Topological Vector Spaces, Distributions and Kernels

This text for upper-level undergraduates and graduate students focuses on key notions and results in functional analysis. Extending beyond the boundaries of Hilbert and Banach space theory, it explores aspects of analysis relevant to the solution of partial differential equations. The three-part treatment begins with topological vector spaces and spaces of functions, progressing to duality and spaces of distribution, and concluding with tensor products and kernels. The archetypes of linear partial differential equations (Laplace's, the wave, and the heat equations) and the traditional problems (Dirichlet's and Cauchy's) are this volume's main focus. Most of the basic classical results appear here. There are 390 exercises, several of which contain detailed information that will enable readers to reconstruct the proofs of some important results. Content: Editoral Page Page v Copyright Page Page vi Preface Pages ix-xi François Treves PART I Topological Vector Spaces. Spaces of Functions Pages 2-5 1 Filters. Topological Spaces. Continuous Mappings Pages 6-13 2 Vector Spaces. Linear Mappings Pages 14-19 3 Topological Vector Spaces. Definition Pages 20-30 4 Hausdorff Topological Vector Spaces. Quotient Topological Vector Spaces. Continuous Linear Mappings Pages 31-36 5 Cauchy Filters. Complete Subsets. Completion Pages 37-49 6 Compact Sets Pages 50-56 7 Locally Convex Spaces. Seminorms Pages 57-69 8 Metrizable Topological Vector Spaces Pages 70-77 9 Finite Dimensional Hausdorff Topological Vector Spaces. Linear Subspaces with Finite Codimension. Hyperplanes Pages 78-84 10 Fréchet Spaces. Examples Pages 85-94 11 Normable Spaces. Banach Spaces. Examples Pages 95-111 12 Hilbert Spaces Pages 112-125 13 Spaces LF. Examples Pages 126-135 14 Bounded Sets Pages 136-149 15 Approximation Procedures in Spaces of Functions Pages 150-160 16 Partitions of Unity Pages 161-165 17 The Open Mapping Theorem Pages 166-173 Part II. Duality. Spaces of Distributions Pages 176-179 18 The Hahn-Banach Theorem Pages 181-194 19 Topologies on the Dual Pages 195-201 20 Examples of Duals among L p Spaces Pages 202-215 21 Radon Measures. Distributions Pages 216-226 22 More Duals: Polynomials and Formal Power Series. Analytic Functionals Pages 227-239 23 Transpose of a Continuous Linear Map Pages 240-252 24 Support and Structure of Distribution Pages 253-266 25 Example of Transpose: Fourier Transformation of Tempered Distributions Pages 267-277 26 Convolution of Functions Pages 278-283 27 Example of Transpose: Convolution of Distributions Pages 284-297 28 Approximation of Distributions by Cutting and Regularizing Pages 298-304 29 Fourier Transforms of Distributions with Compact Support. Pages 305-313 30 Fourier Transforms of Convolutions and Multiplications Pages 314-321 31 The Sobolev Spaces Pages 322-334 32 Equicontinuous Sets of Linear Mappings Pages 335-345 33 Barreled Spaces. The Banach-Steinhaus Theorem Pages 346-350 34 Applications of the Banach-Steinhaus Theorem Pages 351-359 35 Further Study of the Weak Topology Pages 360-367 36 Topologies Compatible with a Duality. The Theorem of Mackey. Reflexivity Pages 368-377 37 Surjections of Fréchet Spaces Pages 378-386 38 Surjections of Fréchet Spaces (continued). Applications Pages 387-394 Part III. Tensor Products. Kernels Pages 396-402 39 Tensor Product of Vector Spaces Pages 403-410 40 Differentiable Functions with Values in Topological Vector Spaces. Tensor Product of Distributions Pages 411-419 41 Bilinear Mappings. Hypocontinuity Pages 420-426 42 Spaces of Bilinear Forms. Relation with Spaces of Linear Mappings and with Tensor Products Pages 427-433 43 The Two Main Topologies on Tensor Products. Completion of Topological Tensor Products Pages 434-445 44 Examples of Completion of Topological Tensor Products: Products ε Pages 446-458 45 Examples of Completion of Topological Tensor Products: Completed π-Product of Two Fréchet Spaces Pages 459-466 46 Examples of Completion of Topological Tensor Products: Completed π-Product with a Space L 1 Pages 467-476 47 Nuclear Mappings Pages 477-487 48 Nuclear Operators in Hilbert Spaces Pages 488-499 49 The Dual of E⊗ ε F. Integral Mappings Pages 500-508 50 Nuclear Spaces Pages 509-525 51 Examples of Nuclear Spaces. The Kernels Theorem Pages 526-534 52 Applications Pages 535-547 Appendix: The Borel Graph Theorem Pages 549-558 Index of Notation Pages 559-560 Subject Index Pages 561-565