ELEMENTS OF THE THEORY OF FUNCTIONS
AND FUNCTIONAL ANALYSIS, VOL. I
METRIC AND NORMED SPACES

This classic work was originally from a 1954 course, by one of the most influential mathematicians of the 20th Century. This translation, from 1957, is renowned for its elegance and clarity. The contents are best summarized in the Translator's Note: This volume is a translation of A.N. Kolmogorov and S.V. Fomin’s Ѐlementy Teorii Funkciĭ I Funkcional’nogo Analiza., I. Metričeskie I Normirovannye Prostranstva. Chapter I is a brief introduction to set theory and mappings. There is a clear presentation of the elements of the theory of metric and complete metric spaces in Chapter II. The latter chapter also has a discussion of the principle of contraction mappings and its applications to the proof of existence theorems in the theory of differential and integral equations. The material on continuous curves in metric spaces is not usually found in textbook form. The elements of the theory of normed linear spaces are taken up in Chapter III where the Hahn-Banach theorem is proved for real separable normed linear spaces… Chapter III also deals with weak sequential convergence of elements and linear functionals and gives a discussion of adjoint operators. The addendum to Chapter III discusses Sobolev’s work on generalized functions which was later generalized further by L. Schwartz. The main results here are that every generalized function has derivatives of all orders and that every convergent series of generalized functions can be differentiated term by term any number of times. Chapter IV, on linear operator equations, discusses spectra and resolvents for continuous linear operators in a complex Banach space. Minor changes have been made, notably in the arrangement in the proof of the Hahn-Banach theorem. A bibliography, listing basic books covering the material in this volume, was added by the translator.