Generalized Functions

Translator's NoteForewordChapter I The Kernel Theorem. Nuclear Spaces. Rigged Hilbert Space 1. Bilinear Functionals on Countably Normed Spaces. The Kernel Theorem 1.1. Convex Functionals 1.2. Bilinear Functionals 1.3. The Structure of Bilinear Functionals on Specific Spaces (the Kernel Theorem) Appendix. The Spaces K, S, and 2 2. Operators of Hilbert-Schmidt Type and Nuclear Operators 2.1. Completely Continuous Operators 2.2. Hilbert-Schmidt Operators 2.3. Nuclear Operators 2.4. The Trace Norm 2.5. The Trace Norm and the Decomposition of an Operator into a Sum of Operators of Rank 1 3. Nuclear Spaces. The Abstract Kernel Theorem 3.1. Countably Hilbert Spaces 3.2. Nuclear Spaces 3.3. A Criterion for the Nuclearity of a Space 3.4. Properties of Nuclear Spaces 3.5. Bilinear Functionals on Nuclear Spaces 3.6. Examples of Nuclear Spaces 3.7. The Metric Order of Sets in Nuclear Spaces 3.8. The Functional Dimension of Linear Topological Spaces 4. Rigged Hilbert Spaces. Spectral Analysis of Self-Adjoint and Unitary Operators 4.1. Generalized Eigenvectors 4.2. Rigged Hilbert Spaces 4.3. The Realization of a Hilbert Space as a Space of Functions, and Rigged Hilbert Spaces 4.4. Direct Integrals of Hilbert Spaces, and Rigged Hilbert Spaces 4.5. The Spectral Analysis of Operators in Rigged Hilbert Spaces Appendix. The Spectral Analysis of Self-Adjoint and Unitary Operators in Hilbert Space 1. The Abstract Theorem on Spectral Decomposition 2. Cyclic Operators 3. The Decomposition of a Hilbert Space into a Direct Integral Corresponding to a Given Self-Adjoint OperatorChapter II Positive and Positive-Definite Generalized Functions 1. Introduction l.l. Positivity and Positive Definiteness 2. Positive Generalized Functions 2.1. Positive Generalized Functions on the Space of Infinitely Differentiable Functions Having Bounded Supports 2.2. The General Form of Positive Generalized Functions on the Space S 2.3. Positive Generalized Functions on Some Other Spaces 2.4. Multiplicatively Positive Generalized Functions 3. Positive-Definite Generalized Functions. Bochner's Theorem 3.1. Positive-Definite Generalized Functions on S 3.2. Continuous Positive-Definite Functions 3.3. Positive-Definite Generalized Functions on K 3.4. Positive-Definite Generalized Functions on ¿ 3.5. Translation-Invariant Positive-Definite Hermitean Bilinear Functionals 3.6. Examples of Positive and Positive-Definite Generalized Functions 4. Conditionally Positive-Definite Generalized Functions 4.1. Basic Definitions 4.2. Conditionally Positive Generalized Functions (Case of One Variable) 4.3. Conditionally Positive Generalized Functions (Case of Several Variables) 4.4. Conditionally Positive-Definite Generalized Functions on ¿ 4.5. Bilinear Functionals Connected with Conditionally Positive-Definite Generalized Functions Appendix 5. Evenly Positive-Definite Generalized Functions 5.1. Preliminary Remarks 5.2. Evenly Positive-Definite Generalized Functions on S1/2 1/2 5.3. Evenly Positive-Definite Generalized Functions on S1/2 1/2 5.4. Positive-Definite Generalized Functions and Groups of Linear Transformations 6. Evenly Positive-Definite Generalized Functions on the Space of Functions of One Variable with Bounded Supports 6.1. Positive and Multiplicatively Positive Generalized Functions 6.2. A Theorem on the Extension of Positive Linear Functionals 6.3. Even Positive Generalized Functions on ¿ 6.4.