Topological Vector Spaces, Distributions and Kernels (eBook)

Front Cover 1
Topological Vector Spaces, Distributions and Kernels 6
Copyright Page 7
Contents 14
Preface 10
Part I: Topological Vector Spaces. Spaces of Functions 18
Chapter 1. Filters. Topological Spaces. Continuous Mappings 23
Chapter 2. Vector Spaces. Linear Mappings 31
Chapter 3. Topological Vector Spaces. Definition 37
Chapter 4. Hausdorff Topological Vector Spaces. Quotient Topological Vector Spaces. Continuous Linear Mappings 48
Hausdorff Topological Vector Spaces 48
Quotient Topological Vector Spaces 50
Continuous Linear Mappings 51
Chapter 5. Cauchy Filters. Complete Subsets. Completion 54
Chapter 6. Compact Sets 67
Chapter 7. Locally Convex Spaces. Seminorms 74
Chapter 8. Metrizable Topological Vector Spaces 87
Chapter 9. Finite Dimensional Hausdorff Topological Vector Spaces. Linear Subspaces with Finite Codimension. Hyperplanes 95
Chapter 10. Fréchet Spaces. Examples 102
Example I. The Space of lk Functions in an Open Subset . of Rn 102
Example II. The Space of Holomorphic Functions in an Open Subset . of Cn 106
Example III. The Space of Formal Power Series in n Indeterminates 108
Example IV. The Space e of e8 Functions in Rn Rapidly Decreasing at Infinity 109
Chapter 11. Normable Spaces. Banach Spaces. Examples. 112
Chapter 12. Hilbert Spaces 129
Chapter 13. Spaces LF. Examples 143
Chapter 14. Bounded Sets 153
Chapter 15. Approximation Procedures in Spaces of Functions 167
Chapter 16. Partitions of Unity 178
Chapter 17. The Open Mapping Theorem 183
Part II: Duality. Spaces of Distributions 192
Chapter 18. The Hahn-Banach Theorem 198
(1) Problems of Approximation 203
(2) Problems of Existence 204
(3) Problems of Separation 206
Chapter 19. Topologies on the Dual 212
Chapter 20. Examples of Duals among Lp Spaces 219
Example I. The Duals of the Spaces of Sequences lp(1 = p < + 8)
Example II. The Duals of the Spaces Lp(.) (1 = p < + 8)
Chapter 21. Radon Measures. Distributions 233
Radon Measures in an Open Subset . of Rn 233
Distributions in an Open Subset of Rn 239
Chapter 22. More Duals: Polynomials and Formal Power Series. Analytic Functionals 244
Polynomials and Formal Power Series 244
Analytic Functionals in an Open Subset . of Cn 248
Chapter 23. Transpose of a Continuous Linear Map 257
Example I. Injections of Duals 260
Example II. Restrictions and Extensions 262
Example III. Differential Operators 264
Chapter 24. Support and Structure of a Distribution 270
Distributions with Support at the Origin 281
Chapter 25.Example of Transpose: Fourier Transformation of Tempered Distributions 284
Chapter 26. Convolution of Functions 295
Chapter 27. Example of Transpose: Convolution of Distributions 301
Chapter 28. Approximation of Distributions by Cutting and Regularizing 315
Chapter 29. Fourier Transforms of Distributions with Compact Support The Paley-Wiener Theorem 322
Chapter 30. Fourier Transforms of Convolutions and Multiplications 331
Chapter 31. The Sobolev Spaces 339
Chapter 32. Equicontinuous Sets of Linear Mappings 352
Chapter 33. Barreled Spaces. The Banach-Steinhaus Theorem 363
Chapter 34. Applications of the Banach-Steinhaus Theorem 368
34.1. Application to Hilbert Spaces 368
34.2. Application to Separately Continuous Functions on Products 369
34.3. Complete Subsets of LG(E F )
34.4. Duals of Montel Spaces 373
Chapter 35. Further Study of the Weak Topology 377
Chapter 36. Topologies Compatible with a Duality. The Theorem of Mackey. Reflexivity 385
The Normed Space EB 387
Examples of Semireflexive and Reflexive Spaces 391
Chapter 37. Surjections of Fréchet Spaces 395
Proof of Theorem 37.1 396
Proof of Theorem 37.2 400
Chapter 38. Surjections of Fréchet Spaces (continued). Applications 404
Proof of Theorem 37.3 404
An Application of Theorem 37.2: A Theorem of E. Borel 407
An Application of Theorem 37.3: A Theorem of Existence of l8 Solutions of a Linear Partial Differential Equation 408
Part III: Tensor Products. Kernels 412
Chapter 39. Tensor Product of Vector Spaces 420
Chapter 40. Differentiable Functions with Values in Topological Vector Spaces. Tensor Product of Distributions 428
Chapter 41. Bilinear Mappings. Hypocontinuity 437
Proof of Theorem 41.1 438
Chapter 42. Spaces of Bilinear Forms. Relation with Spaces of Linear Mappings and with Tensor Products 444
Chapter 43. The Two Main Topologies on Tensor Products. Completion of Topological Tensor Products 451
Chapter 44. Examples of Completion of Topological Tensor Products: Products e 463
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 468
Chapter 45. Examples of Completion of Topological Tensor Products: Completed p-Product of Two Fréchet Spaces 476
Chapter 46. Examples of Completion of Topological Tensor Products: Completed p-Product with a Spaces L1 484
46.1. The Spaces La (E) 484
46.2. The Theorem of Dunford–Pettis 486
46.3. Application to L1 Xp E 490
Chapter 47. Nuclear Mappings 494
Example. Nuclear Mappings of a Banach Space into a Space L1 503
Chapter 48. Nuclear Operators in Hilbert Spaces 505
Chapter 49. The Dual of E.e F. Integral Mappings 517
Chapter 50. Nuclear Spaces 526
Proof of Proposition 50.1 533
Chapter 51. Examples of Nuclear Spaces. The Kernels Theorem 543
Chapter 52. Applications 552
Appendix: The Borel Graph Theorem 566
Bibliography for Appendix 574
General Bibliography 575
Index of Notation 576
Subject Index 578