Tools for infinite dimensional analysis

Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Contents
Preface
1.
Schwartz Space
	1.1. The Schwartz Space
	1.2. The Schwartz Norm Topology
	1.3. Fourier Transform
		1.3.1. The multidimensional setting
	1.4. Physical Context
		1.4.1. Hermite polynomials
		1.4.2. Creation and annihilation operators in L2(R, p(x)dx)
	1.5. Orthonormal Basis, Creation, and Annihilation Operators in L2(R)
		1.5.1. Properties of the creation and annihilation operator
	1.6. Number Operator and the Spaces Sp(R)
	1.7. The Projective Limit Topology on S(R)
	1.8. L2 Norms on S(R)
	1.9. Equivalence of the Topologies
	1.10. The Sequence Space
	1.11. Multidimensional
		1.11.1. The operator approach
		1.11.2. The L2 approach
		1.11.3. Topology on multidimensional Schwartz space
		1.11.4. Multidimensional sequence space
2.
Nuclear Spaces
	2.1. Topological Vector Spaces
		2.1.1. Special sets
		2.1.2. Neighborhoods in topological vector spaces
		2.1.3. Convergence and sequential spaces: A word of caution
			2.1.3.1. Sequential spaces
		2.1.4. Seminorms
		2.1.5. Topologies generated by families of topologies
	2.2. Countably-Normed Spaces
		2.2.1. Open sets in a countably-normed space
		2.2.2. Metrizability
		2.2.3. Bounded sets in countably-normed spaces
		2.2.4. Completeness in countably-normed spaces
	2.3. Nuclear Space Structure
		2.3.1. Projective limit topology
		2.3.2. Bounded sets in a nuclear space
		2.3.3. Compactness and the Heine-Borel property
		2.3.4. Linear operators on nuclear spaces
	2.4. Nuclear Space Construction from a Hilbert Space and an Operator
3.
Dual Space
	3.1. Linear Functionals
		3.1.1. Linear functionals on nuclear spaces
	3.2. Dual Space
		3.2.1. Banach-Steinhaus
		3.2.2. Hahn-Banach Theorems
		3.2.3. Weak topology
		3.2.4. Banach-Alaoglu
		3.2.5. Strong topology
	3.3. The Dual of a Nuclear Space
	3.4. Topologies on H\'
		3.4.1. Weak topology on H\'
		3.4.2. Strong topology on H\'
		3.4.3. Gel\'fand Triples and Rigged Hilbert Spaces
		3.4.4. Strongly and weakly bounded sets in H\'
		3.4.5. Weakly compact and strongly compact sets in H\'
	3.5. Reflexivity
		3.5.1. Bounded sets in H revisited
		3.5.2. Weak convergence on H
	3.6. Convergence and Completeness in H\'
		3.6.1. Convergence of linear operators in H
	3.7. Properties of the Topology on H\' and H-p
	3.8. Inductive Limit Topology
		3.8.1. Inductive limit topology on H\'
		3.8.2. Polars and uniform convergence
	3.9. Sequential Space
		3.9.1. Sequential compactness
	3.10. First Countability (Lack Thereof)
	3.11. Summary
	3.12. Abstract Kernel Theorem
	3.13. White Noise Dual
		3.13.1. The sequence space revisited
	3.14. Schwartz Space Dual
	3.15. Boxes and Continuous Functions
		3.15.1. Continuous functions
4.
Probability Theory on Infinite Dimensional Spaces
	4.1. Probability Theory Fundamentals
		4.1.1. Sigma algebras
			4.1.1.1. Dynkin Class Theorem
		4.1.2. Probability measures
		4.1.3. Outer measures
		4.1.4. Riesz-Markov Theorem
		4.1.5. Characteristic functions and Bochner\'s Theorem
	4.2. Product Sigma Algebras in In nite Dimensions
	4.3. Sigma Algebras on the Dual of a Nuclear Space
	4.4. Probability Measures on In nite Dimensional Spaces
		4.4.1. Gaussian measure in in nite dimensions
		4.4.2. Bochner-Minlos Theorem on sequence space
		4.4.3. Bochner-Minlos Theorem
		4.4.4. White noise space
Appendix: Banach and Hilbert Spaces
	A. Vector Spaces
		A.1. Subspaces
		A.2. Span, Linear Dependence, and Dimension
	B. Normed Linear Spaces
		B.1. Closed Subspaces
		B.2. Banach Spaces
	C. Hilbert Spaces
		C.1. Orthogonality
		C.2. Orthonormal basis
		C.3. Isometric Hilbert Spaces
	D. Dual Space and Riesz Representation Theorem
	E. Operators on Hilbert Spaces
		E.1. The adjoint
		E.2. Closure and Essentially Self-adjoint Operators.
		E.3. Bounded Linear Operators
Bibliography
Index