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Functional Analysis is a comprehensive, 2-volume treatment of a subject lying at the core of modern analysis and mathemati- cal 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 ad- vanced 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.- 2 Closed and Closable Operators. Differential Operators.- 3 The Adjoint Operator.- 4 Defect Numbers of General Operators.- 5 Hermitian and Selfadjoint Operators. General Theory.- 6 Isometric and Unitary Operators. Cayley Transformation.- 7 Extensions of Hermitian Operators to Selfadjoint Operators.- 13 Spectral Decompositions of Selfadjoint, Unitary, and Normal Operators. Criteria of Selfadjointness.- 1 The Resolution of the Identity and Its Properties.- 2 The Construction of Spectral Integrals.- 3 Image of a Resolution of the Identity. Change of Variables in Spectral Integrals. Product of Resolutions of the Identity.- 4 Spectral Decomposition of Bounded Selfadjoint Operators.- 5 Spectral Decompositions for Unitary and Bounded Normal Operators.- 6 Spectral Decompositions of Unbounded Operators.- 7 Spectral of Representation One-Parameter Unitary Groups and Operator Differential Equations.- 8 Evolutionary Criteria of Selfadjointness.- 9 Quasianalytic Criteria of Selfadjointness and Commutability.- 10 Selfadjointness of Perturbed Operators.- 14 Rigged Spaces.- 1 Hilbert Riggings.- 2 Rigging of Hilbert Spaces by Linear Topological Spaces.- 3 Sobolev Spaces in Bounded Domains.- 4 Sobolev Spaces in Unbounded Domains. Classical Spaces of Test Functions.- 5 Tensor Products of Spaces.- 6 The Kernel Theorem.- 7 Completions of a Space with Respect to Two Different Norms.- 8 Semibounded Bilinear Forms.- 15 Expansion in Generalized Eigenvectors.- 1 Differentiation of Operator-Valued Measures and Resolutions of the Identity.- 2 Generalized Eigenvectors and the Projection Spectral Theorem.- 3 Fourier Transformation in Generalized Eigenvectors and the Direct Integral of Hilbert Spaces.- 4 Expansion in Eigenfunctions of Càrleman Operators.- 16 Differential Operators.- 1 Theorem on Isomorphisms for Elliptic Operators.- 2 Local Smoothing of Generalized Solutions of Elliptic Equations.- 3 Elliptic Differential Operators in a Domain with Boundary.- 4 Differential Operators in ?N.- 5 Expansion in Eigenfunctions and Green's Function of Elliptic Differential Operators.- 6 Ordinary Differential Operators.- Bibliographical Notes.- References.
