Functional Analysis

1 Measure Theory.- 1 Operations on Sets. Ordered Sets.- 1.1 Operations on Sets n2.- 1.2 Ordered Sets. The Zorn Lemma.- 2 Systems of Sets.- 2.1 Rings and Algebras of Sets.- 2.2 ?-Rings and ?-Algebras.- 2.3 Generated Rings and Algebras.- 3 Measure of a Set. Simple Properties of Measures.- 4 Outer Measure.- 5 Measurable Sets. Extension of a Measure.- 6 Properties of Measures and Measurable Sets.- 7 Monotone Classes of Sets. Uniqueness of Extensions of Measures.- 8 Measures Taking Infinite Values.- 9 Lebesgue Measure of Bounded Linear Sets.- 10 Lebesgue Measure on the Real Line.- 11 Lebesgue Measure in the N-Dimensional Euclidean Space.- 12 Discrete Measures.- 13 Some Properties of Nondecreasing Functions.- 13.1 Discontinuity Points of Monotone Functions.- 13.2 Jump Function. Continuous Part of a Nondecreasing Function.- 14 Construction of a Measure for a Given Nondecreasing Function. Lebesgue-Stieltjes Measure.- 15 Reconstruction of a Nondecreasing Function for a Given Lebesgue-Stieltjes Measure.- 16 Charges and Their Properties.- 16.1 Concept of a Charge. Decomposition in Hahn’s Sense.- 16.2 Decomposition in Jordan’s Sense.- 17 Relationship between Functions of Bounded Variation and Charges.- 2 Measurable Functions.- 1 Measurable Spaces. Measure Spaces. Measurable Functions.- 2 Properties of Measurable Functions.- 3 Equivalence of Functions.- 4 Sequences of Measurable Functions.- 5 Simple Functions. Approximation of Measurable Functions by Simple Functions. The Luzin Theorem.- 3 Theory of Integration.- 1 Integration of Simple Functions.- 2 Integration of Measurable Bounded Functions.- 3 Relationship Between the Concepts of Riemann and Lebesgue Integrals.- 4 Integration of Nonnegative Unbounded Functions.- 5 Integration of Unbounded Functions with Alternating Sign.- 6 Limit Transition under the Sign of the Lebesgue Integral.- 7 Integration over a Set of Infinite Measure.- 8 Summability and Improper Riemann Integrals.- 8.1 Integrals of Unbounded Functions.- 8.2 Integrals over Sets of Infinite Measure.- 9 Integration of Complex-Valued Functions.- 10 Integrals over Charges.- 10.1 Integrals over Charges.- 10.2 Integral over Complex-Valued Charges.- 11 Lebesgue-Stieltjes Integral and Its Relation to the Riemann-Stieltjes Integral.- 12 The Lebesgue Integral and the Theory of Series.- 4 Measures in the Products of Spaces. Fubini Theorem.- 1 Direct Product of Measurable Spaces. Sections of Sets and Functions.- 2 Product of Measures.- 3 The Fubini Theorem.- 4 Products of Finitely Many Measures.- 5 Absolute Continuity and Singularity of Measures, Charges, and Functions. Radon-Nikodym Theorem. Change of Variables in the Lebesgue Integral.- 1 Absolutely Continuous Measures and Charges.- 2 Radon-Nikodym Theorem.- 3 Radon-Nikodym Derivative. Change of Variables in the Lebesgue Integral.- 4 Mappings of Measure Spaces. Change of Variables in the Lebesgue Integral. (Another Approach).- 5 Singularity of Measures and Charges. Lebesgue Decomposition.- 6 Absolutely Continuous Functions. Basic Properties.- 7 Relationship Between Absolutely Continuous Functions and Charges.- 8 Newton-Leibniz Formula. Singular Functions. Lebesgue Decomposition of a Function of Bounded Variation.- 6 Linear Normed Spaces and Hilbert Spaces.- 1 Topological Spaces.- 2 Linear Topological Spaces.- 3 Linear Normed and Banach Spaces.- 4 Completion of Linear Normed Spaces.- 5 Pre-Hilbert and Hilbert Spaces.- 6 Quasiscalar Product and Seminorms.- 7 Examples of Banach and Hilbert Spaces.- 7.1 The Spaces ?N and ?N.- 7.2 The Space C(Q).- 7.3 The Space M(R).- 7.4 The Space Cm($$
\tilde G
$$).- 7.5 The Space C?($$
\tilde G
$$).- 8 Spaces of Summable Functions. Spaces Lp.- 8.1 Hölder and Minkowski Inequalities. Definition of the Spaces Lp.- 8.2 Everywhere Dense Sets in Lp. Separability Conditions.- 8.3 Different Types of Convergence in Lp.- 8.4 The Space lp.- 8.5 The Space L2(R,d?).- 8.6 Essentially Bounded Functions. The Space L?(R,d?).- 8.7 The Space l?.- 8.8 The Sobolev Spaces.- 7 Linear Continuous Functional and Dual Spaces.- 1 Theorem on an Almost Orthogonal Vector. Finite Dimensional Spaces.- 2 Linear Continuous Functional and Their Simple Properties. Dual Space.- 3 Extension of Linear Continuous Functionals.- 3.1 Extension by Continuity.- 3.2 Extension of a Functional Defined on a Subspace.- 4 Corollaries of the Hahn-Banach Theorem.