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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).
List of contents
I. - Topological vector spaces over a valued division ring I..-
1. Topological vector spaces.-
2. Linear varieties in a topological vector space.-
3. Metrisable topological vector spaces.- Exercises of
1.- Exercises of
2.- Exercises of
3.- II. - Convex sets and locally convex spaces II..-
1. Semi-norms.-
2. Convex sets.-
3. The Hahn-Banach Theorem (analytic form).-
4. Locally convex spaces.-
5. Separation of convex sets.-
6. Weak topologies.-
7. Extremal points and extremal generators.-
8. Complex locally convex 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.-
2. Bornological spaces.-
3. Spaces of continuous linear mappings.-
4. The Banach-Steinhaus theorem.-
5. Hypocontinuous bilinear mappings.-
6. Borel's graph theorem.- 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.-
2. Bidual. Reflexive spaces.-
3. Dual of a Fréchet space.-
4. Strict morphisms of Fréchet spaces.-
5. Compactness criteria.- Appendix. - Fixed points of groups of affine transformations.- 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.-
2. Orthogonal families in a hilbertian space.-
3. Tensor product of hilbertian spaces.-
4. Some classes of operators in hilbertian spaces.- Exercises on
1.- Exercises on
2.- Exercises on
3.- Exercises on
4.- Historical notes.- Index of notation.- Index of terminology.- Summary of some important propertiesof Banach spaces.
Summary
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).
