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.
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ISBN-13
978-1-4612-4128-7 (9781461241287)
DOI
10.1007/978-1-4612-4128-7
Schweitzer Klassifikation
1. Functional Spaces and Problems in the Theory of Approximation.- 1. Metric Spaces.- 2. Compact Sets in Metric Spaces.- 3. Statement of the Main Extremal Problems in the Theory of Approximation. Main Characteristics of the Best Approximations.- 4. The Contraction Mapping Principle.- 5. Linear Spaces.- 6. Normed and Banach Spaces.- 7. Spaces with an Inner Product. Hilbert Spaces.- 8. Problems on the Best Approximation. Orthogonal Expansions and Fourier Series in a Hilbert Space.- 9. Some Extremal Problems in Normed and Hilbert Spaces.- 10. Polynomials the Least Deviating from Zero. Chebyshev Polynomials and Their Properties.- 11. Some Extremal Polynomials.- 2. Linear Operators and Functionals.- 1. Linear Operators in Banach Spaces.- 2. Spaces of Linear Operators.- 3. Inverse Operators. Linear Operator Equations. Condition Measure of Operator.- 4. Spectrum and Spectral Radius of Operator. Convergence Conditions for the Neumann Series. Perturbations Theorem.- 5. Uniform Boundedness Principle.- 6. Linear Functionals and Adjoint Space.- 7. The Riesz Theorem. The Hahn-Banach Theorem. Optimization Problem for Quadrature Formulas. The Duality Principle.- 8. Adjoint, Selfadjoint, Symmetric Operators.- 9. Eigenvalues and Eigenelements of Selfadjoint and Symmetric Operators.- 10. Quadrature Functionals with Positive Definite Symmetric or Symmetrizable Operator and Generalized Solutions of Operator Equations.- 11. Variational Methods for the Minimization of Quadrature Functionals.- 12. Variational Equations. The Vishik-Lax-Milgram Theorem.- 13. Compact (Completely Continuous) Operators in Hilbert Space.- 14. The Sobolev Spaces. Embedding Theorems.- 15. Generalized Solution of the Dirichlet Problem for Elliptic Equations of the Second Order.- 3. Iteration Methods for the Solution of Operator Equations.- 1. General Theory of Iteration Methods.- 2. On the Existence of Convergent Iteration Methods and Their Optimization.- 3. The Chebyshev One-Step (Binomial) Iteration Methods.- 4. The Chebyshev Two-Step (Trinomial) Iteration Method.- 5. The Chebyshev Iteration Methods for Equations with Symmetrized Operators.- 6. Block Chebyshev Method.- 7. The Descent Methods.- 8. Differentiation and Integration of Nonlinear Operators. The Newton Method.- 9. Partial Eigenvalue Problem.- 10. Successive Approximation Method for Inverse Operator.- 11. Stability and Optimization of Explicit Difference Schemes for Stiff Differential Equations.
