An Introduction to Functional Analysis in Computational Mathematics: An Introduction - Tapa dura

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.

Presents the basics of functional analysis, as well as elements of variational equations (on the basis of bi-linear forms), including the Vishik-Lax-Milgram theorem and of generalized solutions of eliptic problems. In terms of functional analysis, such problems of computational mathematics are considered as extremal of problems of approximation theory of various types - the theory of numerical integration, variational methods of minimization of quadratic functional, and Galerkin and Ritz methods of finding solutions to operator equations. Also covered are: the general theory of iteration methods, Chebyshen iteration methods, composition method, along with some elements of nonlinear analysis. Sobolev spaces and embedding theorems are also introduced.