国外优秀数学教材选评（第三章 数学分析和泛函分析）

数学分析是数学专业的最重要的基础课，是理论性较强的微积分，它以严格的极限理论 作为基础。我国的综合性大学数学系学生一年级的课程一般都将数学分析列为重点课程，复旦大学安排三个学期的数学分析课。美国大学数学系一般分两步：先让学生修和非数学专业一样的微积分，即 Calculus（属于 lower division 的课程），再修数学分析（属于 upper division 的课程），所化的时间大致上差不多。到底哪种方法好也没有定论，大概对中学数学基础较差的学生按照美国的办法容易接受一些，事实上我国数学系一年级的数学分析有不少学生是吃不消的。
数学分析的英文书名大致有三种： mathematical analysis, real analysis, advanced calculus. 可能有些差别，一般说来，advanced calculus 和我国的数学分析比较接近，适合没有微积分基础的数学专业学生学习， mathematical analysis 多半是针对学过初等微积分的人， real analysis 更深一些，接近我国的实变函数论。
泛函分析是数学分析的自然延伸，它的基础除了数学分析以外还需要线性代数和实变函数论 和少量的复变函数论。外国的有些数学分析教材中也包含测度和勒贝格积分等泛函分析中需要用的内容，所以很多学校在本科生阶段就不设实变函数课了。

书名：Introducton to Analysis
作者： Arthur Mattuck
出版商： Prentice Hall (1999) ISBN 0-13-081132-7
页数：460
适用范围：大学数学系本科基础数学学生教材
预备知识：微积分初步知识
习题数量：大
习题难度：较大
推荐强度：9.8

书评: 本书是麻省理工学院的 Arthur Mattuck 教授教授这门课程多年经验的基础编写而成的，是一本实分析的优秀入门教程，深受读者欢迎。 本书主要讲述单变量函数的分析理论，侧重于讲述实数理论的基本思想，特别是用分析的方法对函数进行估计。 本书从基本的实数理论讲起，内容主要包括数列与函数的极限和连续性，级数理论，微分理论，Taylor展开， Riemann积分理论， Lebesgue积分理论等等。本书的一个鲜明的特点是，对书中的定理不只是叙述，而是从来源讲起， 对读者以启发为主，侧重于揭示数学思想。 例如，对微积分的两个基本定理，其证明较一般书中繁琐，但是其证明给出了微积分的重要思想， 即积分是微分的无穷积累， 微分是积分的局部化，并且，还分析了两个基本定理之间的关系。另外，书中还给出许多重要的应用。

本书比较适合作为我国综合性大学数学系实分析课程一学年的外文教材，也可以作为 程度较好的数学系本科生进一步深化实分析概念的课外读物。 （王泽军）

国外评论摘选

1) This is an unusual and beautifully written introduction to real analysis. The presentation is carefully crafted and extremely lucid, with wonderfully creative examples and proofs, and a generous sprinkle of subtle humor. The layout of the pages is exceptionally attractive. The author has clearly put a great deal of thought and effort into producing an analysis text of the highest quality. Most of the book concentrates on real-valued functions of a single (real) variable. There is a gradual and careful development of the ideas, with helpful explanations of elementary matters that are often skipped in other books. For instance, prior to the chapter on limits of sequences, the book has a chapter on estimation and approximation, discussing algebraic laws governing inequalities, giving examples of how to use these laws, and developing techniques for bounding sequences and for approximating numbers. Proofs involving epsilons and arbitrarily large n make their first appearance here.
The overall presentation of the book is carefully thought out. Each chapter is broken up into small sections, and each section emphasizes one principle idea or theorem. The proofs of the main theorems are lovely, and give both intuitive explanations and rigorous details. Genuinely interesting examples and problems illuminate the key ideas. Each chapter contains a mix of problems: questions that help students test their grasp of the main points of each section, exercises that are intermediate in scope, and more difficult problems. (A solutions manual is available for instructors from the publisher.)
The careful explanations, even of elementary matters, and two appendices on sets, numbers, logic, and methods of argumentation, make the book suitable for a first analysis course in which students have had no prior exposure to proofs. There is ample material for a one-semester, or in some cases a one-year, course.
In summary, I believe that this is the best introductory real analysis book on the market. Students and instructors alike will find it a joy to read.

