to the English Translation 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. With standard provisos the presentation is self-contained,
exposing about a h- dred famous "named" theorems furnished with complete
proofs and culminating in the Gelfand-Nalmark-Segal construction for
C*-algebras. The first Russian edition was printed by the Siberian
Division of "Nauka" P- lishers in 1983. Since then the monograph has
served as the standard textbook on functional analysis at the University
of Novosibirsk. This volume is translated from the second Russian
edition printed by the Sobolev Institute of Mathematics of the Siberian
Division of the Russian Academy of Sciences- in 1995. It incorporates
new sections on Radon measures, the Schwartz spaces of distributions,
and a supplementary list of theoretical exercises and problems. This
edition was typeset using AMS-'lEX, the American Mathematical Society's
'lEX system. To clear my conscience completely, I also confess that: =
stands for the definor, the assignment operator, signifies the end of
the proof.