Topological Vector Spaces I

One Fundamentals of General Topology.- § 1. Topological spaces.- 1. The notion of a topological space.- 2. Neighbourhoods.- 3. Bases of neighbourhoods.- 4. Hausdorff spaces.- 5. Some simple topological ideas.- 6. Induced topologies and comparison of topologies. Connectedness.- 7. Continuous mappings.- 8. Topological products.- § 2 . Nets and filters.- 1. Partially ordered and directed sets.- 2. Zorn’s lemma.- 3. Nets in topological spaces.- 4. Filters.- 5. Filters in topological spaces.- 6. Nets and filters in topological products.- 7. Ultrafilters.- 8. Regular spaces.- § 3. Compact spaces and sets.- 1. Definition of compact spaces and sets.- 2. Properties of compact sets.- 3. Tychonoff’s theorem.- 4. Other concepts of compactness.- 5. Axioms of countability.- 6. Locally compact spaces.- 7. Normal spaces.- § 4. Metric spaces.- 1. Definition.- 2. Metric space as a topological space.- 3. Continuity in metric spaces.- 4. Completion of a metric space.- 5. Separable and compact metric spaces.- 6. Baire’s theorem.- 7. The topological product of metric spaces.- § 5. Uniform spaces.- 1. Definition.- 2. The topology of a uniform space.- 3. Uniform continuity.- 4. Cauchy filters.- 5. The completion of a Hausdorff uniform space.- 6. Compact uniform spaces.- 7. The product of uniform spaces.- § 6. Real functions on topological spaces.- 1. Upper and lower limits.- 2. Semi-continuous functions.- 3. The least upper bound of a collection of functions.- 4. Continuous functions on normal spaces.- 5. The extension of continuous functions on normal spaces.- 6. Completely regular spaces.- 7. Metrizable uniform spaces.- 8. The complete regularity of uniform spaces.- Two Vector Spaces over General Fields.- § 7. Vector spaces.- 1. Definition of a vector space.- 2. Linear subspaces and quotient spaces.- 3. Bases and complements.- 4. The dimension of a linear space.- 5. Isomorphism, canonical form.- 6. Sums and intersections of subspaces.- 7. Dimension and co-dimension of subspaces.- 8. Products and direct sums of vector spaces.- 9. Lattices.- 10. The lattice of linear subspaces.- § 8. Linear mappings and matrices.- 1. Definition and rules of calculation.- 2. The four characteristic spaces of a linear mapping.- 3. Projections.- 4. Inverse mappings.- 5. Representation by matrices.- 6. Rings of matrices.- 7. Change of basis.- 8. Canonical representation of a linear mapping.- 9. Equivalence of mappings and matrices.- 10. The theory of equivalence.- § 9. The algebraic dual space. Tensor products.- 1. The dual space.- 2. Orthogonality.- 3. The lattice of orthogonally closed subspaces of E*.- 4. The adjoint mapping.- 5. The dimension of E*.- 6. The tensor product of vector spaces.- 7. Linear mappings of tensor products.- § 10. Linearly topologized spaces.- 1. Preliminary remarks.- 2. Linearly topologized spaces.- 3. Dual pairs, weak topologies.- 4. The dual space.- 5. The dual pairs .- 6. Weak convergence and weak completeness.- 7. Quotient spaces and topological complements.- 8. Dual spaces of subspaces and quotient spaces.- 9. Linearly compact spaces.- 10. E* as a linearly compact space.- 11. The topology Tlk.- 12. Tlk-continuous linear mappings.- 13. Continuous basis and continuous dimension.- § 11. The theory of equations in E and E*.- 1. The duality of E and E*.- 2. The theory of the solutions of column-and row-finite systems of equations.- 3. Formulae for solutions.- 4. The countable case.- 5. An example.- § 12. Locally linearly compact spaces.- 1. The structure of locally linearly compact spaces.- 2. The endomorphisms of ?.- 3. The theory of equivalence in ?.- § 13. The linear strong topology.- 1. Linearly bounded subspaces.- 2. The linear strong topology.- 3. The completion.- 4. Topological sums and products.- 5. Spaces of countable degree.- 6. A counterexample.- 7. Further investigations.- Three Topological Vector Spaces.- § 14. Normed spaces.- 1. Definition of a normed space.- 2. Norm isomorphism, equivalent norms.- 3. Banach spaces.- 4. Quotient spaces and topological products.- 5. The dual space.- 6. Continuous linear mappings.- 7. The spaces c0, c, l1 and l?.- 8. The spaces lp, 1 hyperplanes.- 6. The Hahn-Banach theorem for normed spaces. Adjoint mappings.- 7. The dual space of C(I).- Four Locally Convex Spaces. Fundamentals.- § 18. The definition and simplest properties of locally convex spaces.- 1. Definition by neighbourhoods, and by semi-norms.- 2. Metrizable locally convex spaces and (F)-spaces.- 3. Subspaces, quotient spaces and topological products of locally convex spaces.- 4. The completion of a locally convex space.- 5. The locally convex direct sum of locally convex spaces.- § 19. Locally convex hulls and kernels, inductive and projective limits of locally convex spaces.