Topological Vector Spaces

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.