# Topological Vector Spaces, Distributions and Kernels

## 1st Edition

### Pure and Applied Mathematics, Vol. 25

**Author:**François Treves

**Editors:**Paul A. Smith Samuel Eilenberg

**eBook ISBN:**9781483223629

**Imprint:**Academic Press

**Published Date:**1st January 1967

**Page Count:**582

## Description

*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.

## Table of Contents

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 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 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 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 Equation

Part 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. Applications

Appendix: The Borel Graph Theorem

Bibliography for Appendix

General Bibliography

Index of Notation

Subject Index

## Details

- No. of pages:
- 582

- Language:
- English

- Copyright:
- © Academic Press 1967

- Published:
- 1st January 1967

- Imprint:
- Academic Press

- eBook ISBN:
- 9781483223629

## About the Author

### François Treves

### Affiliations and Expertise

Purdue University

## About the Editors

### Paul A. Smith

### Samuel Eilenberg

### Affiliations and Expertise

Columbia University