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Rating: 
Summary: Four stars for the content, five stars for the price.
Review: I am electrical eng who study math all of my life. I had the Arabic version of it which cover more than the English one.I start reading it after Krewsiq book of advance functional analysis. I really like Krewsiq because it is so easy so I cover it within 1 year but I need 2.5 year to finish off this one which is fantastic, I really can't compare it with other books, it is a school of functional analysis and master book that you can't really understand functional analysis without it.
Rating:
Summary: My math master
Review: I am electrical eng who study math all of my life. I had the Arabic version of it which cover more than the English one.I start reading it after Krewsiq book of advance functional analysis. I really like Krewsiq because it is so easy so I cover it within 1 year but I need 2.5 year to finish off this one which is fantastic, I really can't compare it with other books, it is a school of functional analysis and master book that you can't really understand functional analysis without it.
Rating:
Summary: A Great Introductory Functional Analysis Textbook!
Review: I bought this book after I was fascinated by Kolmogorov and Fomin's 'Introductory Real Analysis.' And I found that the content of this book is almost the same as 'Introductory...' Although this book is 2 dollars cheaper than the other, I suggest buying 'Intro...' since the original publication date is later than this book and the typeset is better than this, too.
Rating: 
Summary: Great book, fair reprint
Review: I must say that Kolmogorov is one of my favorite Russian scientists and scientific writers (the others are Arnol'd, Gel'fand and Landau). His approach over real and functional analysis is superb. But I have a serious complaint: why did Dover choose the old and dated Graylock edition to reprint this masterpiece? The most recent Russian edition has a lot of other important topics (missing in Dover's reprint), such as differential calculus in Banach spaces, which is so difficult to find a decent treatment of. The organization of the book is also better in the Russian edition. A good revised and translated reprint of it would deserve not five, but six, seven, ten stars...
Rating:
Summary: It's a classic
Review: I own the book in Spanish language. Some content in the book is not easy stuff, if you study by others books, but Kolmogorov has the gift to make easy things that aren't quite so easy. Perhaps some theory is "old", because all the new books use some diferent approach to the subject, like the chapter dedicated to the Lebesgue integral, the book give the definition of a simple function in a different manner that we use today. The book is a must to have in your library, when you need to work with Functional Analysis.
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Summary: A best-seller forever
Review: The emminent Andrei Kolmogorov wrote the first version of this book by compiling his lecture notes in the 1940's. The overall text of this book dates from the 1950's. When it was first published it became immediately a best-seller, and today it continues to be so. Wonder why? It's easy. Dr. Kolmogorov was one of the finest mathematicians ever, and moreover, he was also recognized as an excellent teacher. So, the combination of first-line expert and first-line lecturer produced this highly appreciated text. By reading it you can see not only the author's capability to introduce delicate subjects (set theory, normed spaces, linear functionals, Lebesgue integral, Fourier integrals, etc.) but also his genius in understanding where the trends of modern mathematics would flow. The more you read this book the more you love it, and the more you love mathematics itself.
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Summary: At the elbow of the Master...
Review: This book is actually two books bound as one. The first part concerns metric spaces and normed linear spaces. The second covers Lebesgue integration. The typesetting and prose are sometimes very tight, but some of the constructions used in the proofs are really amazing. Numerous examples are sprinkled through the text. I would not recommend this book as a first book in functional analysis or in Lebesgue integration. To get the most out of this book, you need to have seen many of the results presented elsewhere to really appreciate the Master's technique used in developing those same results in this book.
Rating:
Summary: A MAGNIFIC BOOK !
Review: This book is divided into two parts. The first part is devoted mainly to metric and normed spaces. There are too a chapter on the essentials of set theory, an addendum on generalized functions, and a chapter on linear operators. The second part is devoted to measure theory, the Lebesgue integral, the theory of square integrable functions(L2) and Hilbert spaces. The second part incorporates exercises to the reader. The clarity of exposition and the elegancy of this book is notorious ! This book can be recommended not only for mathematicians, but for theoretical physicists. Do you know why the Heisenberg picture of quantum mechanics is equivalent to the Schrodinger picture of quantum mechanics ? Mathematically, Heisenberg theory uses the space l2, while Schrodinger theory uses the space L2. A consequence of the Riesz-Fisher theorem is that the spaces l2 and L2 are isomorphic, a result proved in this book. The two theories leads to the same physical results, and in consequence are equivalent, although different in the mathematical content ! You need to buy this book !
Rating:
Summary: Four stars for the content, five stars for the price.
Review: This book is not quite up-to-date, but still very good as a starting point in (functional) analysis. The virtue of Kolmogorov and Fomin is their user-friendly writing style. I am delighted to find their book being available for less than ten dollars.
Rating:
Summary: Still one of the finest.
Review: This highly regarded book came out from the notes of Andrei Kolmogorov's lectures given at Moscow's Lomonosov University in the 1940's, and it still stands as one of the best introductions to real analysis available.The authors introduce step by step all the key concepts needed to get a thorough understanding of the subject and proceed all the way long from set theory to Fredholm integral equations. This book is appreciated not only because the topics it includes but mostly because of the insight with which it was written. It is a pleasure to find through every page of the book the great genious of Kolmogorov who not only mastered most areas of mathematics but who also had an almost unparalleled understanding of what the trends of future mathematics would be. The contents are: Elements of Set Theory; Metric and Topological Spaces; Normed and Topological Linear Spaces; Linear Functionals and Linear Operators; Elements of Differential Calculus in Linear Spaces; Measure, Measurable Functions, Integral; Indefinite Lebesgue Integral, Differentiation Theory; Spaces of Summable Functions; Trigonometric Series, Fourier Transformation; Linear Integral Equations. Full motivation and detailed explanation for each topic. Short bibliography, but that is justified by the fact that the authors themselves were involved in the development of the topics covered. Conclusion: a must-have text for every mathematician or math student.
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