Topological Vector Spaces

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.