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Introductory Real Analysis (Dover Books on Mathematics) 1st Edition

4.3 out of 5 stars 50 customer reviews
ISBN-13: 978-0486612263
ISBN-10: 0486612260
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Editorial Reviews

Language Notes

Text: English, Russian (translation)

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Product Details

  • Series: Dover Books on Mathematics
  • Paperback: 416 pages
  • Publisher: Dover Publications; 1st edition (June 1, 1975)
  • Language: English
  • ISBN-10: 0486612260
  • ISBN-13: 978-0486612263
  • Product Dimensions: 1 x 5.8 x 8.5 inches
  • Shipping Weight: 15.5 ounces (View shipping rates and policies)
  • Average Customer Review: 4.3 out of 5 stars  See all reviews (50 customer reviews)
  • Amazon Best Sellers Rank: #240,372 in Books (See Top 100 in Books)

Customer Reviews

Top Customer Reviews

Format: 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<...
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Format: 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?
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Format: Paperback
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.
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