2) The book is slow to begin but it does a great job in explaining all the concepts. The author explains the proofs and theorems and it introduces some intermediate ideas to understand the theorems and definitions. The book contains a lot of exercise of different nature and difficulty. It covers a great range of subjects but not enough on the Rn. The book is basic in it contain, it is not difficult to read and follow. It can serve as an introduction to analysis. I would recommend it if you want an introduction to analysis.

书名：Mathematical Analysis, Second Edition
作者： Tom M. Apostol
出版商： Addison Wesley (1974), 机械工业出版社影印
页数：492
适用范围：大学数学系本科生
预备知识：高中数学
习题数量：中
习题难度：中等
推荐强度：9.8

[作者简介] Tom M. Apostol, 美国数学家，生于犹他州。他于1946年在华盛顿大学西雅图分校获得数学硕士学位，于1948年在加州大学伯克利分校获得数学博士学位，1962年起任加州理工学院教授，美国数学会、美国科学发展协会（A.A.A.S）会员。对初等数论和解析数论有研究，他的著作很多，除本书外，还著有《Calculus, One-Variable Calculus with an Introduction to Linear Algebra》、 《Calculus, Multi-Variable Calculus and Linear Algebra with Applications》等。

书评: 本书第一章以公理化的方式引入了实数系和复数系，接下来介绍了集合论和点集拓扑的一些基本概念和内容，为后面微积分理论的展开打好基础。从第四章开始，作者开始介绍极限、连续和导数等微积分的基本概念。在第六章作者引入了有界变差函数与可求长曲线的概念，接着就对Riemann-Stieltjes积分进行了介绍，而Riemann积分则是它的特例。第八第九章是对级数和函数序列知识的讲解。第十章介绍Lebesgue积分，第十一章介绍Fourier级数以及Fourier积分，第十二章介绍多元微分学，第十三章介绍隐函数与极值问题，接下来的两章是关于多重Riemann积分与Lebesgue积分的介绍，最后一章介绍了复变函数的Cauchy定理以及留数的计算。 本书是一部现代数学名著：自20世纪70年代面世以来，一直受到西方学术界、教育界的广泛推崇，被许多知名大学指定为教材。作为一本大学数学系的本科教材，本书仔细而又不累赘地向读者介绍了微积分的思想，涵盖了数学分析绝大部分的基本知识点，并配有覆盖各级难度的练习题，适用于初次接触数学分析的读者。无论对于教学还是自学，都不失为一本理想的教材。另一方面，本书对于实分析和复分析中的部分内容也有所介绍，这其实也是很多美国大学数学教材（Mathematical Analysis或者Advanced Calculus）内容设置的共同点。例如作者在第十章有对Lebesgue积分的介绍。不过与一般实分析教材里的思路不同，作者采用了Riesz-Nagy的方法引入了Lebesgue积分，此方法直接着眼于函数及其积分，从而避免了对于测度论知识的要求；同时作者还进行了简化、延伸和调整，以适应大学本科水平的教学。 （徐晓津）

国外评论摘选

1) If you're the type of person who likes crisp and clear proofs but don't want to have the proofs be as skinny as Rudin's then this is the perfect book. Apostol's writing style is not only accessible and clear but the organization of the text is excellent too. There are plenty of problems with a good mix of difficulty levels. He also throws in an example here and there to give you firm footing on some difficult topics. If I had to recommend one analysis text this would be it.

2) I own analysis texts by Apostol, Rudin, Bear, Fulks, Protter, and Kosmala. This one by Apostol gets my vote as the best all-around text on the subject. It's rigorous, elegant, readable, and has just the right amount of explanatory text. This would be my first choice as an undergraduate textbook, a self-study text, or as a supplemental reference to another text. I also recommend Bear for his elegance and witty style, and Kosmala for his thorough explanations. But if you are going to buy only one, make it this one.

