Customer Reviews

Real Analysis (Dover Books on Mathematics)

5 star:		(3)
4 star:		(3)
3 star:		(0)
2 star:		(1)
1 star:		(2)

 

 

Yes this book is an excellent introduction to real analysis and assumes minimal preparation beyond linear algebra and two semesters of calculus. I have been looking everywhere for a clear and lucid treatment of the Banach Fixed Point Theorem, and here it is. At a time when most math textbooks (even paperback) cannot be had for less than $50, Dover has been a godsend for those of us who want to collect a decent mathematics library. If it were not for Dover Publications, I would not have this text in my library, nor would I be able to afford it. Please visit Dover Publications website to view all the other fine mathematics texts which they offer at bargain prices. Incidentally, all these fine Dover mathematics books that can be had for chump change are reprints of original texts that have long been out of print and hard to find. To a previous reviewer who critized Dover Publications, I would simply say: Don't bite the hand that feeds you.

As an undergraduate math major with knowledge of only some linear algebra and elementary calculus of one and several variables, I found this text to be interesting and challenging. The chapter on metric spaces serves as a good introduction to concepts in point-set topology, while providing motivation for such studies. While the proofs are rigorous and complete, sometimes the developments seem to lack motivation. This can be annoying when attempting the exercises, but motivation for such developments could easily be provided by examples from other texts or a professor. After studying Stewart's "Calculus" and Bartle and Sherbert's introductory analysis text, I find the rigor and thoroughness of this text most refreshing. For instance, rather than assuming the completeness property of the reals, the authors develop the reals as an equvalence class on the rationals, and proceed to prove the completeness property. I am certain that anyone interested in learning analysis could benefit greatly from this text, especially in combination with other analysis texts.

I don't see why everyone buys other books on the subject for ten times the price when they can just get this one. This book is REALLY good. It really teaches the material in a way which is both clear and understandable and completely rigorous. Moreover, the excercizes, which range from easy to difficult, are all quite instructive. I'd recommend this book to anyone.

This book can serve as an important bridge between books like baby Rudin and big Rudin. Like baby Rudin, this book assumes only the basics from calculus and linear algebra (it is fairly self-contained) and covers the basics on convergence, continuity, differentiation, uniform convergence, etc. It then goes on to cover many topics in the first half of big Rudin like Lebesgue integration, Banach spaces, and Hilbert spaces. The style and tone of the book is sophisticated, and prepares the reader for the arid tone of big Rudin. On the other hand, this book always tries to develop topics in the most elementary way. For example, the Lebesgue theory is developed via the Daniell method on R^n and then, in a brief separate section, the general theory is sketched, leaving many proofs to the reader. I liked this approach, because working in R^n is comfortable and the proofs extend to the general case in an obvious way. Another example is the Riesz representation theorem, which is done on the real line with a very intuitive proof. In contrast, big Rudin is really a book to marvel at once you already know something about its contents. This book is ideal preparation for big Rudin because after reading it, you will know in essence what Rudin wants to say and basically why it is true. But big Rudin will show you how these results extend to more general settings with extremely elegant (although sometimes baffling) proofs. You should also note that when I was at Chicago they were using this book, so the big guys and gals must like it too.

This is the first real analysis text I've studied, and so I should begin by saying that, for all I know, every real analysis book suffers from the flaws I'm about to describe. With that said, I'll point out a few things that make this volume totally inappropriate for self study.

1) In three hundred pages, there are two illustrations, and they're both on the same page. Not only is there a terrific absence of pictures and graphs, but even tables, charts and lists are nowhere to be found. It might be argued that the authors did not wish for students to rely too heavily on visual intuition, given that enlightening visual interpretations simply do not exist for some of the concepts presented. But there are very clearly arguments in this book that are drenched in geometric terminology. Connectedness, compactness, the development of the Lebesgue measure, metric spaces, ... these issues would be made infinitely clearer by a couple of well placed pictures.

