Functional Analysis

Cover FUNCTIONAL ANALYSIS Copyright © 1972 Wolters-Noordhoff Publishing ISBN: 90 0! 90980 9 Library of Congress Catalog Card Number: 75-90855 CONTENTS EDITOR'S FOREWORD TO THE RUSSIAN EDITION CHAPTER I: FUNDAMENTAL CONCEPTS OF FUNCTIONAL ANALYSIS § 1. Linear Systems 1. Concept of a linear system 2. Linear dependence and independence 3. Linear manifolds and convex sets § 2. Linear topological, metric, normed and Banach spaces 1. Linear topological space 2. Locally convex space 3. Metric linear space 4. Normed linear space 5. Examples of normed linear spaces 6. Completeness of metric spaces. Banach space 7. Compact sets 8. Separable spaces § 3. Linear functionals 1. Concept of a linear functional 2. Continuous linear functionals 3. Extension of continuous linear functionals 4. Examples of linear functionals § 4. Conjugate spaces 1. Duality of linear systems 2. Conjugate space to a normed linear space. 3. Weak and weak* topology 4. Properties of a sphere in a conjugate Banach space 5. Factor space and orthogonal complements 6. Reflexive Banach spaces § 5. Linear operators 1. Bounded linear operators 2. Examples of bounded linear operators. integral operators Interpolation theorems 3. Convergence of a sequence of operators 4. inverse operators 5. Space of operators. Ring of operators 6. Resolvent of a bounded linear operator. Spectrum 7. Adjoint operator 8. Completely continuous operators 9. Operators with an everywhere dense domain of definition. Linear equations. 10. Closed unbounded operato 11. Remark on complex spaces § 6. Spaces with a basis 1. Completeness and minimality of a system of elements. 2. Concept of a basis 3. Criteria for bases 4. Unconditional bases 5. Stability of a basis CHAPTER II: LINEAR OPERATORS IN HILBERT SPACE § 1. Abstract Hubert space 1. Concept of a Hilbert space 2. Examples of Hilbert spaces 3. Orthogonality. Projection onto a subspace 4. Linear functionals 5. Weak convergence 6. Orthonormal systems § 2. Bounded linear operators in a Hilbert space 1. Bounded linear operators. Adjoint operators. Bilinear forms 2. Unitary operators 3. Self-adjoint operators 4. Self-adjoint completely continuous operators 5. Completely continuous operators 6. Projective operators § 3. Spectral expansion of seif-adjoint operators 1. Operations on seif-adjoint opera 2. Resolution of the identity. The spectral function 3. Functions of a seif-adjoint ope 4. Unbounded seif-adjoint operators 5. Spectrum of a seif-adjoint operator. 6. Theory of perturbations 7. Multiplicity of the spectrum of a seif-adjoint operator. 8. Generalized eigenvectors. § 4. Symmetric operators 1. Concept of a symmetric operator, deficiency indices 2. SeIf-adjoint extensions of symmetric operators 3. SeIf-adjoint extensions of semi-bounded operators 4. Dissipative extensions § 5. Ordinary differential operators 1. SeIf-adjoint differential expressions 2. Regular case 3. Singular case 4. Criteria for self-adjointness of the operator Ao on (- \infinity, \infinity). 5. Nature of the spectrum of self-ac/joint extensions 6. Expansion in terms of eigenfunctions 7. Examples 8. Inverse Sturm-Liouville problem § 6. Elliptic differential operators of second order 1. Self-adjoint elliptic differential expressions 2. Minimal and maximal operators. L-harmonic functions 3. Self-adjoint extensions corresponding to basic boundary value problenis. § 7. Hubert scale of spaces 1. Hilbert scale and its properties 2. Example of a Flilbert scale. The spaces W2 3. Operators in a Hubert scale 4. Theorems about traces CHAPTER III: LINEAR DIFFERENTIAL EQUATIONS IN A BANACH SPACE § 1. Linear equations with a bounded operator 1. Linear equations of first order. Cauchy problem 2. Homogeneous equations with a constant operator 3. Case of a Hilbert space 4. Equations of second order 5. Homogeneous equation with a variable operator § 2. Equation with a constant unbounded operator. Semi-groups 1. Cauchy problem 2. Uniformly correct Cauchy problem 3. Generating operator and its resolvent 4. Weakened Cauchy problem 5. Abstract parabolic equation. Analytic semi-groups 6. Reverse Cauchy problem 7. Equations in a Hi/bert space 8. Examples of well posed problems for partial differential equations 9. Equations in a space with a basis. Continual integrals § 3. Equation with a variable unbounded operator 1. Homogeneous equation 2. Case of an operator A (t) with a variable domain of definition 3. Non-homogeneous equation 4. Fractional powers of operators CHAPTER IV: NONLINEAR OPERATOR EQUATIONS Introductory remarks § 1. Nonlinear operators and functionals 1. Continuity and boundedness of an operator 2. Differentiability of a nonlinear operator 3. Integration of abstract functions 4. Urysohn operator in the spaces C and Lp 5. Operator f. 6. Hammerstein operator 7. Derivatives of higher order 8. Potential operators § 2. Existence of solutions 1. Method of successive approximations 2. Principle of contractive mappings 3. Uniqueness of a solution 4. Equations with completely continuous operators. Schauder principle 5. Use of the theory of completely continuous vector fields 6. Variational method. 7. Transformation of equations 8. Examples. Decomposition of operators. § 3. Qualitative methods in the theory of branching of solutions 1. Extension of solutions, implicit function theorem 2. Branch points 3. Points of bifurcation, linearization principle 4. Examples from mechanics. 