Topological Vector Spaces, Distributions and Kernels

Topological Vector Spaces, Distributions and Kernels

Pure and Applied Mathematics, Vol. 25

1st Edition - January 1, 1967

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  • Author: François Treves
  • eBook ISBN: 9781483223629

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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

Product details

  • No. of pages: 582
  • Language: English
  • Copyright: © Academic Press 1967
  • Published: January 1, 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

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