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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.
List of contents
12 General Theory of Unbounded Operators in Hilbert Spaces.- 1 Definition of an Unbounded Operator. The Graph of an Operator.- 1.1 Definitions.- 1.2 Graphs of Operators.- 2 Closed and Closable Operators. Differential Operators.- 2.1 Closed Operators.- 2.2 Closable Operators.- 2.3 Differential Operators.- 3 The Adjoint Operator.- 3.1 Definition and Properties of the Adjoint Operator.- 3.2 The Second Adjoint Operator.- 3.3 The Closed Graph Theorem.- 4 Defect Numbers of General Operators.- 4.1 Deficient Subspaces.- 4.2 Defect Numbers.- 5 Hermitian and Selfadjoint Operators. General Theory.- 5.1 Hermitian Operators.- 5.2 Criterion of Selfadjointness.- 5.3 Semibounded Operators.- 6 Isometric and Unitary Operators. Cayley Transformation.- 6.1 Defect Numbers of Isometric Operators.- 6.2 Direct Cayley Transformation.- 6.3 Inverse Cayley Transformation.- 7 Extensions of Hermitian Operators to Selfadjoint Operators.- 7.1 The Construction of Extensions.- 7.2 Von Neumann Formulas.- 13 Spectral Decompositions of Selfadjoint, Unitary, and Normal Operators. Criteria of Selfadjointness.- 1 The Resolution of the Identity and Its Properties.- 1.1 The Resolution of the Identity.- 1.2 Theorem on Extension.- 2 The Construction of Spectral Integrals.- 2.1 Integrals of Simple Functions.- 2.2 Integrals of Bounded Measurable Functions.- 2.3 Integrals of Unbounded Measurable Functions.- 2.4 Other Properties of Spectral Integrals.- 3 Image of a Resolution of the Identity. Change of Variables in Spectral Integrals. Product of Resolutions of the Identity.- 3.1 Image of a Resolution of the Identity.- 3.2 Product of Resolutions of the Identity.- 4 Spectral Decomposition of Bounded Selfadjoint Operators.- 4.1 The Spectral Theorem.- 4.2 Functions of Operators and Their Spectrum.- 5 Spectral Decompositions for Unitary and Bounded Normal Operators.- 5.1 Spectral Theorem for Unitary Operators.- 5.2 Spectral Theorem for Normal Operators.- 6 Spectral Decompositions of Unbounded Operators.- 6.1 Selfadjoint Operators.- 6.2 Stone s Formula.- 6.3 Commuting Operators.- 6.4 The Function E?.- 6.5 The Case of Normal Operators.- 7 Spectral of Representation One-Parameter Unitary Groups and Operator Differential Equations.- 7.1 Stone s Theorem.- 7.2 Operator Differential Equations.- 8 Evolutionary Criteria of Selfadjointness.- 8.1 The Schrödinger Criterion of Selfadjointness.- 8.2 The Hyperbolic Criterion of Selfadjointness.- 8.3 The Parabolic Criterion of Selfadjointness.- 9 Quasianalytic Criteria of Selfadjointness and Commutability.- 9.1 The Quasianalytic Criterion of Selfadjointness.- 9.2 Other Criteria of Selfadjointness.- 9.3 Commutability of Operators.- 10 Selfadjointness of Perturbed Operators.- 14 Rigged Spaces.- 1 Hilbert Riggings.- 1.1 Positive and Negative Norms.- 1.2 Operators in Chains.- 2 Rigging of Hilbert Spaces by Linear Topological Spaces.- 2.1 Topological Spaces.- 2.2 Projective Limits of Spaces.- 2.3 Riggings Constructed by Using Projective Limits.- 3 Sobolev Spaces in Bounded Domains.- 3.1 The ?-Function.- 3.2 Embeddings of Sobolev Spaces.- 4 Sobolev Spaces in Unbounded Domains. Classical Spaces of Test Functions.- 4.1 The ?-Function.- 4.2 Embeddings of Weighted Sobolev Spaces.- 4.3 The Classical Spaces of Test Functions.- 5 Tensor Products of Spaces.- 5.1 Tensor Products of Spaces.- 5.2 Tensor Products of Operators.- 5.3 Tensor Products of Chains.- 5.4 Projective Limits.- 6 The Kernel Theorem.- 6.1 Hilbert Riggings.- 6.2 Nuclear Riggings.- 6.3 Bilinear Forms.- 6.4 One More Kernel Theorem.- 7 Completions of a Space with Respect to Two Different Norms.- 7.1 Completions with Respect to Two Different Norms.- 7.2 Examples.- 8 Semibounded Bilinear Forms.- 8.1 Lemma on Hilbert Riggings.- 8.2 Positive Forms.- 8.3 Semibounded Forms.- 8.4 Form Sums of Operators.- 15 Expansion in Generalized Eigenvectors.- 1 Differentiation of Operator-Valued Measures and Resolutions of the Identity.- 1.1 Differentiation of Operator-Valued Measures.- 1.2 Differentiation of a Re