2) The concepts are totally unmotivated. No attention is paid to explaining why some theorem might be of importance (with the notable exception of the fixed point theorem). More than once, a hierarchy of definitions and theorems will be laboriously developed, only to unexpectedly disappear and never resurface, leaving the reader wondering what the point was.

3) No solutions to any of the exercises are provided. This is partially admissible given that many of the problems consist of "prove that X is true", and a solution would occupy half a page of work. But there are certainly more computational problems for which answers should be given.

4) There is no narration whatsoever. Outside the preface, the reader is never addressed directly, and rarely acknowledged to exist. It is fully possible to understand the proof of a theorem without having any intuitive, satisfying understanding of the theorem. When presenting unexpected and paradoxical results, the authors take no time to explain things, apparently considering the proof to be sufficient explanation. The reader has no idea, ahead of time, which ideas are critical and which are secondary. The reader has no sense of where the discussion is going, nor is the reader given any historical context of the mathematics. These are issues that actual human beings care about. The authors have written something closer to a reference text, appropriate for robots. Enthusiasm and excitement are absent; the text is dry, uniform, unmotivated and boring.

With that said, the text is very rigorous and thorough, although I have read that there are errors in some of the proofs. It's a cheap purchase and, despite the authors' intentions, you will learn some Real Analysis by going through it.

Hi, I am the previous reviewer of the "so sad it is dover" review. I feel deep respect for the review of Daniel R Greenfield. And I fully agree that this book is excellent! And of course, Dover books are cheap, so this book is definitely worth its money. Though I want to clarify my Dover critics a little bit more. To all those guys who admire Dover books , I wonder whether they use these books as a reference or as a text to study from. I can swear you : it is not so much fun to study high level mathematics in such a dense text! Too my opinion it is essential that you need enough space between the different statements of a complex proof. I believe that the great authors of this book deserve better then a cheap Dover edition. But since Dover owns the rights of these authors, nobody can create this text in a better edition.

User friendly editions however seems to be a common problem for all abstract math books. Why is it that we, students of pure mathematics, have to learn from such user-unfriendly editions.
For instance look at the size : I think user friendly books should have have a size of at least 9.8 x 7.1 inch. For physics,chemistry or calculus you find enough books of this size, with with enough spacing and nice motivating pictures, just to alleviate the learning process...But apparently if you want tot study higher level math, no such books exist, as far as I know ...

This review is a call (even a cry) to editors of advanced math books just to make their editions more user-friendly. If some readers agree that there is a need for this, just vote by clicking the "review was helpfull - yes" button. I hope some editors will then see that there is a market place for user friendly books on advanced math. Even if you have to pay a little more for a book of bigger size, please realize how it will make your life so much easier. Your learning productivity would increase a lot, so better editions are definitely woth its money!!!! Compare it with a professional worker who wants the best and most productive tools.
Unfortunately these critics do not apply to Dover books only, even a lot of more expensive editors just bring their texts in the same user-unfriendly format. On the exiting subject of abstract math, up until now, I even did not find the ideally formatted book.

If some readers agree with my, please vote by telling this review was helpfull (click "yes" button). Hopefully this will inspire editors for future editions. If you really like to study from Dover Editions, please vote by using the review was helpfull-No button. If most people vote "No", I am probably the exceptional guy and will stop complaining about Dover editions.

This book is excellent. It treats all the basic principles of abstract real analysis. It also covers some more advanced topics. It is very clear written and reads extremely wel. I just feel sad this book is a Dover book. Dover books contain high quality, however these books try to put as much text on one page in order to keep the price down. As a consequence, it is not so confortable to read. Especially the proofs need more whitespace. But this is a well known fact for all Dover books, if you are already familiar with Dover books and like them, you will definitely have no complaints against this one.

contains almost no illustrations or examples. No answer key either. Worst math book ever.

That Shane Guy is "DEAD ON". No examples, No pictures, there is a reason this is the cheapest Real Analysis book around. ...Where is the list of errors for this book, because for every Y pages there are X many errors, s.t. X = alot.