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82 of 85 people found the following review helpful
5.0 out of 5 stars Very good exposition, great problems
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...
Published on April 3, 2004 by David Hagar

versus
2.0 out of 5 stars Not good as rudin
I brought as a Companion for self-study baby rudin as first my exposure to calculus ,However ,I find the exposition of this book is much harder to understand than Baby rudin. But I mean rudin is a very good text on explanation clearly enough
Published 1 month ago by Crushheavy


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82 of 85 people found the following review helpful
5.0 out of 5 stars Very good exposition, great problems, April 3, 2004
By 
This review is from: Real Mathematical Analysis (Undergraduate Texts in Mathematics) (Hardcover)
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.
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62 of 63 people found the following review helpful
5.0 out of 5 stars Improves on the classic, May 27, 2006
By 
C. Dolan (New York, NY) - See all my reviews
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This review is from: Real Mathematical Analysis (Undergraduate Texts in Mathematics) (Hardcover)
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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24 of 25 people found the following review helpful
5.0 out of 5 stars excellent, June 6, 2003
By 
Dima (United States) - See all my reviews
This review is from: Real Mathematical Analysis (Undergraduate Texts in Mathematics) (Hardcover)
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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18 of 20 people found the following review helpful
5.0 out of 5 stars A thorough text for an advance undergraduate, June 26, 2003
By 
"muon1183" (Berkeley, CA USA) - See all my reviews
This review is from: Real Mathematical Analysis (Undergraduate Texts in Mathematics) (Hardcover)
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.
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10 of 10 people found the following review helpful
5.0 out of 5 stars Pugh is wonderful. Rudin is good too, but both texts working together is the best., May 5, 2008
This review is from: Real Mathematical Analysis (Undergraduate Texts in Mathematics) (Hardcover)
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.
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8 of 8 people found the following review helpful
5.0 out of 5 stars What a breath of fresh air!, December 29, 2006
By 
Shooter McGavin (Atlanta, GA United States) - See all my reviews
This review is from: Real Mathematical Analysis (Undergraduate Texts in Mathematics) (Hardcover)
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.
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11 of 12 people found the following review helpful
5.0 out of 5 stars One of the Best Books in Analysis, or even Maths, June 28, 2010
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.
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3 of 3 people found the following review helpful
5.0 out of 5 stars good intro real analysis - Kindle edition decent, August 26, 2012
By 
Christine "xristy" (Austin, TX, United States) - See all my reviews
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Pugh's book is an introduction to Real Analysis and should not be confused as covering the same material at the same level as Rudin's Principles which keeps getting cited by reviewers. Pugh's book is aimed at students with less sophistication and preparation than that implied by Rudin. Pugh provides a lot of helpful comments along the way and very helpful diagrams that actually communicate some of the subtleties of real analysis. More sophisticated or experienced students may find Pugh too chatty and less experienced students may find Rudin too brusque and less informative. They are for two different audiences.

As for the Kindle version, it is one of the better examples I've found thus far of a book with heavy mathematical content being pretty reasonably formatted using mobi eReader format. There are still line wrap issues where inline equations are broken across lines (these do not occur in the typeset physical versions) and the figures are still inserted as graphics with white backgrounds that conflict with sepia and black backgrounding in the eReader. Further, such graphics do not resize with changes in the font size and if a graphic is zoomed it is not in high resolution. In other words, you will not get as satisfying a reading experience with the Kindle version as you would with the physical version or a print replica.

The print replica versions sometimes supplied in Kindle are PDFs and they can resize and zoom properly. All text, equations and graphics are equally resized. Either the eReader formats such as mobi and ePub don't support actually typesetting technical materials or they are not being properly used to do so with virtually all technical books currently for sale as Kindle - other than the print replica editions.
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2 of 2 people found the following review helpful
4.0 out of 5 stars Problems, Copious Problems!, July 22, 2013
First, let me start off by saying that I do not rate this book highly for the way Pugh writes. I'm one of those people who would much rather have a mathematics book be simple, straightforward, and concise. If you are similar to me in this manner, I highly suggest "The Principles of Mathematical Analysis" by Rudin. Beautifully put together, it definitely stands as one of my favorite books on mathematics.

The reason for my high rating comes from the sets of problems. At the end of each chapter comes a set of problems, whose cardinality is far greater than those in Rudin's book, in which the difficulties range from trivial to rather challenging. The answers to some problems the author claims not to know! So if there is any reason to pick up this book it is for the problem sets.
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2 of 2 people found the following review helpful
5.0 out of 5 stars I love analysis now, April 26, 2013
By 
Kodoku (Toronto Canada) - See all my reviews
This book was used as the textbook for a third year specialist-stream (honours) one-semester course, MAT357, in real analysis at the University of Toronto. We covered all of chapters 2 (topology on metric spaces), 4 (function spaces) and 6 (measure theory and the lebesgue integral).

There are two things that I think make this book great.

1) The exercises. When you complete an exercise in this book, you learn something. It's not some hand-holding through a calculation, or a repetition of an example from the book. You actually gain a piece of insight into the nature of what you're studying. Most of the problems are difficult, but generally require nothing more than an understanding of the definitions and some mathematical maturity. It's relatively rare that there is some lame trick that you either see or don't see, and therefore either solve or don't solve the problem. In other words, the problems are difficult in a good way, and force you to learn in a way that couldn't be accomplished by studying the theorems/proofs for any amount of time.

2) The presentation of the material. In some sense, it's there mostly for the sake of defining the terms used in the problems. You read a problem, check any unfamiliar definitions, and are good to go. If you're stuck, you can look at the proofs of theorems using the concepts to see how reasoning involving those definitions works. Great effort appears to have been made in making the proofs themselves illuminating, rather than just establishing the truth of the claim being proven. And of course, a huge emphasis has been placed on pictures. The picture of a sequence of functions converging uniformly to their limit versus pointwise non-uniform convergence is immediately illuminating. Finally, everything feels like it belongs there. There's no material that feels like a waste of time or irrelevant to the chapter, and every exercise seems to offer a valuable insight.

At the end of the day, after proving 20-40 exercises from each of those chapters, it feels like I get it.
I scored a 50% in second year calculus, yet got very close to 100% in honours third year analysis thanks to this text (though a good prof didn't hurt either).
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Real Mathematical Analysis (Undergraduate Texts in Mathematics)
Real Mathematical Analysis (Undergraduate Texts in Mathematics) by C. C. Pugh (Hardcover - December 12, 2003)
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