Locally Convex Spaces

ISBN-13: 9783322905611 / Angielski / Miękka / 2014 / 550 str.

Hans Jarchow

Opis

The present book grew out of several courses which I have taught at the University of Zurich and at the University of Maryland during the past seven years. It is primarily intended to be a systematic text on locally convex spaces at the level of a student who has some familiarity with general topology and basic measure theory. However, since much of the material is of fairly recent origin and partly appears here for the first time in a book, and also since some well-known material has been given a not so well-known treatment, I hope that this book might prove useful even to more advanced readers. And in addition I hope that the selection ofmaterial marks a sufficient set-offfrom the treatments in e.g. N. Bourbaki 4], 5], R.E. Edwards 1], K. Floret-J. Wloka 1], H.G.Garnir-M.De Wilde-J.Schmets 1], AGrothendieck 13], H.Heuser 1], J. Horvath 1], J. L. Kelley-I. Namioka et al. 1], G. Kothe 7], 10], A P. Robertson- W.Robertson 1], W.Rudin 2], H.H.Schaefer 1], F.Treves l], A Wilansky 1]. A few sentences should be said about the organization of the book. It consists of 21 chapters which are grouped into three parts. Each chapter splits into several sections. Chapters, sections, and the statements therein are enumerated in consecutive fashion.

Spis treści

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$$ and $$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.