Locally Convex Spaces

I: Linear Topologies.- 1 Vector Spaces.- 1.1 Generalities.- 1.2 Elementary Constructions.- 1.3 Linear Maps.- 1.4 Linear Independence.- 1.5 Linear Forms.- 1.6 Bilinear Maps and Tensor Products.- 1.7 Some Examples.- 2 Topological Vector Spaces.- 2.1 Generalities.- 2.2 Circled and Absorbent Sets.- 2.3 Bounded Sets. Continuous Linear Forms.- 2.4 Projective Topologies.- 2.5 A Universal Characterization of Products.- 2.6 Projective Limits.- 2.7 F-Seminorms.- 2.8 Metrizable Tvs.- 2.9 Projective Representation of Tvs.- 2.10 Linear Topologies on Function and Sequence Spaces.- 2.11 References.- 3 Completeness.- 3.1 Some General Concepts.- 3.2 Some Completeness Concepts.- 3.3 Completion of a Tvs.- 3.4 Extension of Uniformly Continuous Maps.- 3.5 Precompact Sets.- 3.6 Examples.- 3.7 References.- 4 Inductive Linear Topologies.- 4.1 Generalities.- 4.2 Quotients of Tvs.- 4.3 Direct Sums.- 4.4 Some Completeness Results.- 4.5 Inductive Limits.- 4.6 Strict Inductive Limits.- 4.7 References.- 5 Baire Tvs and Webbed Tvs.- 5.1 Baire Category.- 5.2 Webs in Tvs.- 5.3 Stability Properties of Webbed Tvs.- 5.4 The Closed Graph Theorem.- 5.5 Some Consequences.- 5.6 Strictly Webbed Tvs.- 5.7 Some Examples.- 5.8 References.- 6 Locally r-Convex Tvs.- 6.1 r-Convex Sets.- 6.2 r-Convex Sets in Tvs.- 6.3 Gauge Functionals and r-Seminorms.- 6.4 Continuity Properties of Gauge Functionals.- 6.5 Definition and Basic Properties of Lc,s.- 6.6 Some Permanence Properties of Lc,s.- 6.7 Bounded, Precompact, and Compact Sets.- 6.8 Locally Bounded Tvs.- 6.9 Linear Mappings Between r-Normable Tvs.- 6.10 Examples.- 6.11 References.- 7 Theorems of Hahn-Banach, Krein-Milman, and Riesz.- 7.1 Sublinear Functionals.- 7.2 Extension Theorem for Lcs.- 7.3 Separation Theorems.- 7.4 Extension Theorems for Normed Spaces.- 7.5 The Krein-Milman Theorem.- 7.6 The Riesz Representation Theorem.- 7.7 References.- II: Duality Theory for Locally Convex Spaces.- 8 Basic Duality Theory.- 8.1 Dual Pairings and Weak Topologies.- 8.2 Polarization.- 8.3 Barrels and Disks.- 8.4 Bornologies and ?-Topologies.- 8.5 Equicontinuous Sets and Compactologies.- 8.6 Continuity of Linear Maps.- 8.7 Duality of Subspaces and Quotients.- 8.8 Duality of Products and Direct Sums.- 8.9 The Stone-Weierstrass Theorem.- 8.10 References.- 9 Continuous Convergence and Related Topologies.- 9.1 Continuous Convergence.- 9.2 Grothendieck's Completeness Theorem.- 9.3 The Topologies ?t and ?.- 9.4 The Banach-Dieudonné Theorem.- 9.5 B-Completeness and Related Properties.- 9.6 Open and Nearly Open Mappings.- 9.7 Application to B-Completeness.- 9.8 On Weak Compactness.- 9.9 References.- 10 Local Convergence and Schwartz Spaces.- 10.1 ?-Convergence. Local Convergence.- 10.2 Local Completeness.- 10.3 Equicontinuous Convergence. The Topologies ?t and ?.- 10.4 Schwartz Topologies.- 10.5 A Universal Schwartz Space.- 10.6 Diametral Dimension. Power Series Spaces.- 10.7 Quasi-Normable Lcs.- 10.8 Application to Continuous Function Spaces.- 10.9 References.- 11 Barrelledness and Reflexivity.- 11.1 Barrelled Lcs.- 11.2 Quasi-Barrelled Lcs.- 11.3 Some Permanence Properties.- 11.4 Semi-Reflexive and Reflexive Lcs.- 11.5 Semi-Montel and Montei Spaces.- 11.6 On Fréchet-Montel Spaces.- 11.7 Application to Continuous Function Spaces.- 11.8 On Uniformly Convex Banach Spaces.