Customer Reviews

Real Mathematical Analysis

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Real analysis is a genre with an established classic (Rudin) and a plethora of available books and resources. Unfortunately, most analysis books cost a great deal of money so the average reader will only purchase one or two texts. In evaluating which book(s) to purchase two questions should be asked:

1.) Why purchase this book rather than the classic of the genre?

2.) Is this book appropriate for me?

So why buy this book rather than Rudin? It has great exposition (as does Rudin), very well chosen problems (as does Rudin), but Pugh manages to improve on the standard by supplementing his written explanations with diagrams and pictures that Rudin mostly lacks. Additonally, the price stands at something less than half the cost of Rudin's book.

Who is this book appropriate for? This text delves into the topological underpinnings of analysis. It is not an "analysis-lite" textbook a la Ken Ross's Elementary Analysis. It is a rigorous treatment of the subject, and it has a comprehensive feel to it, covering topics like Lebesgue measure and integration, and multivariable analysis in addition to the normal topics one would expect. In short, it is appropriate for somebody who is seeking the challenges and rewards of a full treatment of what for many is a difficult subject.

It is a very good book that does not shy away from difficult material that no amount of explanation or good writing will make easy to learn, but of all the analysis books I've seen, this comes the closest.

As a previous reviewer has noted, Walter Rudin's Principles of Mathematical Analysis is the standard textbook for a rigorous analysis course. Rudin's book is very good because of the level of rigour and abstraction, the bredth of material covered, the way it forces the reader to fill in the blanks, and because of the challenging exercises throughout. In my opinion, Pugh has managed to improve on the classic in every aspect.

First of all, he does not develop all the concepts in same order as Rudin - first he develops the real number system, a few basic things about Cauchy sequences, and then moves onto continuity. Then he goes into a lengthy chapter on topology, which, in my humble opinion, is where the book first outshines Rudin. He defines compactness in terms of the convergence of subsequences, which is much more natural than the covering definition. He later proves that the two conditions are equivalent. In the third chapter, he develops differentiation and integration, much in the way Rudin does. In the fourth chapter, develops series and sequences (of functions). In the fifth chapter, he develops multivariable calculus, and the in the sixth chapter, he develops measure theory and the Lebesgue integral. Since there are fewer chapters than there are in Rudin's book, I think he develops the subject matter in a more natural, cohesive manner.

Rudin's book is excellent through the series and sequences of function. It is generally agreed that the book tails off after the seventh chapter, that is, he does not do as good a job with multivariable calculus and Lebesgue Theory. Pugh manages to do a good job throughout, so in addition to having a better chapter in topology, he is better than Rudin in those areas. I also believe that his treatment of series and sequences of functions is more interesting: Rudin treats them, for the most part, as distinct mathematical objects, and only briefly makes reference to the space of functions, whereas Pugh centers the chapter around the idea of function spaces (the heart of real analysis, really). Furthermore, Pugh uses illustrations (not too many, but enough) to illustrate certain concepts, and in fact, to simplify certain proofs. He also emphasizes the utility of geometric thinking in developing proofs, something which Rudin does not do. Furthermore, Rudin is notoriously terse; I think Pugh does a better job motivating and explaining the material without being "chatty" (the cardinal sin in mathematical exposition), while not insulting the reader's intelligence, that is, you are expected to fill in certain gaps on your own.

I would also like to emphasize the quality of the exercises in this book. There are many, many exercises - more than PMA, in fact. None of them are trivial. Many of them are quite challenging, on par with those in Rudin's book. Unlike Rudin, though, Pugh includes a fair share of easier, but still interesting exercises, which I think are essential for really getting a grasp on the material. He also has some problems, I think, which are a good bit harder than any of Rudin's, which is saying a lot, so there is something for everyone here.

Overall, I think this is the best book out there for an intro to analysis course. The price is also quite reasonable, considering how much math books tend to cost.

This is one of the best books on introductory real analysis that I have looked at. Before I found this book, I have been reading another work on real analysis which was also very good, but was far less comprehensive. Not only does this book present a precise exposition of concepts and theorems, it also gives illustrations to better explain the ideas and plenty of excercises at the end of each chapter. For example, the author does not only say what a "covering" means, but he gives an illustration of it. The style of exposition is fine and relaxed, but the rigor of presentation of theorems and proofs is not in the least compromised. I would think that this book will be of enormous help to anyone trying to make a transition from concrete to more abstract mathematical reasoning.

Having taken Pugh's honors analysis course, in which he used this book, I can strongly reccommend it to any student interested in the subject of analysis, especially students seeking to learn more than the average introductory real analysis book contains. Pugh's book contains advanced theorems and topics not often found in undergraduate level texts. Additionally, the problems are well thought out and tend to be of a high level of difficulty.

