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Real Mathematical Analysis (Undergraduate Texts in Mathematics) 1st Edition

4.2 out of 5 stars 25 customer reviews
ISBN-13: 978-0387952970
ISBN-10: 0387952977
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Product Details

  • Series: Undergraduate Texts in Mathematics
  • Hardcover: 440 pages
  • Publisher: Springer; 1 edition (November 14, 2003)
  • Language: English
  • ISBN-10: 0387952977
  • ISBN-13: 978-0387952970
  • Product Dimensions: 6.1 x 1 x 9.2 inches
  • Shipping Weight: 1.6 pounds (View shipping rates and policies)
  • Average Customer Review: 4.2 out of 5 stars  See all reviews (25 customer reviews)
  • Amazon Best Sellers Rank: #339,095 in Books (See Top 100 in Books)

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Top Customer Reviews

Format: 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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Format: 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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Format: Hardcover Verified Purchase
First, beware: Amazon offers this book as a "print on demand" (POD) product, so the print quality of the text is not going to look as nice as that copy of the book your university library has. The library's copy would have been printed in a more civilized time -- around 12 years ago -- before the POD scourge ravaged the planet. In particular about half of the 133 illustrations in the book are going to look like complete crap. This is why I'm returning my copy to the crap factory from whence it came, because I've had enough of POD scum. I take some measure of pride in the books that I have on my shelf, and am always willing to pay extra for a hardcover textbook; but what's the use if what's between the covers looks like it was printed with an eighty-dollar inkjet set on economy mode?

Amazon's villainous POD offering is not why I knocked two stars off Pugh's shoulder, however. Here's what cost one star: in the middle of the proof of the Implicit Function Theorem it says, "In general, the idea is that the remainder R depends so weakly on y that we can switch it to the left hand side of (8), absorbing it in the y term." Okay, that sort of hand-waving "argument" might pass muster in a freshman calculus book, but it really has no place in a book titled "Real Mathematical Analysis." I had to wonder why, at first glance, Pugh's proof of the theorem seemed so short, with more prose than symbols. Now I know why. Then Pugh goes on to prove the Inverse Function Theorem using the Implicit Function Theorem, and it's even shorter, with the statement "Except for a little fussy set theory, this completes the proof" being the cherry on top.
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Format: 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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