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.
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14 of 15 people found the following review helpful:
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.
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