I wish that I had discovered Pugh in my first semester of undergraduate analysis. The assigned text was Rudin and it was a great choice. The exposition there is excellent. The exercises are incredibly well done. Pugh covers just about the same material as Rudin, and in the same rigor, but is more likely to give you paragraphs before and after important theorems/definitions that help to clarify things. I must admit I am not too familiar with the first half of Pugh's text as I didn't discover it until I was well in chapter 10 of Rudin ~~ chapter 5 of Pugh. But, if the first chapters are as good as the fourth and fifth, you can get just as much from Pugh as from Rudin, if not more.

Sometimes, you get a picture (this would have been really helpful back when I was learning what an open cover was). Other times, Pugh actually gives a better presentation. For instance, when discussion the rank theorem, Rudin's statement of it is hard to follow. The proof is about as difficult. Pugh, however, introduces C' equivalence and then gives an alternate statement of the theorem which is much more intuitive. AND some pictures after the proof. Some think having pictures in analysis books is bad--Pugh gives evidence otherwise.

It is difficult to say which text has better exercises as I have not attempted them all. But Pugh definitely has more of them. I think the best thing for any undergraduate to do is to just own both books. Rudin is the standard for a good reason. Pugh's or someone else's exposition may become the standard in the future, but Rudin will always be an excellent reference. Doing Rudin's exercises will help prepare you for your qualifying exams if you ever take them. Pugh has some UC Berkeley good prelim exam questions in his book which prepare you for future math endeavors as well. So I say just buy both. But if you can only buy one.... probably get Pugh because he's cheaper. Or you can get International Edition Rudin for cheaper still.

I dont know how to say how good this book is: it not only teaches us the technical aspects of mathematics, it also teaches us intuitions, ideas behind the proofs, styles, and philosophy.

I would like to also start with a comparison of the classic baby Rudin:

While Rudin's little book is also a real gem, I would say Pugh's belongs to a slightly higher level (due to its problems mainly and topic selection and coverage). Rudin's book could either be used in 1st year one semester as a strong first course in basic analysis (1st 7 chapters) for extremely motivated and hardworking students (such as those at MIT, Harvard, Princeton, and many other good institutions), or ought to be supplemented in an honour's undergrad real analysis (in 2nd year or 3rd year). Its presentation of Lebesgue theory is rather incomplete and no one virtually uses it for lebesgue theory. On the other hand, pugh has a full chapter on it, covering almost all the standard undergrad lebesgue materials.

Pugh's book on the other hand can be the last reading before attempting Folland and Big Rudin. Knowing pugh well and having solved its problems would make Folland and big Rudin not hard, whereas little rudin may not surve this purpose that well. Many Rudin's problems are hard but standard (Prove, Show this. Very few is it true? what about?), whereas Pugh's is more thought provoking (Is it true? What about? What do you think? which mimics a key part in maths research).

Moreover. mathematics is not just about formalism and logic, especially in analysis ang geometry. The ideas and our feeling about how the objects behave are at least equally important. (Anyone can write proofs well with sufficient training; yet not everyone feels that a measurable function is no more complicated than continuous ones in a sense; why lebesgue's definition of length and integral are powerful; weirstrass approximation is as simple as "taking expectations of functionals", etc.) Amazingly Pugh's book trains people to this direction very well.

Every once in a while, a mathematics book comes along that gets it right. While most math departments are filled with irrelevant, arcane texts; every once in a while a text will appear that is genuinely fun to read. Casella/Berger's fantastic statistics text is one of these; Pugh's Analysis is another one.

I cannot overstate how much I enjoyed this book - no matter what book your math department is using for undergraduate analysis, I would recommend picking this one up and reading it on your own.

This is an excellent introductory text on real analysis. It is very approachable, and he does a very good job at supplementing the traditional "definition-theorem-proof" style with intuitive explanations and wonderfully descriptive diagrams (the diagrams are one of the strongest points of this book -- and are something that are sadly left out of many otherwise good books on analysis).

My only (minor) complaint is with the layout/formatting of the book -- it is very jumbled together, the typesetting is poor, and it looks like it was printed on a low-resolution $10 printer.

Other than that, it is an excellent companion to a more in-depth/advanced treatment. As far as more "advanced" books go, I would recommend -- Apostol's "Mathematical Analysis" and/or Shilov's "Real and Complex Analysis" -- both of which are incredibly well written and informative.

The style is friendly and fun, and the presentation is really intuitive! My personal favorite!

Great textbook, great afterchapter exercises! However, since the exercises are a bit challenging, it would have been better if solutions or hints for solutions are provided. By the way, I would be grateful if anyone could tell me where I can find solutions for the exercises.