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Topological Vector Spaces

Chapters 1-5. 1st ed. 1987. 2nd printing 2002. Book. …
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Titel: Topological Vector Spaces
Autor/en: N. Bourbaki, Nicolas Bourbaki

ISBN: 3540423389
EAN: 9783540423386
Chapters 1-5.
1st ed. 1987. 2nd printing 2002.
Book.
Sprache: Englisch.
Übersetzt von H. G. Eggleston, S. Madan
Springer

13. November 2002 - kartoniert - 376 Seiten

This is a softcover reprint of the English translation of 1987 of the second edition of Bourbaki's Espaces Vectoriels Topologiques (1981).This [second edition] is a brand new book and completely supersedes the original version of nearly 30 years ago. But a lot of the material has been rearranged, rewritten, or replaced by a more up-to-date exposition, and a good deal of new material has been incorporated in this book, all reflecting the progress made in the field during the last three decades.Table of Contents.Chapter I: Topological vector spaces over a valued field.Chapter II: Convex sets and locally convex spaces.Chapter III: Spaces of continuous linear mappings.Chapter IV: Duality in topological vector spaces.Chapter V: Hilbert spaces (elementary theory). TOC:Topological Vector Spaces over a Valued Division Ring.- Convex Sets and Locally Convex Spaces.- Spaces of Continuous Linear Mappings.- Duality in Topological Vector Spaces.- Hilbertian Spaces (Elementary Theory).
I. - Topological vector spaces over a valued division ring I..- §
1. Topological vector spaces.
- 1. Definition of a topological vector space.
- 2. Normed spaces on a valued division ring.
- 3. Vector subspaces and quotient spaces of a topological vector space; products of topological vector spaces; topological direct sums of subspaces.
- 4. Uniform structure and completion of a topological vector space.
- 5. Neighbourhoods of the origin in a topological vector space over a valued division ring.
- 6. Criteria of continuity and equicontinuity.
- 7. Initial topologies of vector spaces.- §
2. Linear varieties in a topological vector space.
- 1. Theclosure of a linear variety.
- 2. Lines and closed hyperplanes.
- 3. Vector subspaces of finite dimension.
- 4. Locally compact topological vector spaces.- §
3. Metrisable topological vector spaces.
- 1. Neighbourhoods of 0 in a metrisable topological vector space.
- 2. Properties of metrisable vector spaces.
- 3. Continuous linear functions in a metrisable vector space.- Exercises of § 1.- Exercises of § 2.- Exercises of § 3.- II. - Convex sets and locally convex spaces II..- § 1. Semi-norms.
- 1. Definition of semi-norms.
- 2. Topologies defined by semi-norms.
- 3. Semi-norms in quotient spaces and in product spaces.
- 4. Equicontinuity criteria of multilinear mappings for topologies defined by semi-norms.- § 2. Convex sets.
- 1. Definition of a convex set.
- 2. Intersections of convex sets. Products of convex sets.
- 3. Convex envelope of a set.
- 4. Convex cones.
- 5. Ordered vector spaces.
- 6. Convex cones in topological vector spaces.
- 7. Topologies on ordered vector spaces.
- 8. Convex functions.
- 9. Operations on convex functions.
- 10. Convex functions over an open convex set.
- 11. Semi-norms and convex sets.- § 3. The Hahn-Banach Theorem (analytic form).
- 1. Extension of positive linear forms.
- 2. The Hahn-Banach theorem (analytic form).- §
4. Locally convex spaces.
- 1. Definition of a locally convex space.
- 2. Examples of locally convex spaces.
- 3. Locally convex initial topologies.
- 4. Locally convex final topologies.
- 5. The direct topological sum of a family of locally convex spaces.
- 6. Inductive limits of sequences of locally convex spaces.
- 7. Remarks on Fréchet spaces.- §
5. Separation of convex sets.
- 1. The Hahn-Banach theorem (geometric form).
- 2. Separation of convex sets in a topological vector space.
- 3. Separation of convex sets in a locally convex space.
- 4. Approximation to convex functions.- §
6. Weak topologies.
- 1. Dual vector spaces.
- 2. Weak topologies.
- 3. Polar sets and orthogonal subspaces.
- 4. Transposition of a continuous linear mapping.
- 5. Quotient spaces and subspaces of a weak space.
- 6. Products of weak topologies.
- 7. Weakly complete spaces.
- 8. Complete convex cones in weak spaces.- §
7. Extremal points and extremal generators.
- 1. Extremal points of compact convex sets.
- 2. Extremal generators of convex cones.
- 3. Convex cones with compact sole.- §
8. Complex locally convex spaces.
- 1. Topological vector spaces over C.
- 2. Complex locally convex spaces.
- 3. The Hahn-Banach theorem and its applications.
