Introductory Real Analysis

I am currently a first year graduate math student. I have had advanced calculus (we used Introduction to Analysis by W. Wade, covered chapters 1-7) and basic topology as an undergrad, and I'm working through Principles of Mathematical Analysis right now for class. On the side, I decided to try to learn some more advanced analysis. I found that my undergraduate courses were good enough to start reading Kolmogorov and Fomin on my own (after all, the preface states that Adv Calc is a prereq for the text). The definitions and theorems are clean and consise, and there are plenty of good examples to help you along with the concepts. At the end of most sections, there are several instructive problems to think about to help you along. Some of the methods and notations are a little dated, but for the price of the text, that can be easily ignored. This is a wonderful introduction and I'd recommend it for anyone who is interested in graduate level mathematics.

I find this a great introduction to real analysis. Contrary to what one reviewer has suggested, I think the book is fairly rigorous. It is true that some details are omitted, but they can always be filled up by the reader. In fact, this is the one of the most fun parts of reading the book!To give a concrete example: One reviewer has suggested that the theorem "Every infinite set has a countable subset" is proved without stating that the axiom of choice is required. This is certainly a serious lapse of rigour, BUT, in a later page, the author explains the axiom of choice (and several equivalent assertions) and also touches upon the fact that there are some very deep set theoretic questions, not yet fully resolved, concerning this axiom. He goes on to say "The axiom of choice will be assumed in this book. In fact, without it, we will be severely hampered for making various set-theoretic constructions". It is evident that the above theorem is one such construction.This book emphasizes an intuitive approach to the subject, something which in my opinion is neglected by far too many books. Rigour is necessary but never sufficient to acheive proficiency in math!

Years ago I used this book as a supplementary text for a course in functional analysis and measure theory. When I learned that it was being republished by Dover I immediately bought my own copy. It is a thoroughly readable book with lots of examples to illustrate concepts. The chapters on measure theory and the Lebesgue integral were exceptional. And the chapters on linear functionals and operators also very good. On the downside the introductory chapter on definitions of concepts like open and closed sets and the treatment of compactness and the Heine-Borel theorem could have been presented more clearly (I preferred Dieudonne's presentation in Foundations of Modern Analysis). I strongly recommend this book as excellent value for money.

This is a most beautiful exposition of Analysis going upto graduate level! The wonderful thing about the book is the examples, examples, examples! Every definition and many of the theorems are followed by concrete examples, many of them closely related to familiar notions such as the real line or R^n. He begins measure theory by constrution of measures on plane sets, then proceeds to generalize, one example of the conrcrete approach in the book. Kolmogorov also provided us with the axiomatics of Functional Analysis in 3 clear chapters. I heartily recommend this book as a stop-over before ,say, a study of Rudin's Real and Complex whose expostion (especially of measure theory) is as abstract as it is beautiful.And then of course, enough cannot be said about the price.....!

There are several excellent textbooks for Real Analysis, just to name a few: Royden's, Folland's and Rudin's. Also, there are several excellent Functional Analysis textbooks. To combine both subjects seamlessly, I have to say this one is the best. The price of this book is "Wow......!!!" great! Your instructor will most likely assign you to read Royden's textbook. Unless you can get hold of Prof. Chernoff's note (Berkeley), you should definitely buy this one and study it thoroughly. I have three copies of it. Don't ask me why, now? You will know after you study it. One is for my library, one is for my office. One is for my toilet room. Well, I like to enjoy good math when I ......