3) I've never been a big fan of Apostol. He tends to make things more difficult than they really are. Some of the reviewers commented that they are impressed with the elegance of the proofs, which makes me wonder if they are as confused as Apostol. As an example let's consider his proof of the FTC. There is an easy and elegant proof which you find in most books, but Apostol tries to be cute and gives an obscure and ugly proof. Mathematics is an art, and Mr. Apostol is no Picasso.

书名：Principle of Mathematical Analysis, 3rd edition
作者： Walter Rudin
出版商： McGraw-Hill (1976), 机械工业出版社影印
页数：334
适用范围：数学系一、二年级学生与理工科高年级学生
预备知识：高中数学，最好具备微积分的初步知识
习题数量：287 道习题，较大
习题难度： 较难，但是很多有难度的题目有提示
推荐强度：9.5

[作者简介] Walter Rudin 1953年于杜克大学获得教学博士学位。曾先后执教于麻省学院、罗切斯特大学、威斯康星大学麦迪逊分校、耶鲁大学等。他的主要研究领域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《Real and Complex Analysis》，这些教材已被翻译成13种语言，在世界各地广泛使用，以本书作为教材的名校有加利福尼亚大学伯克利分校、哈佛大学、麻省理工学院等。

书评: 本书前二章介绍了从高中数学到大学数学过渡中的基本知识：实数与复数理论，基础拓扑理论。第三章介绍数列与级数。第四章介绍函数的连续性。第五章介绍微分的概念。第六章介绍Riemann-Stieltjes积分的概念。第七章介绍了数学分析中很重要的一个概念：函数序列与函数项级数的一致收敛性。在第八章作者列举了几个特殊的函数项级数，如幂级数、Fourier级数等作专门讨论。第九章介绍多变量函数。第十章介绍了微分形式的积分。在最后第十一章对勒贝格积分作了初步的介绍。
本书内容相当精练，结构简单明了，这是Rudin著作的一大特色。例如在第六章积分部分，作者直接介绍了Riemann-Stieltjes积分，而一般数学分析课程中的Riemann积分就是它的特例。书中的习题经过了精心挑选，有助于学生掌握数学分析的基本概念及提高逻辑推理的技巧。本书第3版经过了增删与修订，更加符合学生的阅读习惯与思考方式。 本书适合作数学系学生学习数学分析课程的参考书，也适合作为具有一定微积分知识的理工科高年级学生提高分析水平与能力的教材。 本书是一部现代数学名著，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、函数序列与函数项级数、多变量函数以及微分形式的积分等章节。
[零星感悟] 作者从学生的角度出发来考察问题的接受难易程度，并在整本书的结构上做了精心的安排和调整。 比如说，从理论上讲，从有理数的概念出发引入实数的概念是非常正常和符合逻辑的，但是Rudin通过以往的教学经历发现学生对这样的做法不容易接受，因此Rudin从有序集与具有上（下）确界的性质入手来介绍实数，显得简洁而具有新意。 在第九章多变量函数中，一个关键的问题就是反函数存在定理的证明。记得以前看过的书上证明都比较复杂。在此书中，Rudin利用压缩映射的不动点理论，大大简化了证明过程。 (刘东弟)

