯顿大学数学系获博士学位。毕业后从事金融工作的同时继续编写《普林斯顿分析讲座》（Princeton Lectures in Analysis），该系列强调分析分支之间的统一性以及分析在其他数学领域的适用性。
书评：
Every once in a while, I am struck by how often mathematics textbooks sound just like each other. Glance at the table of contents of your typical analysis or algebra textbook, and you can be 90% sure of seeing exactly the same sequence of topics each time. There are honorable exceptions of course, and I'm always glad to see one, because they indicate that (at least some) mathematicians are still actively thinking about what should be taught, in what order, and how.
So here is the first volume in the Princeton Lectures on Analysis, entitled Fourier Analysis: an Introduction and written by Elias M. Stein and Rami Shakarchi. The series wants to serve as an integrated introduction to the core areas in analysis. The following volumes will treat complex analysis (volume 2), measure theory, integration, and hilbert spaces (volume 3), and selected other topics (volume 4). The basic pre-requisite for the series seems to be a standard undergraduate introduction to analysis covering the basic theory of convergence, derivatives, and the Riemann integral. Some basic familiarity with the complex numbers and elementary functions (e.g., complex exponentials) is assumed. So the book is aimed at graduate students and maybe advanced undergraduates.
The new series begins with Fourier analysis because the authors feel that this subject plays a central role in modern analysis and because it has played an important historical role. It is also much more concrete than abstract measure theory or functional analysis. Finally, the authors plan to use results from volume one in the following volumes, emphasizing that analysis is a coherent whole rather than a collection of disjointed topics.
The first book covers the basic theory of Fourier series, Fourier transforms in one and more dimensions, and finite Fourier analysis. The last topic allows the authors to present, as an application, the proof of Dirichlet's theorem on primes in arithmetic progressions. The result would make a great book for independent study courses with advanced undergraduates, and, I think, would also be useful for graduate courses. It's definitely worth a look.
每隔一段时间，我都会为数学教材之间的相似程度而震惊。浏览一下你的分析或代数经典教材的目录，有90%的把握能看到完全相同的内容设计。当然，也有例外，我总是很高兴看到这样的例外，这些例外是值得尊敬的，因为它们表明（至少有些）数学家仍在积极思考应该以何种顺序或何种方式来教授数学知识。
这是普林斯顿大学分析讲座系列书籍的第一卷，名为《Fourier Analysis: an Introduction》，由Elias M. Stein与Rami Shakarchi撰写。这个系列希望综合性地介绍分析学的核心领域。接下来的几卷会介绍复分析（第二卷），测度论，积分，以及Hilbert空间（第三卷），与一些其他内容（第四卷）。
阅读此系列书籍基本的先决条件是本科生级别的分析基础，包括收敛性，微分与黎曼积分，此外同样也需要熟悉复数与初等函数（如指数函数）。所以这本书是面向研究生与高年级本科生的。
这一新系列从Fourier分析开始，因为作者们认为这部分内容是现代分析领域的核心，并且也同样具有重要的历史意义。它同样也比抽象的泛函分析更加具体。最后，作者们想要在系列中后几卷中使用本书的结论，以强调分析是一个连贯的整体，而不是一些琐碎主题的拼凑。
本书包括Fourier级数的基本理论，一维与高维情形的Fourier变换，和有限Fourier分析。最后一个主题使得作者能够作为应用证明描述等差数列上素数分布的Dirichlet定理。本书是一本很好的教材，适合高年级本科生来研习，同样也对研究生的课程有所帮助，它绝对值得一看。
点评人：Fernando Q. Gouvêa（科尔比学院数学教授）
This book is the best book on real analysis I have ever studied. It does a wonderful job in bridging undergraduate level with graduate level analysis. I have not seen any book that makes measure and Lebesgue theory so easy to understand.
The books begin by defining what a measure is all about. And the description is so intuitive and geometrical that you would wonder why you weren't taught it this way before. The book then goes into Lebesgue theory and all of it suddenly becomes so easy.
The book has plenty of wonderful examples and a good set of over 30 problems per chapter.
