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.