国外评论摘选

1) OK... Deep breaths everybody...
It is not possible to overstate how good this book is. I tried to give it uncountably many stars but they only have five. Five is an insult. I'm sorry Dr. Rudin...
This book is a good reference but let me tell you what its really good for. You have taken all the lower division courses. You have taken that transition to proof writing class in number theory, or linear algebra, or logic, or discrete math, or whatever they do at your institution of higher learning. You can tell a contrapositive from a proof by contradiction. You can explain to your grandma why there are more real numbers than rationals. Now its time to get serious.
Get this book. Start at page one. Read until you come to the word Theorem. Do not read the proof. Prove it yourself. Or at least try. If you get stuck read a line or two until you see what to do.
Thrust, repeat.
If you make it through the first six or seven chaptors like this then there shall be no power in the verse that can stop you. Enjoy graduate school. You half way there.
Now some people complain about this book being too hard. Don't listen to them. They are just trying to pull you down and keep you from your true destiny. They are the same people who try to sell you TV's and lobodemies.
The material is not motivated. Not motivated? Judas just stick a dagger in my heart. This material needs no motivation. Just do it. Faith will come. He's teaching you analysis. Not selling you a used car. By the time you are ready to read this book you should not need motivation from the author as to why you need to know analysis. You should just feel a burning in you chest that can only be quenched by arguments involving an arbitrary sequence ${x_n$ that converges to $x \in X.$
Finally, some people complain about the level of abstraction, which let me just say is not that high. If you want to see abstraction grab a copy of Spanier's 'Algebraic Topology' and stare at it for about an hour. Then open 'Baby Rudin' up again. I promise you the feeling you get when you sit in a hottub for like twenty minutes and then jump back in the pool. Invigorating.
No but really. Anyone who passes you an analysis book that does not say the words metric space, and have the chaptor on topology before the chaptor on limits is doing you no favors. You need to know what compactness is when you get out of an analysis course. And it's lunacy to start talking about differentiation without it. It's possible, sure, but it's a waste of time and energy. To say a continuous function is one where the inverse image of open sets is open is way cooler than that epsilon delta stuff. Then you prove the epsilon delta thing as a theorem. Hows that for motivation?
Anyway, if this review comes off a combative that's because it is. It's unethical to use another text for an undergraduate real analysis class. It insults and short changes the students. Sure it was OK before Rudin wrote the thing, but now? Why spit on your luck? And if you'r a student and find the book too hard? Try harder. That's the point. If you did not crave intellectual work why are you sitting in an analysis course? Dig in. It will make you a better person. Trust me.
Or you could just change your major back to engineering. It's more money and the books always have lots of nice pictures.
In conclusion: Thank you Dr. Rudin for your wonderfull book on analysis. You made a man of me.

2) What has been said below is all true. Rudin really does have some excellent moments in this book, except perhaps the chapter on Lebesque integration, which is one of the crappiest expositions of the topic I have found so far. Get the texts by E. Stein (Real analysis), or Bartle's small book on lebesgue integration. There is even a probability text by A.Shiryaev (Probability, 2nd edition) which has a truly amazing treatment of the Lebesgue integral. Anyhow, the rest of the book is excellent, concepts of single-variable analysis are very well explained, the proofs are short and enlightening. The multivariable calculus is also very well explained. I gave it 3 stars because it's not a good book for self-study. There are hardly any explanations, the beginning student will likely get really frustrated. In order to enjoy this book you either have to know analysis already, so this would be a second text, or you have to take a course that uses Rudin as textbook. Then you won't have any problems, since the teacher will probably end up explaining the stuff in class. I have heard complaints even by some of the world's best mathematicians at Princeton against the fact that Rudin's text is so terse. To me that's not even impressive. It's just arrogance or laziness on the behalf of the author. So don't feel too bad if you read it and find that things aren't explained very well; that's because they aren't! The false sense of reward comes from banging your head against the wall before finding the answer, and being thankful you finally got somewhere rather than committing suicide (only thing is, if anything, you may have just reinvented the wheel...) But by then, you'll have wasted a lot of time already. If you have nothing else to do, and are incredibly patient, this is no problem at all, otherwise, it's a real waste of your time. You could also be a genius, in which case, none of these points are even an issue; then you can prove all theorems in the world, so congratulations, I look forward to meeting you! Oh yeah, and there are no solutions to the problems provided to check if your answers are right or not, so good luck.