- 4. Weak topologies on complex vector spaces.- Exercises on § 2.- Exercises on § 3.- Exercises on § 4.- Exercises on § 5.- Exercises on § 6.- Exercises on § 7.- Exercises on § 8.- III. - Spaces of continuous linear mappings III..- § 1. Bornology in a topological vector space.
- 1. Bornologies.
- 2. Bounded subsets of a topological vector space.
- 3. Image under a continuous mapping.
- 4. Bounded subsets in certain inductive limits.
- 5. The spaces EA (A bounded).
- 6. Complete bounded sets and quasi-complete spaces.
- 7. Examples.- § 2. Bornological spaces.- § 3. Spaces of continuous linear mappings.
- 1. Thespaces ?? (E; F).
- 2. Condition for ?? (E; F) to be Hausdorff.
- 3. Relations between ? (E; F) and ? (Ê; F).
- 4. Equicontinuous subsets of 2112 (E; F).
- 5. Equicontinuous subsets of E'.
- 6. The completion of a locally convex space.
- 7. S-bornologies on ? (E; F).
- 8. Complete subsets of ?? (E; F).- § 4. The Banach-Steinhaus theorem.
- 1. Barrels and barrelled spaces.
- 2. The Banach-Steinhaus theorem.
- 3. Bounded subsets of ? (E; F) (quasi-complete case).- § 5. Hypocontinuous bilinear mappings.
- 1. Separately continuous bilinear mappings.
- 2. Separately continuous bilinear mappings on a product of Fréchet spaces.
- 3. Hypocontinuous bilinear mappings.
- 4. Extension of a hypocontinuous bilinear mapping.
- 5. Hypocontinuity of the mapping (u, v) ? v o u.- § 6. Borel's graph theorem.
- 1. Borel's graph theorem.
- 2. Locally convex Lusin spaces.
- 3. Measurable linear mappings on a Banach space.- Exercises on § 1.- Exercises on § 2.- Exercises on § 3.- Exercises on § 4.- Exercises on § 5.- Exercises on § 6.- IV. - Duality in topological vector spaces IV..- § 1. Duality.
- 1. Topologies compatible with a duality.
- 2. Mackey topology and weakened topology on a locally convex space.
- 3. Transpose of a continuous linear mapping.
- 4. Dual of a quotient space and of a subspace.
- 5. Dual of a direct sum and of a product.- § 2. Bidual. Reflexive spaces.
- 1. Bidual.
- 2. Semi-reflexive spaces.
- 3. Reflexive spaces.
- 4. The case of normed spaces.
- 5. Montel spaces.- § 3. Dual of a Fréchet space.
- 1. Semi-barrelled spaces.
- 2. Dual of a locally convex metrizable space.
- 3. Bidual of a locally convex metrizable space.
- 4. Dual of a reflexive Fréchet space.
- 5. The topology of compact convergence on the dual of a Fréchet Space.
- 6. Separately continuous bilinear mappings.- § 4. Strict morphisms of Fréchet spaces.
- 1. Characterizations of strict morphisms.
- 2. Strict morphisms of Fréchet spaces.
- 3. Criteria for surjectivity.- § 5. Compactness criteria.
- 1. General remarks.
- 2. Simple compactness of sets of continuous functions.
- 3. The Eberlein and Smulian theorems.
- 4. The case of spaces of bounded continuous functions.
- 5. Convex envelope of a weakly compact set.- Appendix. - Fixed points of groups of affine transformations.
- 1. The case of solvable groups.
- 2. Invariant means.
- 3. Ryll-Nardzewski theorem.
- 4. Applications.- Exercises on § 1.- Exercises on § 2.- Exercises on § 3.- Exercises on § 4.- Exercises on § 5.- Exercises on Appendix.- Table I. - Principal types of locally convex spaces.- Table II. - Principal homologies on the dual of a locally convex space.- V. - Hilbertian spaces (elementary theory) V..- § 1. Prehilbertian spaces and hilbertian spaces.
- 1. Hermitian forms.
- 2. Positive hermitian forms.
- 3. Prehilbertian spaces.
- 4. Hilbertian spaces.
- 5. Convex subsets of a prehilbertian space.
- 6. Vector subspaces and orthoprojectors.
- 7. Dual of a hilbertian space.- § 2. Orthogonal families in a hilbertian space.
- 1. External hilbertian sum of hilbertian spaces.
- 2. Hilbertian sum of orthogonal subspaces of a hilbertian space.
- 3. Orthonormal families.
- 4. Orthonormalisation.- § 3. Tensor product of hilbertian spaces.
- 1. Tensor product of prehilbertian spaces.
- 2. Hilbertian tensor product of hilbertian spaces.
- 3. Symmetric hilbertian powers.
- 4. Exterior hilbertian powers.
- 5. Exterior Multiplication.- § 4. Some classes of operators in hilbertian spaces.
- 1. Adjoint.
- 2. Partially isometric linear mappings.
- 3. Normal endomorphisms.
- 4. Hermitian endomorphisms.
- 5. Positive endomorphisms.
- 6. Trace of an endomorphism.
- 7. Hilbert-Schmidt mappings.
- 8. Diagonalization of Hilbert-Schmidt mappings.
- 9. Trace of a quadratic form with respect to another.- Exercises on § 1.- Exercises on § 2.- Exercises on § 3.- Exercises on § 4.- Historical notes.- Index of notation.- Index of terminology.- Summary of some important properties of Banach spaces.

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