- 5 General Form of Linear Continuous Functionals in Some Banach Spaces.- 5.1 The Concept of a Schauder Basis.- 5.2 The Space Dual to lp (1 < p < ?).- 5.3 The Space Dual to l1.- 5.4 The Space Dual to l?. Banach Limit.- 5.5 The Space Dual to LP(R, d?) (1 < p < ?).- 5.6 The Spaces Dual to L1(R, d?) and L?(R, d?).- 5.7 The Space Dual to C(Q).- 6 Embedding of a Linear Normed Space in the Second Dual Space. Reflexive Spaces.- 7 Banach-Steinhaus Theorem. Weak Convergence.- 7.1 Banach-Steinhaus Theorem.- 7.2 Weak Convergence of Linear Continuous Functional.- 7.3Weak convergence in (C([a, b]))?. The Helly Theorems.- 7.4 Weak Convergence in a Linear Normed Space.- 8 Tikhonov Product. Weak Topology in the Dual Space.- 8.1 Tikhonov Product of Topological Spaces.- 8.2 Weak Topology in the Dual Space.- 9 Orthogonality and Orthogonal Projections in Hilbert Spaces. General Form of a Linear Continuous Functional.- 9.1 Orthogonality. Theorem on the Projection of a Vector onto a Subspace.- 9.2 Orthogonal Sums of Subspaces.- 9.3 Linear Continuous Functionals in Hilbert Spaces.- 10 Orthonormal Systems of Vectors and Orthonormal Bases in Hilbert Spaces.- 10.1 Orthonormal Systems of Vectors. The Bessel Inequality.- 10.2 Orthonormal Bases in H. The Parseval Equality.- 10.3 Orthogonalization of a System of Vectors.- 10.4 Examples of Orthogonal Polynomials.- 10.5 Orthonormal Systems of Vectors of Arbitrary Cardinality.- 8 Linear Continuous Operators.- 1 Linear Operators in Normed Spaces.- 2 The Space of Linear Continuous Operators.- 3 Product of Operators. The Inverse Operator.- 3.1 Product of Operators.- 3.2 Normed Algebras.- 3.3 The Inverse Operator.- 4 The Adjoint Operator.- 5 Linear Operators in Hilbert Spaces.- 5.1 Bilinear Forms.- 5.2 Selfadjoint Operators.- 5.3 Nonnegative Operators.- 5.4 Projection Operators.- 5.5 Normal Operators.- 5.6 Unitary and Isometric Operators.- 6 Matrix Representation of Operators in Hilbert Spaces.- 6.1 Linear Operators in a Separable Space.- 6.2 Selfadjoint Operators.- 6.3 Nonnegative Operators.- 6.4 Orthoprojectors.- 6.5 Isometric Operators.- 6.6 Jacobian Matrices.- 7 Hilbert-Schmidt Operators.- 7.1 Absolute Norm.- 7.2 Integral Hilbert-Schmidt Operators.- 8 Spectrum and Resolvent of a Linear Continuous Operator.- 9 Compact Operators. Equations with Compact Operators.- 1 Definition and Properties of Compact Operators.- 2 Riesz-Schauder Theory of Solvability of Equations with Compact Operators.- 3 Solvability of Fredholm Integral Equations.- 3.1 Some Classes of Integral Operators.- 3.2 Solvability of Fredholm Integral Equations of the Second Kind.- 3.3 Integral Equations with Degenerate Kernels.- 4 Spectrum of a Compact Operator.- 5 Spectral Radius of an Operator.- 5.1 Power Series with Operator Coefficients.- 5.2 Spectral Radius of a Linear Continuous Operator.- 5.3 Method of Successive Approximations.- 6 Solution of Integral Equations of the Second Kind by the Method of Successive Approximations.- 10 Spectral Decomposition of Compact Selfadjoint Operators. Analytic Functions of Operators.- 1 Spectral Decomposition of a Compact Selfadjoint Operator.- 1.1 One Property of Hermitian Bilinear Forms.- 1.2 Theorem on Existence of an Eigenvector for a Selfadjoint Compact Operator.- 1.3 Spectral Theorem for a Compact Selfadjoint Operator.- 2 Integral Operators with Hermitian Kernels.- 2.1 Spectral Decomposition of a Selfadjoint Integral Operator.- 2.2 Bilinear Decomposition of Hermitian Kernels.- 2.3 Hilbert-Schmidt Theorem.- 2.4 Integral Operators with Positive Definite Kernels. The Mercer Theorem.- 3 The Bochner Integral.- 4 Analytic Functions of Operators.- 11 Elements of the Theory of Generalized Functions.- 1 Test and Generalized Functions.- 1.1 Space of Test Functions D (?N).- 1.2 Operators of Averaging.- 1.3 Decomposition of the Unit.- 1.4 Space of Generalized Functions D?(?N).- 1.5 Order of a Generalized Function.- 1.6 Support of a Generalized Function.- 1.7 Regularization.- 2 Operations with Generalized Functions.- 2.1 Operations in D?(?N). Definitions.- 2.2 Multiplication of Generalized Functions by a Smooth Function.- 2.3 Change of Variables in Generalized Functions.- 2.4 Differentiation of Generalized Functions.- 3 Tempered Generalized Functions. Fourier Transformation.- 3.1 The Space S(?N) of Test (Rapidly Decreasing) Functions.- 3.2 The Space S? (?N) of (Tempered) Generalized Functions.- 3.3 Fourier Transformation.- Bibliographical Notes.- References.