书名：Advanced Calculus, 2nd Edition
作者： Patrick M. Fitzpatrick
出版商： Brooks/Cole (2005), 机械工业出版社影印
页数：590
适用范围：数学系与理工科其他专业的本科生
预备知识：高中数学
习题数量：较大
习题难度： 具有一定难度
推荐强度：9.3

[作者简介] Patrick M. Fitzpatrick拥有格兰特大学博士学位，是纽约大学科朗研究所和芝加哥大学的博士后，1975年进入马里兰大学College Park分校任教，现在是数学系教授和系主任，同时它还是巴黎大学和佛罗伦萨大学的客座教授。他的研究方向是非线性泛函分析，在该方向著有50多篇论文。
书评: 本书以清晰、简洁的方式介绍了数学分析的基本概念：第一部分讲述单变量函数的微积分，包括实数理论、数列的收敛、函数的连续姓和极限、函数的导数和积分、多项式逼近等；第二部分把微积分的概念推广到多维欧几里得空间，讨论多变量函数的偏导数、反函数、隐函数及其应用、曲线积分和曲面积分等。 数学分析已经根植于自然科学和社会科学的各个学科分支之中，微积分作为数学分析的基础，不仅要为全部数学方法和算法工具提供方法论，同时还要为人们灌输逻辑思维的方法，本书在实现这一目标中取得了引人注目的成果。本书一方面按传统的和严格的演绎形式介绍微积分的所有主题，另一方面强调主题的相关性和统一性，使读者受到数学科学思维的系统训练。 本书的一大特点是除了包含必不可少的论题，如实数、收敛序列、连续函数与极限、初等函数、微分、积分、多元函数微积分等以外，还包含其他一些重要的论题，如求积分的逼近方法、Weierstrass逼近定理、度量空间等。例如本书专门用一章讨论度量空间，从而把在欧几里得空间讨论微积分时使用的许多概念和导出的结果扩展到更抽象的空间中，引导读者作广泛深入的思考。 另外，与第一版相比，第二版增加了200多道难易不等的习题。全书贯穿了许多具有启发性的例题，并且本版还为教学考虑进行了许多实质性的改动，例如将选学材料与前后内容的关联度降到最低，单独放置，既不影响教学和读者自学的进度，又能让读者集中攻破一些难点，这样使得全书的叙述更简洁、更自然。本书曾于2003-2004年作为马里兰大学教材。 （高威）

国外评论摘选

1) A great book. Starts with two very good chapters on linear algebra, adapted to the needs of calculus, and then proceeds to introduce you to the contemporary way to do multivariate calculus, including existence theorems connected to completeness. Very thorough treatment of integration, including integration of forms on manifolds, up to the Stokes theorem, built upon a fine chapter on differential manifolds, exterior differential forms, riemannian metrics, etc. Good illustrations and beautiful typesetting add to the joy of reading it. Plenty of exercises and chapters on applications to physics and differential geometry.

2) This is the best book on mathematics I've ever come across. The superbly written text succeeds in guiding the reader in an easy, clear-cut, graceful way through the realm of what he modestly calls Advanced Calculus. Some minor misprints are to regret, but they don't even come close to blurring the fact that this is - no doubt about that - an unsurpassable masterpiece.

3) As Spivak's Calculus on Manifolds, this book is labeled with a very modest title. It should be something as All you wanted to know about analysis on manifolds but were afraid to ask. This book is a must-reading for the analyst. It covers everything from the most basic vector space concepts up to the fundamental theorems of classical mechanics, running through multivariate calculus, exterior calculus, integration of forms, and many topics more, always keeping a very modern and rigorous style.
The undergraduate may find it a little difficult, but the effort is worth it. For the graduate student and the working mathematician it is an almost-daily reference.

4) This book is out of print, but is available from Sternberg's website. Search on his full name at Google.

