Inner Product Structures

1/General Topology. Topological Spaces.- 1.0 Introduction.- 1.1 Sets. Functions.- 1.2 Topology and Topological Spaces.- 1.3 Compactness in Topological Spaces.- 1.4 Metric Spaces. Examples and Some Properties.- 1.5 Measures of Noncompactness in Metric Spaces.- 1.6 Some Historical Remarks.- 2/Banach Spaces and Complete Inner Product Spaces.- 2.0 Introduction.- 2.1 Linear Spaces. Sets in Linear Spaces.- 2.2 Normed Linear Spaces and Banach Spaces.- 2.3 The Extension Theorems.- 2.4 Linear Operators on Banach Spaces. Classes of Linear Operators.- 2.5 Three Basic Theorems of Linear Functional Analysis.- 2.6 Inner Product Spaces. Definitions and Some Examples.- 2.7 Von Neumann Generalized Direct Sums.- 2.8 Tensor Products of Banach Spaces and of Complete Inner Product Spaces.- 3/Orthogonality and Bases.- 3.0 Introduction.- 3.1 Orthogonality in Linear Spaces with an Inner Product.- 3.2 Bases in Complete Inner Product Spaces.- 3.3 Subspaces in Spaces with an Inner Product. The Orthogonal Decomposition.- 3.4 Some Applications of the Fréchet–Riesz Representation Theorem.- 3.5 Some Examples of Bases in Concrete Complete Inner Product Spaces.- 3.6 Perturbation of Bases in Complete Inner Product Spaces.- 3.7 Some Classes of Bases (Hardy Bases) and the Theory of Communication.- 4/Metric Characterizations of Inner Product Spaces.- 4.0 Introduction.- 4.1 Inner Product Structures on Linear Spaces.- 4.2 Inner Product Structures and Complexification.- 4.3 The Fréchet and Jordan–von Neumann Characterization of Inner Product Spaces.- 4.4 The Ficken Characterization of Inner Product Spaces.- 4.5 Closed Maximal Linear Subspaces and Inner Product Structures.- 4.6 Loewner’s Ellipses. Ellipsoids.- 4.7 Ellipses and Inner Product Spaces.- 4.8 The Integral Form of the Parallelogram Law.- 4.9 Topological Inner Productability.- 4.10 Local Norm Characterizations of Inner Product Structures.- 4.11 Other Norm Characterizations of Inner Product Structures.- 4.12 Orthogonality in Normed Linear Spaces and Characterizations of Inner Product Spaces.- 4.13 Approximation Theory and Characterizations of Inner Product Spaces.- 4.14 Chebyshev Centers and Inner Product Structures.- 4.15 On Some Norms on Two-Dimensional Spaces.- 4.16 Parameters Associated with Normed Linear Spaces and Inner Product Structures.- 4.17 The Modulus of Convexity and the Modulus of Smoothness and Inner Product Spaces.- 4.18 Spaces Isomorphic to Inner Product Spaces.- 4.19 Inner Product Spaces and Classes of Metric Spaces.- 4.20 Other Metric Characterizations of Inner Product Spaces.- 4.21 Angles and Complete Inner Product Spaces.- 5/Banach Algebras.- 5.0 Introduction.- 5.1 Definition of Banach Algebras and Some Examples.- 5.2 Ideals in Banach Algebras.- 5.3 The Spectrum of an Element in a Complex Banach Algebra with Identity.- 5.4 The Gelfand Representation. The Representation and Structure of Commutative Banach Algebras.- 5.5 The Representations of B*-Algebras with Identity.- 5.6 Approximate Identities in Banach Algebras.- 5.7 Classes of Elements in Banach Algebras.- 6/Bounded and Unbounded Linear Operators.- 6.0 Introduction.- 6.1 Classes of Bounded Linear Operators on Complete Inner Product Spaces.- 6.2 Normal, Unitary and Partial Isometry Operators.- 6.3 Semispectral and Spectral Families of Radon Measures.- 6.4 Unbounded Operators.