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