Functional Analysis Vol. I

"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.- 2 Systems of Sets.- 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.- 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.- 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.- 9 Integration of Complex-Valued Functions.- 10 Integrals over 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.- 8 Spaces of Summable Functions. Spaces Lp.- 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.- 4 Corollaries of the Hahn-Banach Theorem.- 5 General Form of Linear Continuous Functionals in Some Banach Spaces.- 6 Embedding of a Linear Normed Space in the Second Dual Space. Reflexive Spaces.- 7 Banach-Steinhaus Theorem. Weak Convergence.- 8 Tikhonov Product. Weak Topology in the Dual Space.- 9 Orthogonality and Orthogonal Projections in Hilbert Spaces. General Form of a Linear Continuous Functional.- 10 Orthonormal Systems of Vectors and Orthonormal Bases in Hilbert Spaces.- 8 Linear Continuous Operators.- 1 Linear Operators in Normed Spaces.- 2 The Space of Linear Continuous Operators.- 3 Product of Operators. The Inverse Operator.- 4 The Adjoint Operator.- 5 Linear Operators in Hilbert Spaces.- 6 Matrix Representation of Operators in Hilbert Spaces.- 7 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.- 4 Spectrum of a Compact Operator.- 5 Spectral Radius of an Operator.- 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.- 2 Integral Operators with Hermitian Kernels.- 3 The Bochner Integral.- 4 Analytic Functions of Operators.- 11 Elements of the Theory of Generalized Functions.- 1 Test and Generalized Functions.- 2 Operations with Generalized Functions.- 3 Tempered Generalized Functions. Fourier Transformation.- Bibliographical Notes.- References.