Topological Vector Spaces

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).

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.