Topological Vector Spaces I

ISBN-13: 9783642649905 / Angielski / Miękka / 2011 / 456 str.

Gottfried Kothe;D.J.H. Garling

Opis

It is the author's aim to give a systematic account of the most im portant ideas, methods and results of the theory of topological vector spaces. After a rapid development during the last 15 years, this theory has now achieved a form which makes such an account seem both possible and desirable. This present first volume begins with the fundamental ideas of general topology. These are of crucial importance for the theory that follows, and so it seems necessary to give a concise account, giving complete proofs. This also has the advantage that the only preliminary knowledge required for reading this book is of classical analysis and set theory. In the second chapter, infinite dimensional linear algebra is considered in comparative detail. As a result, the concept of dual pair and linear topologies on vector spaces over arbitrary fields are intro duced in a natural way. It appears to the author to be of interest to follow the theory of these linearly topologised spaces quite far, since this theory can be developed in a way which closely resembles the theory of locally convex spaces. It should however be stressed that this part of chapter two is not needed for the comprehension of the later chapters. Chapter three is concerned with real and complex topological vector spaces. The classical results of Banach's theory are given here, as are fundamental results about convex sets in infinite dimensional spaces."

Spis treści

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.