Elias Stein (one of the authors) is a very renowned mathematician, and one need not worry about the accuracy of the proofs in the book--they are bullet-proof, and at the same time succinct.
If you are struggling with W. Rudin's book on Analysis, this book is a MUST for you.
这本书是我所学过的关于实分析的最好的书。它在衔接本科阶段与研究生阶段的知识方面做得很好。我从未看过一本使测度论和Lebesgue理论如此容易理解的教材。
这本书从如何定义一个测度开始。而且这种描述是如此的直观和易于理解，以至于你会想为什么你之前不是被这样教的。随后它开始介绍Lebesgue理论，并且这一切突然变得十分简单。
这本书有很多精彩的例子，每章有30多个问题。
Elias Stein（作者之一）是一位著名的数学家，因此读者不必担心书中证明的准确性——这些证明既严谨又简洁。
如果你正努力阅读W. Rudin的分析教材，这本书对你就必不可少了。
资料整理：方诗雨
翻译：赵旭彤
《Measure Theory》
作者： Donald L. Cohn
出版商：Birkhäuser
出版年：2013
ISBN：9781461469551
适用范围：高年级本科生，研究生
推荐强度：9
作者简介：Donald L. Cohn，萨福克大学数学与计算机科学系教授，获哈弗大学硕士学位与博士学位，主要研究领域：复变函数、Banach空间中的极值问题、Banach空间几何、近似理论等。
书评：
In this second edition, Cohn has updated his excellent introduction to measure theory (1st ed. 1980, reprinted in 1993, MathSciNet MR578344) and has made this great textbook even better. Those readers unfamiliar with Cohn’s style will discover that his writing is lucid. It is a treat to behold such a wonderfully put-together book that has certainly been carefully edited and copy-edited, and seems to be (almost incredibly for a book this size) free from typos.
Cohn’s text appears to be not as well-known as, say Folland’s Real Analysis or Rudin’s Real and Complex Analysis — both of which contain introductions to measure theory but then move on to cover distinct terrains. Beyond the basics of measure theory with a dose of functional analysis thrown in for good measure: Folland treats Fourier transforms, distribution theory and probability; while Rudin seamlessly transitions to a full course on complex (and some harmonic) analysis ending with introductions to H^p-Spaces and Banach algebras. Another famous textbook with comparable content would be Royden’s Real Analysis, but the latter (whose third edition I am familiar with; there is now a fourth edition) is horribly blasé when it comes to statements and proofs of various theorems. For example, there were often, especially towards the latter half of Royden’s text, instances of redundant hypotheses in theorems, and a careful study or (re)discovery of their proofs would often lead to cleaner and tighter material. There are those who believe this “flexibility” to be among Royden’s advantages. In contrast, Cohn is precise in both his statements and proofs throughout the text. All his Is have been dotted and his Ts crossed. This I (currently) believe to be a virtue, especially when exposing the art of analysis to a first-year graduate student, or even a precocious senior. Good habits should be taught early!
Cohn’s textbook, as the title promises, contains a good deal more about measure theory proper — with a view to applications in probability theory and stochastic processes. I can imagine that his attention to detail and mathematical precision may lead more than a few (graduate student) readers to find Cohn somewhat dry. The first edition started right off the bat with abstract measure theory: the definition of a sigma algebra, measures, outer measures, etc. The second edition has some more motivation in the form of an introductory chapter that reviews the Riemann integral, a few classic pitfalls that led to Lebesgue’s theory, and an outline of the main topics and structure of the text. Pedagogically speaking this seems to me an excellent strategy, though my own introduction to measure theory was abstract measure theory first; only much later was there talk of Lebesgue and Riemann. Among other new additions to the second edition, there are concise and useful appendices on the Banach-Tarski paradox and the Henstock-Kurzweil integral.
To echo Mark Kac’s infamous apocrypha, Cohn adds some soul in the form of his entirely new Chapter 10 on Probability Theory. In under 70 pages he covers fundamental results such as the strong law of large numbers, the central limit theorem, the martingale convergence theorem, the construction of Brownian motion and Kolmogorov’s consistency theorem. This is treated as a chapter where a large collection of previously built measure-theoretic tools are (finally?) applied. The reviewer would liked to have seen a section on Ergodic Theory in this new chapter — to provide the budding researcher with another attractive area of application with various strands of active ongoing research interest. A chapter on Geometric Measure Theory would also provide a nice complement to the material developed in this text. But one shouldn’t make one’s wish list, or the text, too long.
