Real Analysis

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.