Introductory Real Analysis (Dover Books on Mathematics) [Paperback]

A. N. Kolmogorov (Author), S. V. Fomin (Author), Richard A. Silverman (Translator)

Book Description

ISBN-10: 0486612260 | ISBN-13: 978-0486612263 | Publication Date: June 1, 1975 | Edition: 1st

Self-contained and comprehensive, this elementary introduction to real and functional analysis is readily accessible to those with background in advanced calculus. It covers basic concepts and introductory principles in set theory, metric spaces, topological and linear spaces, linear functionals and linear operators, and much more. 350 problems. 1970 edition.

Editorial Reviews

Language Notes

Text: English, Russian (translation)

Product Details

• Paperback: 416 pages
• Publisher: Dover Publications; 1st edition (June 1, 1975)
• Language: English
• ISBN-10: 0486612260
• ISBN-13: 978-0486612263
• Product Dimensions: 8.2 x 5.6 x 0.8 inches
• Shipping Weight: 15.5 ounces (View shipping rates and policies)
• Average Customer Review: 4.1 out of 5 stars  See all reviews (35 customer reviews)
• Amazon Best Sellers Rank: #51,381 in Books (See Top 100 in Books)
  • #7 in Books > Education & Reference > Foreign Language Nonfiction > Russian

Customer Reviews

Most Helpful Customer Reviews

86 of 88 people found the following review helpful:

3.0 out of 5 stars Are all Silverman's "translations" like this one?, May 26, 2004

By A Customer

This review is from: Introductory Real Analysis (Dover Books on Mathematics) (Paperback)

First, let us be precise in reviewing this book. It is NOT a book by Kolmogorov/Fomin, but rather an edited version by Silverman. So, if you read the first lines in the Editor's Preface, it states, "The present course is a freely revised and restyled version of ... the Russian original". Further down it continues, "...As in the other volumes of this series, I have not hesitated to make a number of pedagogical and mathematical improvements that occurred to me...". Read it as a big red warning flag. Alas, I would have to agree with the reader from Rio de Janeiro. I've been working through this book to rehash my knowledge of measure theory and Lebesgue integration as a prerequisite for stochastic calculus. And I've encountered many results of "mathematical improvements" that occurred to the esteemed "translator". Things are fine when topics/theorems are not too sophisticated (I guess not much room for "improvements"). Not so when you work through some more subtle proofs. Most mistakes I discovered are relatively easy to rectify (and I'm ignoring typos). But the latest is rather egregious. The proof of theorem 1 from ch. 9 (p.344-345) (about the Hahn decomposition induced on X by a signed measure F) contains such a blatant error, I am very hard pressed to believe it comes from the original. That book survived generations of math students at Moscow State, and believe me, they would go through each letter of the proofs. Astounded by such an obvious nonsense, I grabed the only other reference book on the subject I had at hand, "Measure Theory" by Halmos. The equivalent there is theorem A, sec. 29 (p.121 of Springer-Verlag edition), which has a correct proof.
For those interested in details, Silverman's proof states that positive integers are strictly ordered: k1<k2<...<kn<... which is nonsense. Consider subsets B1 and B2 such that F(B1)=3, F(B2)=2, for example. That gives you both k1=k2=1. The flaw of course is not in this slip-up per se, but rather in the logic of the proof which follows from the above ordering of k's. Compare vs Halmos text to see the difference. It may seem insignificant to certain translators, but in mathematics, out of all of sciences, subtle details change an elegant proof to a flagrant nonsense.
Unfortunately, I don't have the Russian original. Instead, I'm trying to get the other, hopefully real translation, "Elements of the Theory of Functions and Functional Analysis". BTW, this is the actual title of the original, not "Introductory Real Analysis". Which apparently is causing significant confussion amoung past and present readers. To give you a background info, the Russian original is (or has been, at least) used as a textbook for a third-year subject for (hard-core) math students. Meaning, in the preceding two years they would complete a pre-requisite four-semester calculus course. For example, criteria of convergence of series and their properties is an assumed knowledge in presentation of Lebesgue integral. So, I think most of the critique from earlier reivews is a bit misdirected. The original book is a great starting book into functional analyis/Lebesgue integration and differentiation, but proofs require solid understanding of fundamentals of calculus.
The best part about Kolmogorov's text is the clarity of conceptual structure of the presented subject a reader would gain, if he/she puts some effort. You would gain a thorough understanding, not just a knowledge of the subject. There is quite a difference between the two, and not that many authors succeed in delivering that.
But to gain that from Kolmogorov, I would suggest the other, "unimproved" but real, translation.

67 of 71 people found the following review helpful:

4.0 out of 5 stars Strong "introduction", February 5, 2001

By 

Kevin P. Costello (Riverside, CA) - See all my reviews

This review is from: Introductory Real Analysis (Dover Books on Mathematics) (Paperback)

Overall, this book is a very strong "introduction" (I use the word grudgingly, see below) to real analysis. Topics range from the basics of set theory through metrics, operators, and Lebesgue measures and integrals. Particularly well done are the section on linear maps and operators, which include excellent generalizations as well as the usual concrete examples. The book usually includes a large number of examples and exercises on each topic which aid in the understanding of the material (though in a few instances, most notably the introduction to measure, it would have been more helpful to have examples as the theory was being developed instead of spending 20 pages getting through the theorems and only then giving a few examples).

The main problem for this book, however, is that it is located at an awkward level in terms of its assumptions of what students have seen before. Most of the material covered is that of a first analysis course, and the book is probably usually used as such. The authors, however, sometimes make assumptions that students have had exposure to some of the concepts before, claiming that "the reader has probably already encountered the familiar Heine-Borel theorem", for example. One particularly annoying case was when the authors gave as an example that the set of polynomials with rational coefficients is dense in the set of continuous functions, and left it at that. Are we supposed to have encountered Weierstrass's theorem before we take our first analysis course?

22 of 25 people found the following review helpful:

5.0 out of 5 stars Highly Motivated, April 12, 2000

By 

Marete - See all my reviews

This review is from: Introductory Real Analysis (Dover Books on Mathematics) (Paperback)

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.....!