5. Equations with potential operators 6. Appearance of large solutions 7. Equation of branching 8. Construction of solutions in the form of a series CHAPTER V: OPERATORS IN SPACES WITH A CONE § 1. Cones in linear spaces 1. Cone in a linear system 2. Partially ordered spaces 3. Vector lattices, minihedral cones 4. K-spaces 5. Cones in a Banach space 6. Regular cones 7. Theorems on the realization of partially ordered spaces § 2. Positive linear functionals 1. Positive functionals 2. Extension of positive linear fun ctionals 3. Uniformly positive fun ctionals 4. Bounded functionals on a cone § 3. Positive linear operators 1. Concept of a positive operator 2. Affirmative eigenvalues 3. Positive operators on a minihedral cone 4. Non-homogeneous linear equation 5. invariant functionals and eigenvectors of conjugate operators 6. Inconsistent inequalities § 4. Nonlinear operators 1. Basic concepts 2. Existence of positive solutions 3. Existence of a non-zero positive solution 4. Concave operators 5. Convergence of successive approximations CHAPTER VI: COMMUTATIVE NORMED RINGS (BANACH ALGEBRAS) § 1. Basic concepts 1. Commutative normed rings.*) 2. Examples of normed rings 3. Normed fields. 4. Maximal ideals and multiplicative functionals 5. Maximal ideal space 6. Ring boundary of the space R 7. Analytic functions on a ring 8. Invariant subspaces of R'. 9. Rings with involution § 2. Group rings. Harmonic analysis 1. Group rings 2. The characters of a discrete group and maximal ideals of a group ring. 3. Compact groups. Principle of duality 4. Locally compact groups 5. Fourier transforms 6. Hypercomplex systems § 3. Regular rings 1. Regular rings 2. Closed ideals 3. The ring C(S) and its subrings CHAPTER VII: OPERATORS OF QUANTUM MECHANICS § 1. General statements of quantum mechanics 1. State and physical magnitudes of a quantum-mechanical system. Representations of algebraic systems 3. Coordinates and impulses 4. Energy operator. Schrôdinger equation 5. Concrete quantum-mechanical systems 6. Transition from quantum mechanics to classica/ mechanics. § 2. Self-adjointness and the spectrum of the energy operator 1. Criterion for self-adjointness 2. Nature of the spectrum of a radial Schrädinger operator 3. Nature of the spectrum of a one-dimensional Schrödinger operator 4. Nature of the spectrum of a three-dimensional Schrodinger operator § 3. Discrete spectrum, eigenfunctions 1. Exact solutions 2. General properties of the solutions of the Schrödinger equation 3. Quasi-classical approximation for solutions of the one-dimensional Schrödinger equation 4. Calculation of eigen values in one-dimensional and radial symmetric cases 5. Perturbation theory § 4. Solution of the Cauchy problem for the Schrödinger equation 1. General information 2. Theory of perturbations. 3. Physical interpretation 4. Quasi-classical asymptotics of the Green's function 5. Passage to the limit as h—> 0 6. Quasi-classical asyniptotics of a solution of the Dirac equation § 5. Continuous spectrum of the energy operator and the problem of scattering 1. Formulation of the problem 2. Basis for the formulation of the problem and its solution 3. Amplitude of scattering and its equation 4. Case of spherical symmetry 5. General case 6. inverse problem of the theory of scattering CHAPTER VIII: GENERALIZED FUNCTIONS § 1. Generalized functions and operations on them 1. Introductory remarks 2. Notation 3. Generalized functions 4. Operations on generalized functions 5. Differentiation and integration of generalized functions. 6. Limit of a sequence of generalized functions 7. Local properties of generalized functions 8. Direct product of generalized functions 9. Convolution of generalized functions 10. Genera/form of generalized functions 11. Kernel Theorem § 2. Generalized functions and divergent integrals 1. Regularization of divergent integrals 2. Regularization of the functions x^2, x^3, x^-n and their linear combinations 3. Regularization of functions with algebraic singularities 4. Regularization on a finite segment. 5. Regularization at infinity 6. Non-canonical regularizations 7. Generalized functions x^2+, x^2_, and functions which are analogous tothem as function of the parameter \lambda. 8. Homogeneous generalized functions 9. Table of derivatives of some generalized functions 10. Differentiation and integration of arbitrary order 11. Expression of some special functions in the form of derivatives of fractional order. § 3. Some generalized functions of several variables 1. The generalized function r^\lambda 2. Generalized functions connected with quadratic forms 3. Generalizedfunctions (P+iO)^\lambda and (P—iO)^\lambda. 4. Generalized functions of the form 5. Generalized functions on smooth surfaces § 4. Fourier transformation of generalized functions 1. The space S and generalized functions of exponential growth. 2. Fourier transformation of generalizedfunctions of exponential growth 3. Fourier transformation of arbitrary generalized functions 4. Table of Fourier transforms of generalized functions of one variable. 6. Positive definite generalized functions § 5. Radon transformation 1. Radon transformation of test functions and its properties 2. Radon transformation of generalized functions § 6. Generalized functions and differential equations 1. Fundamental solutions 2. Fundamental solutions for some differential equations 3. Construction of fundamental solutions for elliptic equations 4. Fundamental solutions of homogeneous regular equations 5. Fundamental solution of the Cauchy problem § 7. Generalized functions in a complex space 1. Generalizedfunctions of one complex variable 2. Generalized functions of m complex variables BIBLIOGRAPHY INDEX OF LITERATURE ACCORDING TO CHAPTERS