书名：Advanced Calculus, 5th Edition
作者： Wilfred Kaplan
出版商： Addison Wesley (1991), 电子工业出版社影印
页数：741
适用范围：理工类本科高年级学生与研究生
预备知识：高中数学和初步的微积分学基础
习题数量：大
习题难度： 有难度
推荐强度：9.5

[作者简介] Wilfred Kaplan于1939年在Harward大学师从Hassler Whitney 获得博士学位，后任教于Michigan大学。
书评:
本书除了全面地介绍微积分的知识，还介绍了线性代数、矢量分析、复变函数、以及常微分方程、偏微分方程等方面的知识。全书共分为10章：前两章介绍了线性代数和偏微分；第三章介绍了散度、旋度和一些基本的恒等式，还介绍了n维空间中的张量；第四、五章介绍了积分理论，包括定积分、重积分、曲线积分、曲面积分、Stokes公式等；第六章介绍级数理论；第七章介绍Fourier级数理论；第八章介绍复变函数的解析理论；第九章介绍了常微分方程理论；第十章介绍了偏微分方程。本书内容丰富，编写深入简出，在每一章都有相当篇幅的内容打了*号，这些内容属于基础理论的深化与拓广，可供教师教学时选用，或供基础好的学生选读。 本书的前身是作者应他的一位工程学同事的建议所著，目的是让工科学生在掌握初等微积分的基础上进一步扩充数学知识，提高数学水平与能力。初稿写成后，曾用于工科大三学生的教学。付诸印刷后，被Michigan大学指定为理工科高年级学生的教材。 因为本书的写作初衷是提供给工科学生，并且作者认识到数值方法具有实用价值和帮助读者更深入的了解微积分理论，所以本书不仅介绍了理论知识，还涉及到相关数值方法，这也是本书的一个特点。 本书另一个特点是十分方便读者自学自测。比如说，本书中的定义都有明确标示，所有的重要结果都作为定理以公式的形式给出；书中不仅提供了大量难易不同的习题，更给出了习题答案；此外，还提供了大量的参考文献，并在每章的末尾给出了推荐阅读的书目。 本书为方便教师安排教学进度，注意在各章有机联系的同时，尽量减少每章节对前面章节知识的依赖程度。作者还在序言中为一学期每周四小时的课时提供了具体的教学安排建议。 (高威)

国外评论摘选

1) This book is simply the best that I have found for math texts. Kaplan does not expect much from the reader; he explains basically everything besides Calculus I material. Kaplan's writing is lively and is (relatively) easy to read. He gets to the point and keeps everything easy to follow. I am still in awe about how much material (look below) he was able to fit into this relatively small book and still keep it so clear. The examples are clear and concise. The problems in the book compliment the understanding of the material; they start out easy and guide the reader to do more difficult problems. This book is MORE THAN SUFFICIENT FOR SELF-STUDY.

2) Any student who is taking analysis/advanced calculus course should read chapter 2 of this book, especially if he is confused or is struggling on the excellent but relatively abstract/concise texts of Rudin, Apostal, Bartle, Marsden et al. I've never seen a book which can explain the concept of Jacobian and Inplicit function theory in such a clear way!!

3) It is good and clear book. Excellent for undergrad students who want to dig into calculus a bit deeper. But it is too easy for an advanced undergrad or a grad student in any technical field. I recommend the books published by Springer.

书名：Advanced Calculus
作者： Lynn H. Loomis, Shlomo Sternberg
出版商： Jones & Bartlett Pub (1989)
页数：592
适用范围：大学数学系本科生教材
预备知识：高中数学
习题数量：大
习题难度： 中等
推荐强度：9.6

[作者简介]Shlomo Sternberg，美国Harvard大学教授，他于1957年在约翰霍普金斯大学获得博士学位。Shlomo Sternberg是一位杰出的数学家，尤其因他在微分几何上的贡献而闻名。