- 1. The locally convex hull of locally convex spaces.- 2. The inductive limit of vector spaces.- 3. The topological inductive limit of locally convex spaces.- 4. Strict inductive limits.- 5. (LB)-and (LF)-spaces. Completeness.- 6. The locally convex kernel of locally convex spaces.- 7. The projective limit of vector spaces.- 8. The topological projective limit of locally convex spaces.- 9. The representation of a locally convex space as a projective limit.- 10. A criterion for completeness.- § 20. Duality.- 1. The existence of continuous linear functionals.- 2. Dual pairs and weak topologies.- 3. The duality of closed subspaces.- 4. Duality of mappings.- 5. Duality of complementary spaces.- 6. The convex cover of a compact set.- 7. The separation theorem for compact convex sets.- 8. Polarity.- 9. The polar of a neighbourhood of ?.- 10. A representation of locally convex spaces.- 11. Bounded and strongly bounded sets in dual pairs.- § 21. The different topologies on a locally convex space.- 1. The topology TM of uniform convergence on M.- 2. The strong topology.- 3. The original topology of a locally convex space; separability.- 4. The Mackey topology.- 5. The topology of a metrizable space.- 6. The topology Tc of precompact convergence.- 7. Polar topologies.- 8. The topologies Tf and Tlf.- 9. Grothendieck’s construction of the completion.- 10. The Banach-Diedonné theorem.- 11. Real and complex locally convex spaces.- § 22. The determination of various dual spaces and their topologies.- 1. The dual of subspaces and quotient spaces.- 2. The topologies of subspaces, quotient spaces and their duals.- 3. Subspaces and quotient spaces of normed spaces.- 4. The quotient spaces of l1.- 5. The duality of topological products and locally convex direct sums.- 6. The duality of locally convex hulls and kernels.- 7. Topologies on locally convex hulls and kernels.- Five Topological and Geometrical Properties of Locally Convex Spaces.- § 23. The bidual space. Semi-reflexivity and reflexivity.- 1. Quasi-completeness.- 2. The bidual space.- 3. Semi-reflexivity.- 4. The topologies on the bidual.- 5. Reflexivity.- 6. The relationship between semi-reflexivity and reflexivity.- 7. Distinguished spaces.- 8. The dual of a semi-reflexive space.- 9. Polar reflexivity.- § 24. Some results on compact and on convex sets.- 1. The theorems of Šmulian and Kaplansky.- 2. Eberlein’s theorem.- 3. Further criteria for weak compactness.- 4. Convex sets in spaces which are not semi-reflexive. The theorems of Klee.- 5. Krein’s theorem.- 6. Pták’s theorem.- § 25. Extreme points and extreme rays of convex sets.- 1. The Krein-Milman theorem.- 2. Examples and applications.- 3. Variants of the Krein-Milman theorem.- 4. The extreme rays of a cone.- 5. Locally compact convex sets.- § 26. Metric properties of normed spaces.- 1. Strict convexity.- 2. Shortest distance.- 3. Points of smoothness.- 4. Weak differentiability of the norm.- 5. Examples.- 6. Uniform convexity.- 7. The uniform convexity of the lp and Lp spaces.- 8. Further examples.- 9. Invariance under topological isomorphisms.- 10. Uniform smoothness and strong differentiability of the norm.- 11. Further ideas.- Six Some Special Classes of Locally Convex Spaces.- § 27. Barrelled spaces and Montel spaces.- 1. Quasi-barrelled spaces and barrelled spaces.- 2. (M)-spaces and (FM)-spaces.- 3. The space H(G).- 4. (M)-spaces of locally holomorphic functions.- § 28. Bornological spaces.- 1. Definition.- 2. The structure of bornological spaces.- 3. Local convergence. Sequentially continuous mappings.- 4. Hereditary properties.- 5. The dual, and the topology Tc0.- 6. Boundedly closed spaces.- 7. Reflexivity and completeness.- 8. The Mackey-Ulam theorem.- § 29. (F)- and (DF)-spaces.- 1. Fundamental sequences of bounded sets. Metrizability.- 2. The bidual.- 3. (DF)-spaces.- 4. Bornological (DF)-spaces.- 5. Hereditary properties of (DF)-spaces.- 6. Further results, and open questions.- § 30. Perfect spaces.- 1. The ?-dual. Examples.- 2. The normal topology of a sequence space.- 3. Sums and products of sequence spaces.- 4. Unions and intersections of sequence spaces.- 5. Topological properties of sequence spaces.- 6. Compact subsets of a perfect space.- 7. Barrelled spaces and (M)-spaces.- 8. Echelon and co-echelon spaces.- 9. Co-echelon spaces of type (M).- 10. Further investigations into sequence spaces.- § 31. Counterexamples.- 1. The dual of l?.- 2. Subspaces of l? and l1 with no topological complements.- 3. The problem of complements in lp and Lp.- 4. Complements in (F)-spaces.- 5. An (FM)-space.- 6. An (LB)-space which is not complete.- 7. An (F)-space which is not distinguished.- Author and Subject Index.