- 6.5 Closed and Closable Operators.- 6.6 The Graph of Linear Operators and Some Applications.- 6.7 Hermitian, Selfadjoint and Essentially Selfadjoint Operators.- 6.8 Some Examples of Selfadjoint and Essentially Selfadjoint Operators.- 6.9 Selfadjoint Extensions.- 6.10 Extensions of Semibounded Linear Operators.- 6.11 Unbounded Normal Operators and Some Related Classes of Operators.- 6.12 Some Decomposition Theorems.- 7/Ideals of Operators on Complete Inner Product Spaces and on Banach Spaces.- 7.0 Introduction.- 7.1 Some Terminology and Notations.- 7.2 Ideals of Operators on Complete Inner Product Spaces.- 7.3 The Banach Spaces Cp.- 7.4 Ideal Sets and Ideals of Compact Operators.- 7.5 Banach Ideals. Classes of Summing Operators.- 7.6 Grothendieck’s Fundamental Theorem.- 7.7 On the Coincidence of Classes of Absolutely p-Summing Operators.- 7.8 Types, Cotypes and Rademacher Averages in Banach Spaces.- 8/Operator Characterizations of Inner Product Spaces.- 8.0 Introduction.- 8.1 O-Negative Definite Functions and Inner Product Spaces.- 8.2 Some Inequalities and a Characterization of Inner Product Spaces.- 8.3 Nonexpansive Mappings and the Extension Problem.- 8.4 Fixed Point Sets for Nonexpansive Mappings and Inner Product Structures.- 8.5 Support Mappings and Inner Product Structures.- 8.6 Smooth Functions on Banach Spaces and Inner Product Structures.- 8.7 Classes of Functions on Banach Spaces and Inner Product Structures.- 8.8 Linear Operators and Inner Product Structures.- 8.9 Algebraic Characterizations of Inner Product Structures.- 8.10 Hermitian Decomposition of a Banach Space and Inner Product Spaces.- 8.11 Classes of Hermitian Elements and Inner Product Structures.- 8.12 A Variational Characterization of Inner Product Structures.- 8.13 Von Neumann Spectral Sets and a Characterization of Inner Product Spaces.- 8.14 A Series-Immersed Isomorphic Characterization of Complete Inner Product Spaces.- 8.15 A Symmetric-Invariant Characterization of L2[0,1]..- 9/Probability Theory and Inner Product Structures.- 9.0 Introduction.- 9.1 Probabilities on Banach Spaces.- 9.2 Bernoulli and Gaussian Random Independent Variables and Inner Product Structures.- 9.3 Biconvex Functions and a Characterization of Complete Inner Product Spaces.- 9.4 Other Probabilistic Characterizations of Inner Product Structures.- 10/Positive Definite Functions, Functions of Positive Type and Inner Product Structures.- 10.0 Introduction.- 10.1 Positive Definite Functions. Definitions and Some Examples.- 10.2 The Coincidence of Classes of Positive Definite Functions and Functions of Positive Type on Locally Compact Abelian Groups.- 10.3 Completely Positive Maps. Stinespring’s Theorem.- 10.4 The Nevanlinna Problem.- 10.5 The Monotone Functions of C. Loewner.- 11/Reproducing Kernels and Inner Product Spaces. Applications.- 11.0 Introduction.- 11.1 Reproducing Kernels. Basic Properties.- 11.2 Linear Functionals and Linear Operators on Spaces with Reproducing Kernels.- 11.3 Some Properties of Reproducing Kernels.- 11.4 Functional Completion of a Space, of Functions. The Existence of Complete Inner Product Spaces with Reproducing Kernels.- 11.5 Some Examples of Complete Inner Product Spaces with Reproducing Kernels.- 11.6 Operations on the Reproducing Kernel Functions (the Sum, Products and Limits of Reproducing Kernels).