Special mention should be made of Cohn’s excellent Chapter 8: “Polish spaces and Analytic Sets’, which covers ground that is often missed in most analysis texts, but is indispensable for analysts/probabilitsts who would like to work on complete separable metric spaces. This material is not altogether absent from the literature. It may be found, e.g., in the first chapter of K. R. Parthasarathy’s Probability Measures on Metric Spaces (1967), reprinted by AMS Chelsea Publishing; or in S. M. Srivastava’s A Course on Borel Sets (Springer, 1998), which has a more leisurely exposition.
To summarize, this is a wonderful text to learn measure theory from and I strongly recommend it.
在第二版中，Cohn已经更新了他对测度论的介绍（第一版，于1980年编写，1993年再版），并使这本优秀的教材变得更好。那些对Cohn的风格不熟悉的读者会觉得他的文字清晰易懂。阅读这本精心编排的书是一种享受，而且似乎（对于这么大的一本书来说几乎难以置信）没有打字错误。
Cohn的书似乎不像Folland的《Real Analysis》或是Rudin的《Real and Complex Analysis》那么有名，这两本书都介绍了测度论，但随后介绍了不同的内容。除了测度论的基础知识外还有着大量的泛函分析的内容，以便建立起对测度论更好的理解：Folland介绍了Fourier变换，分布理论与概率论；Rudin则自然地过渡到复分析（以及少许调和分析），并在最后介绍了H^p-空间和Banach代数。另一本有着与之相当内容的著名教材是Royden的《Real Analysis》，但后者（我对第三版更加熟悉；而如今已有第四版）在涉及到定理的陈述和证明时无聊得可怕。例如，有很多关于书中定理多余条件的例子，特别是在Royden的书的后半部分，对证明仔细研究，重新思考或许会使内容更加简洁紧凑。也有些人认为这种“灵活性”是Royden的书的优势之一。相比之下，Cohn在整本书中的定理陈述与证明是十分精确的。我（目前）认为这是一种良好的习惯，尤其是当一个一年级的研究生，甚至是一个早熟的大四学生学习分析学的艺术时。好习惯应该趁早教！
Cohn的书，正如书名所承诺的那样，包含了大量关于测度论的内容，以及在概率论和随机过程中的应用。我可以想象，他对细节和数学精确性的关注，可能会让更多（研究生）读者觉得有些枯燥。第一版从抽象的测度论开始：sigma代数、测度、外测度等。第二版则在介绍性的章节中增加了一些研究与学习的动机，复习了Riemann积分并说明了它的局限性，并以此引出Lebesgue理论，并概述了本书的主要内容和结构。从教育的角度来说，这似乎是一种不错的方式，尽管我自己介绍测度论时一定会从抽象的测度论讲起；随后才会谈论Riemann与Lebesgue. 第二版新增的其他内容包括了一些简明却实用的附录，介绍Banach-Tarski悖论和Henstock-Kurzweil积分。
为了呼应Mark Kac的著作，Cohn精心撰写了有关概率论的第10章。在这70多页中，他介绍了一些基础的结论，比如强大数定律，中心极限定理，鞅收敛定理，布朗运动的构造和Kolmogorov一致性定理。这被视为一个将先前构建的测度论工具加以应用的章节。有些读者希望在新的章节中看到遍历论相关的内容——为初出茅庐的研究者提供另一个十分有吸引力的领域和各式各样的研究主题。关于几何测度论的一章也是对其他内容很好的补充。
值得一提的是第8章：波兰空间与解析集，它涵盖了大多数分析教材中经常遗漏的内容，但对于那些希望研究完备可分度量空间的数学家来说，这是必不可少的。
文献中并非完全没有这些内容。例如，可以在K.R.Parthasarathy的《度量空间的概率测度》（1967）的第一章中找到，它由AMS Chelsea出版社再版；或者在s.M.Srivastava的《关于Borel集的课程》（Springer，1998）中找到，相比本书有着更通俗易懂的解释。
总而言之，这是一篇学习测度论的好教材，我强烈推荐它。
点评人：Tushar Das（威斯康星大学拉克罗斯分校数学助理教授）
资料整理：方诗雨
翻译：赵旭彤
《Introduction to Real Analysis》
作者：Robert G. Bartle, Donald R. Sherbert
出版商：Wiley
出版年：2011
ISBN：9780471433316
适用范围：高年级本科生
推荐强度：9
作者简介：Robert G. Bartle（1927-2003），美国数学家，专攻实物分析。他以撰写由约翰·威利父子（John Wiley＆Sons）发行的热门教科书《实在分析的要素》（1964年），《整合的要素》（1966年）和《实在分析简介》（2011年）而闻名。从1955年到1990年，他在伊利诺伊大学数学系任教。从1976年至1978年以及1986年至1990年担任《数学评论》的执行编辑。从1990年至1999年，他在东密歇根大学任教。1997年，他的论文“重返黎曼积分” 获得了美国数学协会的写作奖。
Donald R. Sherbert，生于1935年，伊利诺伊大学香槟分校荣誉教授。于1957年在威斯康星大学攻读学士学位，于1962年在斯坦福大学攻读博士学位1985年获LAS院长卓越教学奖，1996年，获本科教学卓越奖。
书评：