书评:
本书第零章是关于集合、映射等基础知识，接下来对向量空间作了介绍；第三章引入了微分的概念，接下来作者又对紧性、完备性和点积空间进行了介绍。第六章是有关微分方程的简单讲解，第七章介绍了多重线性函数，第八章引入了积分，第九第十章介绍了可微流形以及流形上的微积分问题，第十一章介绍了外微分，第十二章介绍了位势理论，而最后一个章节对微积分在经典力学上的应用作了介绍，向读者展现了数学的威力。 本书是一部优秀的分析教材。与一般的微积分教材不同，它大体上可以分为两个部分：第一部分介绍了赋范向量空间上的微分知识，第二部分主要介绍了可微流形上的微积分知识。本书既有基础的章节，例如第一第二章对于向量空间的介绍，也有对于读者而言要求比较高的内容，比如第九章中关于切空间和李导数的概念。作者在用朴实的语言向读者介绍微积分的概念和思想的同时，也尽可能地展现了不同的观点：例如对于隐函数存在定理的证明，作者就给出了三种证明方法，揭示了数学的魅力。本书的另一大特色在于丰富的习题，练习题的题量大，并且难度不一，作者还把一些重要定理的证明放在了习题中，因此对于读者而言，尽可能多地完成书后习题可以更好的把握和巩固知识，提高分析能力。 本书可以根据教学的需要选取部分章节，程度较好的数学系本科生也可选用此书作为微积分的课外读物。 （徐晓津）

书名：Problems and Theorems in Analysis
作者： George Polya and Gabor Szego
出版商： Springer Verlag (1978)
页数：第1卷389页；第2卷391页；共780页
适用范围：数学专业高年级学生与研究生，数学教师与数学工作者
预备知识：数学分析，高等代数，复变函数
习题数量：第1卷776道；第2卷884道。这是一套习题书
习题难度： 难，有的习题甚至为研究者的最新成果，难度很大
推荐强度：9.8

[作者简介] George Polya（1887-1985）匈牙利数学家，早年在苏黎世瑞士联邦理工学院任教，后入美国籍，1942年起在美国Stanford大学任教。Polya 在数学的广阔领域里都有深入的研究，特别在泛函分析、数理统计和组合分析等方面尤为突出。Polya不仅是数学家，也是一为优秀的教育家，他始终把高深的数学研究和数学的普及与教育结合起来。
Gabor Szego（1895-1985）匈牙利数学家，早年在柯尼斯堡大学任教，后入美国籍，也在美国Stanford大学任教。他主要的贡献是在数学分析与数理方程方面。 《分析中的问题与定理》一书是George Polya 与 Gabor Szego 最著名的著作。Polya曾经这样评论他与Szego的合作：这是一段美妙的时光；我们专心致志、充满热情地工作。我们有着同样的背景。我们象同时代其他匈牙利数学家一样，受到Leopold Fejér的影响。我们都是那个为中学生创办的强调解题的刊物Hungarian Mathematical Journal 的读者。我们又对同样的课题、同样的问题感兴趣，但往往是一个人对某一个课题知道得多，而另一个人对其他的课题知道得多。这是一次绝妙的合作。我们的合作成果-《分析中的问题与定理》，是我最好的工作，也是Szego最好的工作。

书评:
本书两卷，共分九个部分。第一部分主要收录无限序列与无限级数方面的问题。第二部分是有关积分的各种问题。第三、第四部分是关于单复变量函数的问题，内容包含了数学系本科生与研究生的复分析课程中的主要问题。第五部分主要涉及代数的零点确定问题。第六部分讲多项式与三角多项式。第七部分为行列式与二次型的问题。第八部分为数论方面的题目。第九部分为数学中与几何有关的一些问题。 本书与其说这是一部教科书，不如说这是一部字典，因为它收录了分析学中的各种问题和定理。这是一本有着突破传统意义的书。它对问题巧妙的系统性安排与归纳，给学生创造了自主性思考的可能，最大程度上启发学生的研究能力和创新能力，这也是它不同于其他一些平庸的习题参考书的地方。作者甚至试图用很多哲学的观点来阐释它所选出的题目的代表性，比如有关特殊和一般的问题，要知道早期著名的数学家迪卡尔曾经说过：我学数学是为了追求最终的哲学。正是这种理念的融入，使得这本书在学术界的地位尤为突出，不只是学生，很多教授和数学工作者都以此书为参考书，并对此书给予了高度的好评。
[零星感悟] 什么是好的教育？给学生一套完善的体系然后让学生在这样的体系下寻找机会自己去发现和解决问题，这样的完善的体系才是好的教育的关键。此习题书不同于其他习题参考书的特点也就在此。它给我们数学系高年级学生与研究生提供了在不同主题下精心安排的问题，启发我们独立思考和研究问题的能力，是一本不可多得的分析习题书籍。 第一部分的习题139让我们明白了很多问题就像两个点决定一条直线一样，是有两个极端的线性组合而得出的结论。 第六部分的习题92让我们明白了掌握一个领域的知识就像了解一个城市的所有交通路线。真正的掌握就是从任何一个出发点，你都可以找到最短的路线达到你想要达到的目的。 (刘东弟)

