Functional Analysis
Springer Basel (Verlag)
978-3-7643-5344-5 (ISBN)
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.
Erscheint lt. Verlag | 28.3.1996 |
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Reihe/Serie | Operator Theory: Advances and Applications |
Zusatzinfo | XIX, 426 p. |
Verlagsort | Basel |
Sprache | englisch |
Maße | 156 x 234 mm |
Gewicht | 950 g |
Themenwelt | Mathematik / Informatik ► Mathematik ► Allgemeines / Lexika |
Mathematik / Informatik ► Mathematik ► Analysis | |
Schlagworte | Algebra • Calculus • Compact operator • Equation • Finite • Fourier transform • Function • Functional Analysis • Funktionalanalysis • generalized function • hilbert space • integral theory • Mathematics • spectral theory • Topology • Variable |
ISBN-10 | 3-7643-5344-9 / 3764353449 |
ISBN-13 | 978-3-7643-5344-5 / 9783764353445 |
Zustand | Neuware |
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