Introductory Real Analysis

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

This volume in Richard Silverman's exceptional series of translations of Russian works in the mathematical science is a comprehensive, elementary introduction to real and functional analysis by two faculty members from Moscow University. It is self-contained, evenly paced, eminently readable, and readily accessible to those with adequate preparation in advanced calculus.
The first four chapters present basic concepts and introductory principles in set theory, metric spaces, topological spaces, and linear spaces. The next two chapters consider linear functionals and linear operators, with detailed discussions of continuous linear functionals, the conjugate space, the weak topology and weak convergence, generalized functions, basic concepts of linear operators, inverse and adjoint operators, and completely continuous operators. The final four chapters cover measure, integration, differentiation, and more on integration. Special attention is here given to the Lebesque integral, Fubini's theorem, and the Stieltjes integral. Each individual section — there are 37 in all — is equipped with a problem set, making a total of some 350 problems, all carefully selected and matched.
With these problems and the clear exposition, this book is useful for self-study or for the classroom — it is basic one-year course in real analysis. Dr. Silverman is a former member of the Institute of Mathematical Sciences of New York University and the Lincoln Library of M.I.T. Along with his translation, he has revised the text with numerous pedagogical and mathematical improvements and restyled the language so that it is even more readable.

Genres: Mathematics, Textbooks, Reference, Nonfiction, Academic, Science

416 pages, Paperback

First published June 1, 1975

About the author

Dr. Andrey Nikolaevich Kolmogorov, Ph.D. (Moscow State University, 1929; Russian: Андре́й Никола́евич Колмого́ров) was a Soviet mathematician and professor at the Moscow State University where he became the first chairman of the department of probability theory two years after the 1933 publication of his book which laid the modern axiomatic foundations of the field. He was a Member of the Russian Academy of Sciences and winner of many awards, including the Stalin Prize (1941), the Lenin Prize (1965), the Wolf Prize (1980), and the Lobachevsky Prize (1986).

Reviews

[William Schram · Rating 4 out of 5]

When approaching this book, I had several assumptions. The first assumption was that this book would be somewhat easy to follow. The second assumption was that it focused on 'Real' analysis as opposed to 'Complex' analysis.

Introductory Real Analysis is a textbook written by A N Kolmogorov and S V Fomin. Richard A Silverman provided the translation and did some editing to change the text. The book is quite thorough in its treatment of the subject.

I enjoyed the book for several reasons. It builds logically on itself. Although you might have to start at the beginning to know where it is going, it is difficult to lose your place. It introduces a theorem and provides the proof to it immediately. The way they presented Set Theory was helpful. Finally, the book includes problems to solve and build your understanding.

The reason I did not like the book is rather simple; it was too advanced for me, despite my reading this version. However, this only shows that I am lacking.

[Mauricio Vargas · Rating 5 out of 5]

Amazing book !!!
This book provides some really challenging exercises that demand you to think rather than memorising properties. The chapter on set theory then is essential to understand the notion of integral and derivative which are exposed in a clear and straightforward way.

[Daniel Solomon · Rating 4 out of 5]

Definitely not an introductory text, in comparison to say Rudin's principles book. Otherwise this book provides good coverage and explanations of basic topology/metric space theory, functional analysis, vector spaces and measure theory in a relatively comprehensible and well written manner.

[Victor Ortega Le Hénanff · Rating 4 out of 5]

Introductory in the sense that it introduces you to analysis, and I assume assuming you have already taken Multivariable Calculus, Linear Algebra, etc. I got this book at a free book bazar at the university I work for, and it has been very useful this semester that I have taken my last analysis class.

Both material, exercises, and the exposition itself is quite difficult, but provides a great thread between concepts that is not usually seen, at least I think, in many other analysis textbooks. You have to sit down and really waste your life away reading this, and even then it might not be enough. I guess it's the same for all math.

I particularly enjoyed the metric space sequence -- that is set in motion as soon as possible, which I think is ideal -- as well the measure theory and Lebesgue integration chapters as well. Good material, great translation (I hope) everyone should read to appreciate the Soviet approach to math.

[Jeev · Rating 4 out of 5]

What the hell! This isn’t introductory !

This took me months to process the first chapter on set theory. Not suitable for non-mathematicians.

Saying that, I have persevered with this book and have now developed an obsession for rigorous proofs

Big up Urysohns lemma !

[Dev Prakash · Rating 5 out of 5]

This is nice book for me

[Greymon War · Rating 5 out of 5]

Best real analysis book, period

[Owen Levens · Rating 1 out of 5]

This is how I imagine heroine withdrawal feels

[Tomáš Ševček · Rating 5 out of 5]

I only read the parts on measure theory (the last four chapters plus the introductory chapter on set theory) as we used this book as the main reference in our measure theory course. I find the exposition to be clearly written and very informative. I did not have any prior experience with measure theory and I was able to follow the text fairly easily. It really helps that planar measure theory is developed first before delving into more general ideas. There are also numerous interesting exercises (no solutions). Some of them contain important concepts such as the Jordan measure/content (even though a direct comparison could have been made in the main text). The only thing missing (at least in my opinion) is the differentiation of parametric Lebesgue integrals, as it is required in functional analysis and the calculus of variations. Other than that, I recommend this book as an introduction to measure theory.