Functional Analysis: Vol 1

"Functional Analysis" is a comprehensive, 2-volume treatment of a subject lying at the core of modern analysis and mathematical physics. The first volume reviews basic concepts such as the measure, the integral, Banach spaces, bounded operators and generalized functions. Volume II moves on to more advanced topics including unbounded operators, spectral decomposition, expansion in generalized eigenvectors, rigged spaces, and partial differential operators. This text provides students of mathematics and physics with a clear introduction into the above concepts, with the theory well illustrated by a wealth of examples. Researchers will appreciate it as a useful reference manual.

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.