See P. N. Ruane's review of the third edition, in which he says I am very impressed with this book by Bartle and Sherbert. The new edition seems to have preserved the book's many virtues.
One of the changes with this edition is that Robert Bartle has passed away; the new edition is dedicated to his memory. Bartle was also the author of a more advanced text, Elements of Real Analysis, which is in the MAA's list of recommendations for undergraduate mathematics libraries. Introduction to Real Analysis is intended as an undergraduate textbook offering a first exposure to (single-variable) real analysis. It even has an appendix on proofs and logic, though the author argues that it is a more useful experience to learn how to construct proofs by first watching and then doing than by reading about techniques of proof. (I agree!)
In the preface, the author says that this edition maintains the same spirit and user-friendly approach as earlier editions. Every section has been examined. Some sections have been revised, new examples and exercises have been added, and a new section on the Darboux approach to the integral has been added to Chapter 7. (The Darboux approach uses upper and lower sums instead of Riemann sums, but the resulting integral is equivalent to the Riemann integral.)
Two interesting features of the earlier editions have been retained. First, notions of topology (metric spaces, compactness, etc.) are postponed until the very last chapter. Second, the authors include an introduction to the generalized Riemann integral (aka the Henstock-Kurzweil or gauge integral).
This is a solid introductory textbook that takes great pains to help students achieve the necessary ability and handling formal arguments. Instructors who are content in staying in a one-variable setting will want to consider it for course adoption.
正如P. N. Ruane对本书第三版的评论所述——这本由Bartle和Sherbert撰写的书给我留下了深刻的影响。新的一版保留了此前版本的许多优点。
这一版的发行一定程度上是为了纪念逝世的Robert G. Bartle. 他也同样是一本更加专业的教材《Elements of Real Analysis》的作者，该书在美国数学协会的本科生数学教材推荐列表中。《Introduction to Real Analysis》是一本本科生级别的教材，更适合那些初次接触（一元）实分析的读者。尽管作者认为，读者第一次阅读本书时，可以尝试自己构建证明的逻辑并自行证明，这样或许比直接阅读证明技巧更有用（我也是这么认为的！），但本书仍有一节关于证明逻辑的附录。
在前言中，作者提到此版延续了初版的思想，对读者十分友好。每一章节都经过仔细审查。作者修改了一些章节，增加了新的例题和习题，并在第7章增添了关于Darboux积分法的内容（Darboux积分法使用Darboux上下和代替Riemann和，但得到的积分与Riemann积分等价）。
初版中两个有趣的地方得以被保留。首先，拓扑学相关的概念（度量空间、紧性等）被安排在最后一章。其次，作者介绍了广义Riemann积分（又称Henstock-Kurzweil积分）。
这是一本扎实的入门教材，它极力帮助学生获得解决问题所需的能力。如果教师们只需要教授一元分析学，那这本书将是一个不错的选择。
点评人：Fernando Q. Gouvêa（缅因州沃特维尔科尔比学院的卡特数学教授，《 MAA评论》的编辑）
书评：
This book provides a solid introduction to real analysis in one variable. The first two chapters introduce the basics of set theory, functions and mathematical induction. Also, the properties of real numbers are introduced here borrowing the concept and properties of field from abstract algebra.
