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# Topological Vector Spaces, Distributions and Kernels

## Pure and Applied Mathematics, Vol. 25

- 1st Edition - June 3, 2016
- Author: François Treves
- Editors: Paul A. Smith, Samuel Eilenberg
- Language: English
- eBook ISBN:9 7 8 - 1 - 4 8 3 2 - 2 3 6 2 - 9

Topological Vector Spaces, Distributions and Kernels discusses partial differential equations involving spaces of functions and space distributions. The book reviews the defini… Read more

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Request a sales quote*Topological Vector Spaces, Distributions and Kernels*discusses partial differential equations involving spaces of functions and space distributions. The book reviews the definitions of a vector space, of a topological space, and of the completion of a topological vector space. The text gives examples of Frechet spaces, Normable spaces, Banach spaces, or Hilbert spaces. The theory of Hilbert space is similar to finite dimensional Euclidean spaces in which they are complete and carry an inner product that can determine their properties. The text also explains the Hahn-Banach theorem, as well as the applications of the Banach-Steinhaus theorem and the Hilbert spaces. The book discusses topologies compatible with a duality, the theorem of Mackey, and reflexivity. The text describes nuclear spaces, the Kernels theorem and the nuclear operators in Hilbert spaces. Kernels and topological tensor products theory can be applied to linear partial differential equations where kernels, in this connection, as inverses (or as approximations of inverses), of differential operators. The book is suitable for vector mathematicians, for students in advanced mathematics and physics.

PrefacePart 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 Topological 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. Frechet Spaces. Examples Example I. The Space of Ck Functions in a Open Subset Ω of Rn Example II. The Space of Holomorphic Functions in an Open Subset Ω of Cn Example III. The Space of Formal Power Series in n Indeterminates Example IV. The Space l of C∞ Functions in Rn, Rapidly Decreasing at Infinity 11. Normable Spaces. Banach Spaces. Examples 12. Hilbert Spaces Examples in Finite Dimensional Spaces Cn 13. Spaces LF. Examples 14. Bounded Sets 15. Approximation Procedures in Spaces of Functions 16. Partitions of Unity 17. The Open Mapping TheoremPart 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 Lp Spaces Example I. The Duals of the Spaces of Sequences lp (1 ≤ p < ∞) Example II. The Duals of the Spaces Lp(Ω) ( 1 ≤ p < + ∞) 21. Radon Measures. Distributions Radon Measures in an Open Subset of Rn Distributions in an Open Subset of Rn 22. More Duals: Polynomials and Formal Power Series. Analytic Functionals Polynomials and Formal Power Series Analytic Functionals in an Open Subset Ω of Cn 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 LG (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 EB 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 EquationPart III. Tensor Products. Kernels 39. Tensor Product of Vector Spaces 40. Diiferentiable 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 Cm(X;E) of Cm 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 Lα(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. TheDualof E⊗εF. Integral Mappings 50. Nuclear Spaces Proof of Proposition 50.1 51. Examples of Nuclear Spaces. The Kernels Theorem 52. ApplicationsAppendix: The Borel Graph Theorem Bibliography for AppendixGeneral BibliographyIndex of NotationSubject Index

- No. of pages: 582
- Language: English
- Edition: 1
- Published: June 3, 2016
- Imprint: Academic Press
- eBook ISBN: 9781483223629

SE

### Samuel Eilenberg

Affiliations and expertise

Columbia UniversityFT

### François Treves

Affiliations and expertise

Purdue University