Topological vector spaces, distributions and kernels

Title page
Preface
Part I: Topological Vector Spaces. Spaces of Functions
	1. Filters. Topological Spaces. Continuous Mappings
	2. Vector Spaces. Linear Mappings
	3. Topological Vector Spaces. Definition
	4. Hausdorff Topological Vector Spaces. Quotient Topological Vector Spaces. Continuous Linear Mappings
		Hausdorff Topological Vector Spaces
		Quotient Topologlcal Vector Spaces
		Continuous Linear Mappings
	5. Cauchy Filters. Complete Subsets. Completion
	6. Compact Sets
	7. Locally Convex Spaces. Seminorms
	8. Metrizable Topological Vector Spaces
	9. Finite Dimensional Hausdorff Topological Vector Spaces. Linear Subspaces with Finite Codimension. Hyperplanes
	10. Fréchet Spaces. Examples
		EXAMPLE 1. The Space of C^∞ Functions in a Open Subset Ω of R^n
		EXAMPLE II. The Space of Holomorphic Functions in an Open Subset Ω of C^n
		EXAMPLE III. The Space of Formal Power Series in n Indeterminates
		EXAMPLE IV. The Space S of C^∞ Functions in R^n, Rapidly Decreasing at Infinity
	11. Normable Spaces. Banach Spaces. Examples
	12. Hilbert Spaces
	13. Spaces LF. Examples
	14. Bounded Sets
	15. Approximation Procedures in Spaces of Functions
	16. Partitions of Unity
	17. The Open Mapping Theorem
Part II: Duality. Spaces of Distributions
	18. The Hahn-Banach Theorem
		(1) Problems of Approximation
		(2) Problems of Existence
		(3) Problems of Separation
	19. Topologies on the Dual
	20. Examples of Duals among L^p Spaces
		EXAMPLE 1. The Duals of the Spaces of Sequences l^p (1<=p<+∞)
		EXAMPLE II. The Duals of the Spaces L^p(Ω<=p<+∞0
	21. Radon Measures. Distributions
		Radon Measures in an Open Subset of R^n
		Distributions in an Open Subset of R^n
	22. More Duals: Polynomials .and FormaI Power Series. Analytic Functionals
		Polynomials and Formal Power Series
		Analytic Functionals in an Open Subset Ω of C^n
	23. Transpose of a Continuous Linear Map
		EXAMPLE I. Injections of Duals
		EXAMPLE II. Restrictions and Extensions
		EXAMPLE III. Differential Operators
	24. Support and Structure of a Distribution
		Distributions with Support at the Origin
	25. Example of Transpose: Fourier Transformation of Tempered Distributions
	26. Convolution of Functions
	27. Example of Transpose: Convolution of Distributions
	28. Approximation of Distributions by Cutting and Regularizing
	29. Fourier Transforms of Distributions with Compact Support. The Paley-Wiener Theorem
	30. Fourier Transforms of Convolutions and Multiplications
	31. The Sobolev Spaces
	32. Equicontinuous Sets of Linear Mappings
	33. Barreled Spaces. The Banach-Steinhaus Theorem
	34. Applications of the Banach-Steinhaus Theorem
		34.1. Application to Hilbert Spaces
		34.2. Application to Separately Continuous Functions on Products
		34.3. Complete Subsets of L_S(E;F)
		34.4. Duals of Montel Spaces
	35. Further Study of the Weak Topology
	36. Topologies Compatible with a Duality. The Theorem of Mackey. Reflexivity
		The Normed Space E_B
		Examples of Semireflexive and Reflexive Spaces
	37. Surjections of Fréchet Spaces
		Proof of Theorem 37.1
		Proof of Theorem 37.2
	38. Surjections of Fréchet Spaces (continued). Applications
		Proof of Theorem 37.3
		An Application of Theorem 37.2: A Theorem of E. Borel
		An Application of Theorem 37.3: A Theorem of Existence of C^∞ Solutions of a Linear Partial Differential Equation
Part III: Tensor Products. Kernels
	39. Tensor Product of Vector Spaces
	40. Differentiable Functions with Values in Topological Vector Spaces. Tensor Product of Distributions
	41. Bilinear Mappings. Hypocontinuity
		Proof of Theorem 41.1
	42. Spaces of Bilinear Forms. Relation with Spaces of Linear Mappings and with Tensor Products
	43. The Two Main Topologies on Tensor Products. Completion of Topological Tensor Products
	44. Examples of Completion of Topological Tensor Products: Products ε
		EXAMPLE 44.1. The Space C^m(X;E) of C^m Functions Valued in a Locally Convex Hausdorff Space E (0<=m<=+∞)
		EXAMPLE 44.2. Summable Sequences in a Locally Convex Hausdorff Space
	45. Examples of Completion of Topological Tensor Products: Completed π-Product of Two Fréchet Spaces
	46. Examples of Completion of Topological Tensor Products: Completed π-Products with a Space L¹
		46.1. The Spaces L^α(E)
		46.2. The Theorem of Dunford-Pettis
		46.3. Application to L¹@_πE
	47. Nuclear Mappings
		EXAMPLE. Nuclear Mappings of a Banach Space into a Space L¹
	48. Nuclear Operators in Hilbert Spaces
	49. The Dual of E@_πF. Integral Mappings
	50. Nuclear Spaces
		Proof of Proposition 50.1
	51. Examples of Nuclear Spaces. The Kernels Theorem
	52. Applications
Appendix: The Borel Graph Theorem
General Bibliography
Index of Notation
Subject Index