Functional Analysis and Numerical Mathematics focuses on the structural changes which numerical analysis has undergone, including iterative methods, vectors, integral equations, matrices, and boundary value problems.
The publication first examines the foundations of functional analysis and applications, including various types of spaces, convergence and completeness, operators in Hilbert spaces, vector and matrix norms, eigenvalue problems, and operators in pseudometric and other special spaces. The text then elaborates on iterative methods. Topics include the fixed-point theorem for a general iterative method in pseudometric spaces; special cases of the fixed-point theorem and change of operator; iterative methods for differential and integral equations; and systems of equations and difference methods.
The manuscript takes a look at monotonicity, inequalities, and other topics, including monotone operators, applications of Schauder's theorem, matrices and boundary value problems of monotone kind, discrete Chebyshev approximation and exchange methods, and approximation of functions.
The publication is a valuable source of data for mathematicians and researchers interested in functional analysis and numerical mathematics.

Translator's NotePreface to the German EditionNotationChapter I Foundations of Functional Analysis and Applications 1. Typical Problems in Numerical Mathematics 1.1 Some General Concepts 1.2 Solutions of Equations 1.3 Properties of the Solutions of Equations 1.4 Extremum Problems with and without Constraints 1.5 Expansions (Determination of Coefficients) 1.6 Evaluations of Expressions 2. Various Types of Spaces 2.1 Hölder's and Minkowski's Inequalities 2.2 The Topological Space 2.3 Quasimetric and Metric Spaces 2.4 Linear Spaces 2.5 Normed Spaces 2.6 Unitary Spaces and Schwarz Inequality 2.7 The Parallelogram Equation 2.8 Orthogonality in Unitary Spaces, Bessel's Inequality 3. Orderings 3.1 Partial Ordering and Complete Ordering 3.2 Lattices 3.3 Pseudometric Spaces 4. Convergence and Completeness 4.1 Convergence in a Pseudometric Space 4.2 Cauchy Sequences 4.3 Completeness, Hilbert Spaces, and Banach Spaces 4.4 Continuity Properties 4.5 Direct Consequences for Hilbert Spaces, Subspaces 4.6 Complete Orthonormal Systems in Hilbert Spaces 4.7 Examples 4.8 Weak Convergence 5. Compactness 5.1 Relative Compactness and Compactness 5.2 Examples of Compactness 5.3 Arzelà's Theorem 5.4 Compact Sets of Functions Generated by Integral Operators 6. Operators in Pseudometric and Other Special Spaces 6.1 Linear and Bounded Operators 6.2 Composition of Operators 6.3 The Inverse Operator 6.4 Examples of Operators 6.5 Inverse Operators of Neighboring Operators 6.6 Condition Number of a Linear, Bounded Operator 6.7 Error Estimates for an Iteration Process 6.8 Riesz's Theorem and Theorem of Choice 6.9 A Theorem by Banach on Sequences of Operators 6.10 Application to Quadrature Formulas 7. Operators in Hilbert Spaces 7.1 The Adjoint Operator 7.2 Examples 7.3 Differential Operators for Functions of a Single Variable 7.4 Differential Operators for Functions of Several Variables 7.5 Completely Continuous Operators 7.6 Completely Continuous Integral Operators 7.7 Estimates for the Remainder Term for Holomorphic Functions 7.8 A Bound for the Truncation Error of Quadrature Formulas 7.9 A Fundamental Principle of Variational Calculus 8. Eigenvalue Problems 8.1 General Eigenvalue Problems 8.2 Spectrum of Operators in a Metric Space 8.3 Inclusion Theorem for Eigenvalues 8.4 Projections 8.5 Extremum Properties of the Eigenvalues 8.6 Two Minimum Principles for Differential Equations 8.7 Ritz's Method 9. Vector and Matrix Norms 9.1 Vector Norms 9.2 Comparison of Different Vector Norms 9.3 Matrix Norms 9.4 From Matrix Theory 9.5 Euclidean Vector Norm and Consistent Matrix Norms 9.6 Other Vector Norms and Subordinate Matrix Norms 9.7 Transformed Norms 10. Further Theorems on Vector and Matrix Norms 10.1 Dual Vector Norms 10.2 Determination of Some Dual Norms 10.3 Powers of Matrices 10.4 A Minimum Property of the Spectral Norm 10.5 Deviation of a Matrix from Normality 10.6 Spectral Variation of Two Matrices 10.7 Selected Problems to Chapter I 10.8 Hints to Selected Problems of Section 10.7Chapter II Iterative Methods 11. The Fixed-Point Theorem for a General Iterative Method in Pseudometric Spaces 11.1 Iterative Methods and Simple Examples 11.2 Iterative Methods for Differential Equations 11.3 The General Fixed-Point Theorem 11.