- 11.7 Interpolation, Extremal and Minimal Problems and Reproducing Kernels.- 11.8 Conformal Mappings and Reproducing Kernel Functions.- 11.9 Invariant Subspaces for Generalized Translations and Reproducing Kernels.- 11.10 Spline Functions and Inner Product Spaces with Reproducing Kernels.- 11.11 Dilation Theory and Reproducing Kernels.- 11.12 Some Applications of Reproducing Kernels.- 12/Inner Product Modules.- 12.0 Introduction.- 12.1 Inner Product Modules. Definition and Some Examples. Bounded Module Maps.- 12.2 Some Representation Theorems.- 12.3 Dilation Theory and Inner Product Modules.- 12.4 Von Neumann Algebra Module.- 12.5 The AW*-Modules of Kaplansky.- 12.6 Classes of Kaplansky’s Inner Product Modules.- 13/Quaternionic Complete Inner Product Spaces.- 13.0 Introduction.- 13.1 The Quaternions.- 13.2 Linear Spaces over Quaternions.- 13.3 The Symplectic Image of a Left Quaternionic Complete Inner Product Space.- 13.4 Classes of Operators on Left Quaternionic Inner Product Spaces.- 13.5 Spectral Theory on Left Quaternionic Complete Inner Product Spaces.- 13.6 Functional Calculus for Operators on Left Quaternionic Complete Inner Product Spaces.- 14/Inner Product Algebras.- 14.0 Introduction.- 14.1 Inner Product Algebras.- 14.2 Complete Inner Product Algebras with Identity.- 14.3 H*-Algebras.- 14.4 Inner Product Algebras and H*-Algebras.- 15/Non-Archimedean, Nonstandard, Intuitionistic and Constructive Inner Product Spaces.- 15.0 Introduction.- 15.1 Non-Archimedean Normed Linear Spaces. Non-Archimedean Inner Product Spaces.- 15.2 Nonstandard Inner Product Spaces.- 15.3 Intuitionistic Complete Inner Product Spaces.- 15.4 Constructive Inner Product Spaces.- 16/Indefinite Inner Product Structures.- 16.0 Introduction.- 16.1 Indefinite Inner Product Linear Spaces.- 16.2 Orthogonality and Orthogonal Decomposition.- 16.3 Linear Operators on Spaces with an Indefinite Inner Product.- 16.4 Some Classes of Spaces with an Indefinite Metric.- 16.5 Modules with an Indefinite Inner Product.- 17/Some Applications of Inner Product Structures.- 17.0 Introduction.- 17.1 Certain Applications of the Cauchy–Buniakowsky Inequality to Some Extremal Problems.- 17.2 Invariant Subspaces for the Shift.- 17.3 Fourier Transforms and the Plancherel Theorem.- 17.4 The Sturm–Liouville Problem and Inner Product Spaces.- 17.5 Measures of Dependence of Random Variables and Inner Product Spaces.- 17.6 Ergodic Theory and Complete Inner Product Spaces.- 17.7 Classes of Stochastic Processes and Inner Product Structures.- 17.8 Inner Products and Differentials. Harmonic and Analytic Differentials.- 17.9 Differential Geometry in Complete Inner Product Spaces.- 17.10 Univalent Functions and Complete Inner Product Spaces.- 17.11 Complete Inner Product Spaces and Roots of Polynomials (and Analytic Functions).- 17.12 Bohr’s Basic Theorem of Almost Periodic Functions.- 17.13 Inner Product Structures of Lie Algebras and Jordan Algebras.- 17.14 Potential Theory and Inner Product Structures.- 17.15 Gravity Theory, Statistical Physics and Dynamics and Inner Product Structures.- 17.16 Quantum Mechanics and Operators on Complete Inner Product Spaces.- 17.17 Number Theory and Complete Inner Product Spaces.- 18/A Collection of Problems.- 18.0 Introduction.- 18.1 Problems on Inner Product Structures.- References.- List of Symbols and Abbreviations.- Author Index.