An Introduction to Functional Analysis in Computational Mathematics

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.