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The book contains the methods and bases of functional analysis that are directly adjacent to the problems of numerical mathematics and its applications; they are what one needs for the understand­ ing from a general viewpoint of ideas and methods of computational mathematics and of optimization problems for numerical algorithms. Functional analysis in mathematics is now just the small visible part of the iceberg. Its relief and summit were formed under the influence of this author's personal experience and tastes. This edition in English contains some additions and changes as compared to the second edition in Russian; discovered errors and misprints had been corrected again here; to the author's distress, they jump incomprehensibly from one edition to another as fleas. The list of literature is far from being complete; just a number of textbooks and monographs published in Russian have been included. The author is grateful to S. Gerasimova for her help and patience in the complex process of typing the mathematical manuscript while the author corrected, rearranged, supplemented, simplified, general­ ized, and improved as it seemed to him the book's contents. The author thanks G. Kontarev for the difficult job of translation and V. Klyachin for the excellent figures.



Chapter 1. Functional Spaces and Problems in the Theory of Approximation

A space is understood in mathematics as a set of any objects (sets of numbers, functions, etc.) with certain relationships established among them, similar to those existing in an elementary three-dimensional space.
V. I. Lebedev

Chapter 2. Linear Operators and Functionals

We defined the notion of operators in Section 1 of Chapter 1 and presented the definition of their continuity. Now we study an important class of these operators and functionals, namely, linear operators and functionals.
V. I. Lebedev

Chapter 3. Iteration Methods for the Solution of Operator Equations

Method of successive approximation for linear equations of the second kind, necessary and sufficient conditions for convergence, error estimate, sufficient conditions for convergence. Iteration methods for solution of linear equations of the first kind. Definition of multistep, one-step, cyclic linear iteration methods. General form of one-step linear iteration methods. Convergence rate.
V. I. Lebedev


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