书名：Functional Analysis
作者： Walter. Rudin
出版商： McGraw-Hill Book Company ISBN: 0-07-054225-2
页数：397
适用范围：数学类专业本科高年级学生和研究生
预备知识：数学分析 复分析 实分析 线性代数
习题数量：中等
习题难度： 中等偏难
推荐强度：9.5

书评： W.Rudin的《泛函分析》是一本分析数学方面的经典名著，多年来一直被国外一些高校用于研究生教学。 全书由三部分组成，第一部分是线性泛函分析基础，作者在线性拓扑空间的框架下建立了开映射定理、 闭图像定理、逆算子定理、共鸣定理和线性泛函延拓定理等基本定理，介绍了赋范线性空间的对偶性、 紧算子的概念与性质。作为这些理论的应用，作者还专辟一章介绍了Stone-Weierstrass定理、插值定理、 不动点定理、紧群上的Harr测度等知识。第二部分介绍了Fourier变换和广义函数理论， 并给出了这些理论在微分方程方面的应用。第三部分在Banach代数的基础上， 介绍Hilbert空间上有界正规算子的谱理论，并进一步建立了无界正规算子的谱定理，最后还介绍了 类算子半群。第一部分是全书的基础，第二部分和第三部分则是可供平行阅读的两个独立部分， 读者可根据需要选择使用。全书叙述严谨，条理清晰，理论的展开较为详尽。 该书既可用作泛函分析课程的教材，也可供数学工作者查阅参考。（童裕孙）

国外评论摘选
1) Hardly can I find words to highlight the goodness of this book. As mentioned by other readers ,it provides elegant, direct and powerfool proofs of the three theorems which constitute the cornserstones of functional analysis (Hanh-Banach, Banach-Steinhaus and Open mapping). These theorems are, in addition, studied in their most general context, namely topological vector spaces.
Specially appealing is its treatment of distributions' theory. It is, as far as I know, the only text which start by defining the rigurous topology on the set of test functions and then obtains the convergence and continuity of functionals (distributions) in terms of this topolgy, which is, indeed, the only way to present and gain insight into these concepts and to reach some results such as completness. In doing otherwise one risk definitions can emerge as artificial and rather arbitrary.
It is, without any doubt, a must have book for those with interest in pure mathematics as well as for those who, eventually, realize that the only way to dominate their area is saling through mathematics.

2) No other book covers the elements of distributions and the fourier transform quite like Rudin's Functional Analysis. This is a must for every budding PDE-er!

3) I enjoy perusing Rudin's Functional Analysis at this stage in my life. It is fairly nice tome for functional analysis, and its general treatment of topological vector spaces (as opposed to the standard Banach space examples studied in a typical functional analysis class) is now well-received.
However, as a student, I was put off by this book. At times, I found it difficult to tie the theory present to the basic examples which were relevant at the time (such as $L^{p$ spaces). For a first time learner, I would suggest the book of Kolmogorov and Fomin (which is a Dover book, by the way), and would wait until later for this book.