The following chapters deal with sequences and series of numbers, limits, continuity, differentiation, integration, sequences and series of function, in this order.
I think the material is presented clearly and the results are proven rigorously throughout the entire book. There are a lot of worked-out examples and many exercises that will test the reader's understanding. Solutions and hints to many (notice, not only the odd ones) of the problems are given in the back of the book. There is also an appendix on logic for those who might need to review the basics, and one on metric spaces and Lebesgue integrals for those students who want to go a bit farther.
In my opinion, this book is not as good as Rudin's book, but it does the job better than many other introductory books on the same topic. For a horrible book see Jiri Lebl's text.
Real analysis is hard, independently of the book you use. It requires a lot of care and hard work. This book does the best it can at clearing the path for you.
这本书是一元分析方面的扎实的入门教材。前两章介绍有关集合论、函数与数学归纳法的基本知识。同时，借由抽象代数中的域的概念介绍了实数的性质。随后几章按顺序讨论了数列与数项级数、极限、连续性、微分、积分、函数列与函数项级数。
我认为全书表达清晰，证明严谨，配以足够的例题与习题来考查读者对正文内容的理解。书后给出了习题的提示与解答（注意，不仅仅包括那些难以解决的习题）。书后有一个关于逻辑的附录，供那些可能需要复习基础知识的人使用，还有一个关于度量空间和Lebesgue积分的附录，供那些想进一步学习的学生使用。在我看来，这本书不如Rudin的书好，但它比许多其他介绍相同内容的书籍做得更好，尤其比Jiri Lebl的教材好多了。
无论你用哪本书，实分析的学习都是很困难的，都需要大量的时间与精力。这本书已经尽其所能为你扫清学习实分析的阻碍了。
点评人：Christian Farina（阿勒格尼县社区学院副教授）
资料整理：方诗雨
翻译：赵旭彤
《Complex Analysis》
作者：John M. Howie
出版商：Springer Verlag
出版年：2003
ISBN：9781852337339
适用范围：本科生
推荐强度：9
作者简介：John M. Howie（1936-2011），苏格兰数学家杰出的半群理论家。1967-1973年任苏格兰考试委员会数学小组成员；1975–1981年任苏格兰数学中央委员会主席；1982-1988及1989-1992年任伦敦数学学会理事会成员，1986–1988及1990–1992年任副主席，1985–1989年任教育委员会主席，1990–1992年公共事务委员会主席；1987–2001年任北方教育学院州长；1987–1993年任苏格兰数学委员会主席；1991-1997年任国际数学科学中心指导委员会主席。
书评：
For better or for worse, I am becoming more and more of an anglophile as I read and study more and more papers and books by mathematical authors from other national origins. It's a bit of a wild opinion, I know, and probably politically incorrect (whence an even more attractive proposition) to hold that there is such a thing as a national character and a corresponding national style. Nonetheless, there is strong evidence that such enviable stylistic elements as clarity and elegance of expression are somehow more common among mathematical authors from the British Isles (but Jean-Pierre Serre is the great Gegenbeispiel to my thesis, of course). In my own student days, now long ago, I recall being enthralled by Burkill's book Lebesgue integration, by Hardy and Wright (i.e. their classic Introduction to the Theory of Numbers), and by Titchmarsh on analytic functions. There was something very special about the way these authors expressed themselves, conveying emotions somehow, even as they headed straight for the heart of the subjects under consideration. Prudence dictates that I desist from naming examples from the other side, i.e. standard sources which despite their importance are all but unreadable. (I'm sure we all have our own candidates.)
