Topological Vector Spaces, Distributions and Kernels

by Treves, Francois

• ISBN13: 9780486453521
• ISBN10: 0486453529
• Format: Paperback
• Copyright: 2006-12-15
• Publisher: Dover Publications

Summary

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. It features basic classical results, plus 390 exercises. 1967 edition.

Table of Contents

Preface	ix
Part I. Topological Vector Spaces. Spaces of Functions	1	(174)
Filters. Topological Spaces. Continuous Mappings	6	(8)
Vector Spaces. Linear Mappings	14	(6)
Topological Vector Spaces. Definition	20	(11)
Hausdorff Topological Vector Spaces. Quotient Topological Vector Spaces. Continuous Linear Mappings	31	(6)
Hausdorff Topological Vector Spaces	31	(2)
Quotient Topological Vector Spaces	33	(1)
Continuous Linear Mappings	34	(3)
Cauchy Filters. Complete Subsets. Completion	37	(13)
Compact Sets	50	(7)
Locally Convex Spaces. Seminorms	57	(13)
Metrizable Topological Vector Spaces	70	(8)
Finite Dimensional Hausdorff Topological Vector Spaces. Linear Subspaces with Finite Codimension. Hyperplanes	78	(7)
Frechet Spaces. Examples	85	(10)
Example I. The Space of lk Functions in a Open Subset Ω of Rn	85	(4)
Example II. The Space of Holomorphic Functions in an Open Subset Ω of Cn	89	(2)
Example III. The Space of Formal Power Series in n Indeterminates	91	(1)
Example IV. The Space y of g∞ Functions in Rn, Rapidly Decreasing at Infinity	92	(3)
Normable Spaces. Banach Spaces. Examples	95	(17)
Hilbert Spaces	112	(14)
Spaces LF. Examples	126	(10)
Bounded Sets	136	(14)
Approximation Procedures in Spaces of Functions	150	(11)
Partitions of Unity	161	(5)
The Open Mapping Theorem	166	(9)
Part II. Duality. Spaces of Distributions	175	(220)
The Hahn-Banach Theorem	181	(14)
Problems of Approximation	186	(1)
Problems of Existence	187	(2)
Problems of Separation	189	(6)
Topologies on the Dual	195	(7)
Examples of Duals among Lp Spaces	202	(14)
Example I. The Duals of the Spaces of Sequences lp (1≤ p < + ∞)	206	(4)
Example II. The Duals of the Spaces Lp(Ω) (1≤p < + ∞)	210	(6)
Radon Measures. Distributions	216	(11)
Radon Measures in an Open Subset of Rn	216	(6)
Distributions in an Open Subset of Rn	222	(5)
More Duals: Polynomials and Formal Power Series. Analytic Functionals	227	(13)
Polynomials and Formal Power Series	227	(4)
Analytic Functionals in an Open Subset ∞ of Cn	231	(9)
Transpose of a Continuous Linear Map	240	(13)
Example I. Injections of Duals	243	(2)
Example II. Restrictions and Extensions	245	(2)
Example III. Differential Operators	247	(6)
Support and Structure of a Distribution	253	(14)
Distributions with Support at the Origin	264	(3)
Example of Transpose: Fourier Transformation of Tempered Distributions	267	(11)
Convolution of Functions	278	(6)
Example of Transpose: Convolution of Distributions	284	(14)
Approximation of Distributions by Cutting and Regularizing	298	(7)
Fourier Transforms of Distributions with Compact Support. The Paley-Wiener Theorem	305	(9)
Fourier Transforms of Convolutions and Multiplications	314	(8)
The Sobolev Spaces	322	(13)
Equicontinuous Sets of Linear Mappings	335	(11)
Barreled Spaces. The Banach-Steinhaus Theorem	346	(5)
Applications of the Banach-Steinhaus Theorem	351	(9)
Application to Hilbert Spaces	351	(1)
Application to Separately Continuous Functions on Products	352	(2)
Complete Subsets of LG(E; F)	354	(2)
Duals of Montel Spaces	356	(4)
Further Study of the Weak Topology	360	(8)
Topologies Compatible with a Duality. The Theorem of Mackey. Reflexivity	368	(10)
The Normed Space EB	370	(4)
Examples of Semireflexive and Reflexive Spaces	374	(4)
Surjections of Frechet Spaces	378	(9)
Proof of Theorem 37.1	379	(4)
Proof of Theorem 37.2	383	(4)
Surjections of Frechet Spaces (continued). Applications	387	(8)
Proof of Theorem 37.3	387	(3)
An Application of Theorem 37.2: A Theorem of E. Borel	390	(1)
An Application of Theorem 37.3: A Theorem of Existence of l∞ Solutions of a Linear Partial Differential Equation	391	(4)
Part III. Tensor Products. Kernels	395	(164)
Tensor Product of Vector Spaces	403	(8)
Differentiable Functions with Values in Topological Vector Spaces. Tensor Product of Distributions	411	(9)
Bilinear Mappings. Hypocontinuity	420	(7)
Proof of Theorem 41.1	421	(6)
Spaces of Bilinear Forms. Relation with Spaces of Linear Mappings and with Tensor Products	427	(7)
The Two Main Topologies on Tensor Products. Completion of Topological Tensor Products	434	(12)
Examples of Completion of Topological Tensor Products: Products ε	446	(13)
Example 44.1. The Space lm(X; E) of lm Functions Valued in a Locally Convex Hausdorff Space E (0 ≤ m ≤ + ∞)	446	(5)
Example 44.2. Summable Sequences in a Locally Convex Hausdorff Space	451	(8)
Examples of Completion of Topological Tensor Products: Completed π-Product of Two Frechet Spaces	459	(8)
Examples of Completion of Topological Tensor Products: Completed π-Products with a Space L1	467	(10)
The Spaces Lα(E)	467	(2)
The Theorem of Dunford-Pettis	469	(4)
Application to L1 π E	473	(4)
Nuclear Mappings	477	(11)
Example. Nuclear Mappings of a Banach Space into a Space L1	486	(2)
Nuclear Operators in Hilbert Spaces	488	(12)
The Dual of E ε F. Integral Mappings	500	(9)
Nuclear Spaces	509	(17)
Proof of Proposition 50.1	516	(10)
Examples of Nuclear Spaces. The Kernels Theorem	526	(9)
Applications	535	(24)
Appendix: The Borel Graph Theorem	549	(8)
Bibliography for Appendix	557	(1)
General Bibliography	558	(1)
Index of Notation	559	(2)
Subject Index	561