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Real Mathematical Analysis

Posted by Carissa Leos on Monday, 10 May 2010
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Great Price "Real Mathematical Analysis" for $46.96 Today



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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Real Mathematical Analysis Overviews

In this new introduction to undergraduate real analysis the author takes a different approach from past presentations of the subject, by stressing the importance of pictures in mathematics and hard problems. The exposition is informal and relaxed, with many helpful asides, examples and occasional comments from mathematicians such as Dieudonne, Littlewood, and Osserman. This book is based on the honors version of a course which the author has taught many times over the last 35 years at Berkeley. The book contains an excellent selection of more than 500 exercises.

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Customer Review


I prefer Bartle and Sherbert - Danial Lee Sellers - Indianapolis, IN
I used this book in my first Real Analysis course and thoroughly disliked it. I seems that everyone else who reviews this book mentions Rudin. I haven't had a chance to read Rudin yet but I prefer Intro to Real Analysis by Robert G. Bartle and Donald R. Sherbert over this book. Many people like Pugh for it's conversational tone but I found it annoying. This might be a good secondary book but I wouldn't recommend this as your first book in real analysis. Pugh makes the cardinal mistake of mathematicians in introducing the most general case first. The most important thing in mathematics is not the most general case but the process of generalization itself. This is like saying the journey is as important as the destination. To generalize one must start with a specific case and then work, layer by layer, to the most general case. That's one reason I prefer Bartle and Sherbert. It starts with functions from R to R and generalizes from there. It takes up where undergraduate calculus leaves off. I also prefer the exercises in Bartle and Sherbert better. They are challenging without being infuriating. They are still general proofing exercises but are specific enough to deal with specific functions, series, sequences, and so forth. I was also annoyed by the way Pugh qualifies his proofs like Chapter 1 Theorem 2: "Proof Easy" ,or theorem 9: "Proof Tricky!", or Chapter 2 Theorem 10:"Proof, Totally natural!". I feel his language is imprecise and sloppy. I feel the section on cuts is superfluous. It seems that cuts are a lot of work and headaches just to prove that everything I learned in elementary school is correct. I was worried that x+0 didn't really equal x but now with cuts, I can rest assured that it does! Whew, what a relief! The only plus to Pugh is the thorough chapter on metric spaces helps put things into a broader context. All and all I dislike Pugh's book and highly recommend Bartle and Sherbert. as the best introduction to Real Analysis.


Excellent problems and diagrams -- great book - Jesse Taylor - North Idaho
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 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.



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