Books

Books : reviews

A. N. Kolmogorov, S. V. Fomin.
Introductory Real Analysis.
Dover. 1968

This volume in Richard Silverman's exceptional series of translations of Russian works in the mathematical sciences 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 Lebesgue 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 a 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.

A. D. Aleksandrov, A. N. Kolmogorov, M. A. Lavrent'ev.
Mathematics: its content, methods and meaning: 2nd edn.
Dover. 1969

This major survey of mathematics, featuring the work of 18 outstanding Russian mathematicians and including material on both elementary and advanced levels, encompasses 20 prime subject areas in mathematics in terms of their simple origins and their subsequent sophisticated development. As Professor Morris Kline of New York University noted, “This unique work presents the amazing panorama of mathematics proper. It is the best answer in print to what mathematics contains both on the elementary and advanced levels.”

Beginning with an overview and analysis of mathematics, the first of three major divisions of the book progresses to an exploration of analytic geometry, algebra, and ordinary differential equations. The second part introduces partial differential equations, along with theories of curves and surfaces, the calculus of variations, and functions of a complex variable. It further examines prime numbers, the theory of probability, approximations, and the role of computers in mathematics. The theory of functions of a real variable opens the final section, followed by discussions of linear algebra and non-Euclidean geometry, topology, functional analysis, and groups and other algebraic systems.

Thorough, coherent explanations of each topic are further augmented by numerous illustrative figures, and every chapter concludes with a suggested reading list. Formerly issued as a three-volume set, this mathematical masterpiece is now available in a convenient and modestly priced one-volume edition, perfect for study or reference.

[Three volumes bound as one]