Topological Vector Spaces I (Paperback, 1983)

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 topol