Real Analysis

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.

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.

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.

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.