Topological Vector Spaces, Distributions and Kernels

Name: Topological Vector Spaces, Distributions and Kernels
Brand: Academic Press
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Academic Press

2nd Edition

Published on 1. January 1967

563 pages

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978-0-08-087337-4 (ISBN)

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Front Cover
Topological Vector Spaces, Distributions and Kernels
Copyright Page
Contents
Preface
Part I: Topological Vector Spaces. Spaces of Functions
Chapter 1. Filters. Topological Spaces. Continuous Mappings
Chapter 2. Vector Spaces. Linear Mappings
Chapter 3. Topological Vector Spaces. Definition
Chapter 4. Hausdorff Topological Vector Spaces. Quotient Topological Vector Spaces. Continuous Linear Mappings
Hausdorff Topological Vector Spaces
Quotient Topological Vector Spaces
Continuous Linear Mappings
Chapter 5. Cauchy Filters. Complete Subsets. Completion
Chapter 6. Compact Sets
Chapter 7. Locally Convex Spaces. Seminorms
Chapter 8. Metrizable Topological Vector Spaces
Chapter 9. Finite Dimensional Hausdorff Topological Vector Spaces. Linear Subspaces with Finite Codimension. Hyperplanes
Chapter 10. Fréchet Spaces. Examples
Example I. The Space of lk Functions in an Open Subset O of Rn
Example II. The Space of Holomorphic Functions in an Open Subset O of Cn
Example III. The Space of Formal Power Series in n Indeterminates
Example IV. The Space e of e8 Functions in Rn Rapidly Decreasing at Infinity
Chapter 11. Normable Spaces. Banach Spaces. Examples.
Chapter 12. Hilbert Spaces
Chapter 13. Spaces LF. Examples
Chapter 14. Bounded Sets
Chapter 15. Approximation Procedures in Spaces of Functions
Chapter 16. Partitions of Unity
Chapter 17. The Open Mapping Theorem
Part II: Duality. Spaces of Distributions
Chapter 18. The Hahn-Banach Theorem
(1) Problems of Approximation
(2) Problems of Existence
(3) Problems of Separation
Chapter 19. Topologies on the Dual
Chapter 20. Examples of Duals among Lp Spaces
Example I. The Duals of the Spaces of Sequences lp(1 = p & + 8)
Example II. The Duals of the Spaces Lp(O) (1 = p & + 8)
Chapter 21. Radon Measures. Distributions
Radon Measures in an Open Subset O of Rn
Distributions in an Open Subset of Rn
Chapter 22. More Duals: Polynomials and Formal Power Series. Analytic Functionals
Polynomials and Formal Power Series
Analytic Functionals in an Open Subset O of Cn
Chapter 23. Transpose of a Continuous Linear Map
Example I. Injections of Duals
Example II. Restrictions and Extensions
Example III. Differential Operators
Chapter 24. Support and Structure of a Distribution
Distributions with Support at the Origin
Chapter 25.Example of Transpose: Fourier Transformation of Tempered Distributions
Chapter 26. Convolution of Functions
Chapter 27. Example of Transpose: Convolution of Distributions
Chapter 28. Approximation of Distributions by Cutting and Regularizing
Chapter 29. Fourier Transforms of Distributions with Compact Support The Paley-Wiener Theorem
Chapter 30. Fourier Transforms of Convolutions and Multiplications
Chapter 31. The Sobolev Spaces
Chapter 32. Equicontinuous Sets of Linear Mappings
Chapter 33. Barreled Spaces. The Banach-Steinhaus Theorem
Chapter 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 LG(E
F )
34.4. Duals of Montel Spaces
Chapter 35. Further Study of the Weak Topology
Chapter 36. Topologies Compatible with a Duality. The Theorem of Mackey. Reflexivity
The Normed Space EB
Examples of Semireflexive and Reflexive Spaces
Chapter 37. Surjections of Fréchet Spaces
Proof of Theorem 37.1
Proof of Theorem 37.2
Chapter 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 l8 Solutions of a Linear Partial Differential Equation
Part III: Tensor Products. Kernels
Chapter 39. Tensor Product of Vector Spaces
Chapter 40. Differentiable Functions with Values in Topological Vector Spaces. Tensor Product of Distributions
Chapter 41. Bilinear Mappings. Hypocontinuity
Proof of Theorem 41.1
Chapter 42. Spaces of Bilinear Forms. Relation with Spaces of Linear Mappings and with Tensor Products
Chapter 43. The Two Main Topologies on Tensor Products. Completion of Topological Tensor Products
Chapter 44. Examples of Completion of Topological Tensor Products: Products e
Example 44.1. The Space lm (X
E ) of lm Functions Valued in a Locally Convex Hausdorff Space E (0 = m = +8)
Example 44.2. Summable Sequences in a Locally Convex Hausdorff Space
Chapter 45. Examples of Completion of Topological Tensor Products: Completed p-Product of Two Fréchet Spaces
Chapter 46. Examples of Completion of Topological Tensor Products: Completed p-Product with a Spaces L1
46.1. The Spaces La (E)
46.2. The Theorem of Dunford-Pettis
46.3. Application to L1 Xp E
Chapter 47. Nuclear Mappings
Example. Nuclear Mappings of a Banach Space into a Space L1
Chapter 48. Nuclear Operators in Hilbert Spaces
Chapter 49. The Dual of E?e F. Integral Mappings
Chapter 50. Nuclear Spaces
Proof of Proposition 50.1
Chapter 51. Examples of Nuclear Spaces. The Kernels Theorem
Chapter 52. Applications
Appendix: The Borel Graph Theorem
Bibliography for Appendix
General Bibliography
Index of Notation
Subject Index

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