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Introductory Real Analysis (Dover Books on Mathematics)

A. N. Kolmogorov, S. V. Fomin, Richard A. Silverman · 2 HN comments
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Amazon Summary
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.
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Jul 16, 2020 · pash on Topology Illustrated (2015)
Topology is a canonical example of "mathematics without numbers" (and indeed of "qualitative" rather than "quantitative" mathematics), so for anyone approaching it with an undergrad or lesser background in math, learning it feels a bit like starting with a clean slate. As benrbray mentions in a sibling comment, there aren't any prerequisites in the usual sense, as you won't need to draw on any reserve of mathematical facts built up in learning previous topics while studying it. But learning it does require the ability to assimilate axioms (and to intuit why they make sense as a starting point), and then the ability to reason about those axioms rigorously to draw out the interesting consequences.

That said, a background in some mathematical topics is useful for learning topology. The subject is rooted in a particularly abstract notion of distance, so it's useful to have some experience reasoning with somewhat less abstract notations of distance like metrics, the most familiar of which is the quite numerical Euclidean metric involving the familiar square root of the sum of squares. Familiarity with metric spaces in general (that is, spaces equipped with any axiomatically acceptable notion of a metric distance) is even more useful because it requires a similar sort of axiomatic reasoning as does topology.

Kolmogorov & Fomin's classic text on analysis, the Dover edition of which is under $12 on Amazon [0], has a good (albeit austere) introduction to topology, leading up to it via axiomatic set theory and metric spaces in the book's first chapters. On the one hand, I hesitate to recommend this book, because it was the text for a course that mercilessly exposed the shortcomings in my mathematical education to that time; on the other hand, I recommend it for precisely that reason. Having later assimilated the material, I do consider the book a very good introduction for anyone who either already has the mathematical maturity to do the book properly or for anyone who wants to gain that ability in the way that everybody who has done does: by staring at the same page for hours on end while working everything out on paper, down to the axioms if necessary, until you stop misunderstanding, and then start understanding.

That said, there surely are gentler introductions to the subject if you just want to get a rough idea of it.


> so for anyone approaching it with an undergrad or lesser background in math, learning it feels a bit like starting with a clean slate

Did you mean without? Algebraic topology would be a pretty standard 4th year undergrad course in the UK.

I've spoken with more than one person who made it through Real Analysis intact by reading through "Introductory Real Analysis" by Kolmogorov and Fomin. There's a Dover version that you can probably find for $12 used... It was where I started, but I know several people who found it invaluable after struggling with other texts.

Introductory Real Analysis (Dover Books on Mathematics)

Come to think of it, there are a lot of good Dover books on mathematics.

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