It is with this perhaps some what controversial axiom posited that I now come to John M. Howie's Complex Analysis.
Well, it's more of the same! Howie's book is a gem. I want to use it the next time I teach complex analysis. Not only do Howie's selection of topics and their sequence correspond perfectly to what I believe to be the ideal approach to this gorgeous subject, the writing style is (again) wonderful. Consider the following sample: Since we shall require Cauchy's Theorem and its consequences for contours that are neither convex nor polygonal, it becomes a duty on the author's part to present a proof of a more general case. Whether there is a corresponding duty on the reader's part is left to individual conscience! There is no doubt, however, that useful skills follow from the mastery of substantial proofs. All in all a wonderful example of sound pedagogy by merely dropping the right hint. So many contemporary texts quickly embrace condescension and proceed mainly to annoy the reader.
As regards technical points, the book is split into twelve chapters, each of which is split into a relatively small number of short and sweet sub-sections which can be easily used to build individual lectures. It's nigh on a perfect textbook in this way. There are also a number of wonderful ideological passages; see e.g. 2.1 Are complex numbers necessary? and 3.1 Why is complex analysis possible? Beyond this there are the right number of exercises for complex analysis: not too many, not too few, a huge number of proofs in them, but also such things as descriptions of multi-valued functions, determinations of specific Möbius transformations, etc., etc. And here is problem 4.18
「Comment, on the mathematical rather than the literary content, of Little Jack Horner sat in a corner. Trying to work out π.
He said, It's the principal logarithm
Of (-1)-i.
」
It doesn't get much better than that!
So, clearly, I think this is a terrific book. I'm going to use it the first chance I get. And I recommend it very, very highly.
不知是好是坏，当我阅读和研究越来越多来自其他国家的数学家的论文与书籍时，我越来越感觉自己像一个英国人。我知道，认为存在民族性格和相应民族风格的观点有些偏激，而且可能并不“政治正确”。尽管如此，有强有力的证据表明，清晰而优雅的语言表达在不列颠群岛的数学家中更为常见。很久以前还在上学时，我曾被Burkill的《Lebesgue integration》、Hardy和Wright的《An Introduction to the Theory of Numbers》和Titchmarsh关于解析函数的书所吸引。这些作者在表达自己的思想时直奔核心，却也有一些不同之处，在字里行间夹杂着自己的感情。为了谨慎起见，我不想举反面例子，即那些标准化且无个人色彩的书籍（我相信我们或多或少都能想到一些），尽管它们很重要，但却几乎读不下去。也许正因为此，我现在才来看John M. Howie的《Complex Analysis》.
Howie的书如同宝石一般。在下次教复分析时，我想将其用作教材。Howie对内容的选择和顺序安排与我理想中的完全一致，他的写作风格也令人赞叹。不妨考虑下面这个例子：由于我们要求将柯西定理及其推论用于一般区域，即既不要求区域为多边形，也不要求边界为凸，因此给出更一般情况的证明成为了作者的责任，而读者是否有相应的责任取决于个人。然而毫无疑问，实用的方法技巧源于对严谨证明的理解与掌握。从而只要给出正确的提示来帮助读者理解，它就能成为一个好的教学案例。因此，许多现代教材很快就接受了以更高层次的视点来阐述定理及其证明，主要是为了帮助读者理解相应内容。至于书的结构，这本书分为十二章，每一章都被分成几个相对简短的小节，甚至每一小节都可以作为专题讲座的主题。通过这种方式，它成为了一本接近完美的教科书。书中还有一些精彩的思想性的段落。如2.1节：复数是必要的吗？以及例3.1：为什么复分析是可能实现的？除此之外，还有适当数量的复分析习题：不多不少，其中有大量的证明题，但也有关于多值函数的描述，特定Möbius变换的定义等。
所以，很明显，我认为这是一本很棒的书。我一有机会就会翻翻它，同时我也极力推荐这本书。
点评人：Michael Berg（洛约拉马利蒙特大学数学教授）
资料整理：方诗雨
翻译：赵旭彤