Customer Reviews

Introductory Real Analysis (Dover Books on Mathematics)

5 star:		(19)
4 star:		(9)
3 star:		(3)
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1 star:		(3)

 

 

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.

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?

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.....!

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.

The advantages of this text have been pointed out by other readers, so I will attempt to exhibit the problems of this book.

There are a lot of mistakes. And by 'a lot', I mean that the careful reader should be able to find at least 5 mathematical mistakes in each chapter. I used this text mainly as a supplement to a fairly advanced analysis course, and we'd often have problems from it used in our problem sets. At first, it appeared as if this were a very well-written text, but once we started with our problem sets, there were at least 2 e-mails sent out per week addressing a concern a student had pointed out. After a while, students stopped e-mailing the professor with their concerns, instead just assuming that they were correct whenever they spotted something weird.

Let's take an example:

Problem 1, pg. 137: Let M be the set of all points x = (x1, x2, ..., xn, ...) in l2 satisfying the condition \sum^{\infty}_{n=1} (n^2) (x_n)^2 \le 1. Prove that M is a convex set, but not a convex body.

The problem with this is that M IS easily a convex body, precisely because x = (0,0,...) is in M.

There are many more big mistakes and little mistakes throughout the exercises, oftentimes destroying the entire POINT of the problem. Take, for example, Problem 1 of pg. 76: Let A be a mapping of a metric space R into itself. Prove that the condition p(Ax,Ay) < p(x,y) (x\ne y) is insufficient for the existence of a fixed point of A.

Now, a counterexample here can be easily produced, even by the most elementary reader. But the exercise quickly becomes worthwhile if we make R complete. It's the little things that count in mathematics, and the small errors like these are clearly detrimental to the student.

But the errors in the text aren't limited to exercises. I was reading independently at the front of the book to get some info on Zorn's Lemma and ordinal numbers, and as I read, I found the following definition:

Let M1 and M2 be two ordered sets of type 01 and 02, respectively. Then we can introduce an ordering in the union M1 U M2 of the two sets by assuming that

1) a and b have the same ordering as in M1 if a,b are in M1
2) a and b have the same ordering as in M2 if a,b are in M2
3) a < b if a in M1, b in M2

The set M1 U M2 ordered in this way is called the ordered sum of M1 and M2, denoted by M1 + M2.

There is a clear problem with this definition pointed out here: http://www.physicsforums.com/showthread.php?t=200985. How is the student expected to learn such important material when even the definitions have loopholes? By the end of our experience with this book, our professor was giving out exercises to correct Kolmogorov/Fomin's incorrect definitions.

Note also that Silverman uses very weird words. For example, 'countably compact' is used instead of 'sequentially compact'.

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 ......

I speak Russian and read it so-so - this is not the original work of Kolmogorov and Fomin but is a "freely" translated version. Unfortunately, "free" is not always "correct".

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.

Don't let the title fool you. This book will take you from sophomore real analysis through the first year of graduate analysis, covering set theory, topology, functionals, integration, and differentiation. This stands out as one of the best math books on my shelf.