- 11.9 On Hilbert Spaces.- 11.10 References.- 12 Sequential Barrelledness.- 12.1 ??-Barrelled and c0-Barrelled Lcs.- 12.2 ?0-Barrelled Lcs.- 12.3 Absorbent and Bornivorous Sequences.- 12.4 DF-Spaces, gDF-Spaces, and df-Spaces.- 12.5 Relations to Schwartz Topologies.- 12.6 Application to Continuous Function Spaces.- 12.7 References.- 13 Bornological and Ultrabornological Spaces.- 13.1 Generalities.- 13.2 ?-Convergent and Rapidly ?-Convergent Sequences.- 13.3 Associated Bornological and Ultrabornological Spaces.- 13.4 On the Topology ?(E', E)bor.- 13.5 Permanence Properties.- 13.6 Application to Continuous Function Spaces.- 13.7 References.- 14 On Topological Bases.- 14.1 Biorthogonal Sequences.- 14.2 Bases and Schauder Bases.- 14.3 Weak Bases. Equicontinuous Bases.- 14.4 Examples and Additional Remarks.- 14.5 Shrinking and Boundedly Complete Bases.- 14.6 On Summable Sequences.- 14.7 Unconditional and Absolute Bases.- 14.8 Orthonormal Bases in Hilbert Spaces.- 14.9 References.- III Tensor Products and Nuclearity.- 15 The Projective Tensor Product.- 15.1 Generalities on Projective Tensor Products.- 15.2 Tensor Product and Linear Mappings.- 15.3 Linear Mappings with Values in a Dual.- 15.4 Projective Limits and Projective Tensor Products.- 15.5 Inductive Limits and Projective Tensor Products.- 15.6 Some Stability Properties.- 15.7 Projective Tensor Products with ?1 (?)-spaces.- 15.8 References.- 16 The Injective Tensor Product.- 16.1 ?-Products and ?-Tensor Products.- 16.2 Tensor Product and Linear Mappings.- 16.3 Projective and Inductive Limits.- 16.4 Some Stability Properties.- 16.5 Spaces of Summable Sequences.- 16.6 Continuous Vector Valued Functions.- 16.7 Holomorphic Vector Valued Functions.- 16.8 References.- 17 Some Classes of Operators.- 17.1 Compact Operators.- 17.2 Weakly Compact Operators.- 17.3 Nuclear Operators.- 17.4 Integral Operators.- 17.5 The Trace for Finite Operators.- 17.6 Some Particular Cases.- 17.7 References.- 18 The Approximation Property.- 18.1 Generalities.- 18.2 Some Stability Properties.- 18.3 The Approximation Property for Banach Spaces.- 18.4 The Metric Approximation Property.- 18.5 The Approximation Property for Concrete Spaces.- 18.6 References.- 19 Ideals of Operators in Banach Spaces.- 19.1 Generalities.- 19.2 Dual, Injective, and Surjective Ideals.- 19.3 Ideal-Quasinorms.- 19.4 ?p-Sequences.- 19.5 Absolutely p-Summing Operators.- 19.6 Factorization.- 19.7 p-Nuclear Operators.- 19.8 p-Approximable Operators.- 19.9 Strongly Nuclear Operators.- 19.10 Some Multiplication Theorems.- 19.11 References.- 20 Components of Ideals on Particular Spaces.- 20.1 Compact Operators on Hilbert Spaces.- 20.2 The Schatten-von Neumann Classes.- 20.3 Grothendieck's Inequality.- 20.4 Applications.- 20.5$$ {P_p}and{N_q} $$on Hilbert Spaces.- 20.6 Composition of Absolutely Summing Operators.- 20.7 Weakly Compact Operators on T(K)-Spaces.- 20.8 References.- 21 Nuclear Locally Convex Spaces.- 21.1 Locally Convex A-Spaces.- 21.2 Generalities on Nuclear Spaces.- 21.3 Further Characterizations by Tensor Products.- 21.4 Nuclear Spaces and Choquet Simplexes.- 21.5 On Co-Nuclear Spaces.- 21.6 Examples of Nuclear Spaces.- 21.7 A Universal Generator.- 21.8 Strongly Nuclear Spaces.- 21.9 Associated Topologies.- 21.10 Bases in Nuclear Spaces.- 21.11 References.- List of Symbols.