This is a concise guide to basic sections of modern functional analysis. Included are such topics as the principles of Banach and Hilbert spaces, the theory of multinormed and uniform spaces, the Riesz-Dunford holomorphic functional calculus, the Fredholm index theory, convex analysis and duality theory for locally convex spaces. More than one hundred famous `named' theorems, culminating in the Gelfand-Naimark-Segal construction for C*-algebras, are treated, and complete proofs are given. This volume, which is regarded already as a standard textbook in functional analysis, has been translated from the second, completely revised and updated Russian edition. It incorporates new sections on the Schwartz spaces of distributions, Radon measures, and a supplementary list of theoretical exercises and problems. Audience: This monograph will be of value to researchers and students who are interested in functional analysis.