Author | Berezansky, Yuri M. author |
---|---|
Title | Functional Analysis [electronic resource] : Vol. I / by Yuri M. Berezansky, Zinovij G. Sheftel, Georgij F. Us |
Imprint | Basel : Birkhรคuser Basel, 1996 |
Connect to | http://dx.doi.org/10.1007/978-3-0348-9185-1 |
Descript | XIX, 426 p. online resource |
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