Locally Convex Spaces
Mathematische LeitfädenHerausgegeben vonem. o. Prof. Dr. phil. Dr. h.c. mult. G. Köthe, Universität Frankfurt/M.,und o. Prof. Dr. rer. nat. G. Trautmann, Universität KaiserslauternLocally Convex Spacesby Dr. phil. Hans JarchowProfessor at the University of Zürich B. G. Teubner Stuttgart 1981
ContentsPart I: Linear TopologiesVector Spaces1.11.21.31.41.51.61.7GeneralitiesElementary ConstructionsLinear MapsLinear IndependenceLinear FormsBilinear Maps and Tensor ProductsSome Examples15161719202225Topological Vector iesCircled and Absorbent SetsBounded Sets. Continuous Linear FormsProjective TopologiesA Universal Characterization of ProductsProjective LimitsF-SeminormsMetrizable TvsProjective Representation of TvsLinear Topologies on Function and Sequence 3.13.23.33.43.53.63.7Some General ConceptsSome Completeness ConceptsCompletion of a TvsExtension of Uniformly Continuous MapsPrecompact SetsExamplesReferences56575961646673
10Contents4Inductive Linear nts of TvsDirect SumsSome Completeness ResultsInductive LimitsStrict Inductive LimitsReferences74767880828486Baire Tvs and Webbed Tvs5.1 Baire Category5.2 Webs in Tvs5.3 Stability Properties of Webbed Tvs5.4 The Closed Graph Theorem5.5 Some Consequences5.6 Strictly Webbed Tvs5.7 Some Examples5.8 References6Locally r-Convex 69799r-Convex Setsr-Convex Sets in TvsGauge Functionals and r-SeminormsContinuity Properties of Gauge FunctionalsDefinition and Basic Properties ofLc r sSome Permanence Properties of Lc,sBounded, Precompact, and Compact SetsLocally Bounded TvsLinear Mappings Between r-Normable 119124Theorems of Hahn-Banach, Krein-Milman, and Riesz7.17.27.37.47.57.67.7Sublinear FunctionalsExtension Theorem for LcsSeparation TheoremsExtension Theorems for Normed SpacesThe Krein-Milman TheoremThe Riesz Representation TheoremReferences125127130132133137144
Contents11Part II: Duality Theory for Locally Convex Spaces8Basic Duality Theory8.18.28.38.48.58.68.78.88.98.109Dual Pairings and Weak TopologiesPolarizationBarrels and DisksBornologies and -TopologiesEquicontinuous Sets and CompactologiesContinuity of Linear MapsDuality of Subspaces and QuotientsDuality of Products and Direct SumsThe Stone-Weierstrass tinuous Convergence and Related Topologies9.1 Continuous Convergence9.2 Grothendieck's Completeness Theorem9.3 The Topologies y and y9.4 The Banach-Dieudonne Theorem9.5 fi-Completeness and Related Properties9.6 Open and Nearly Open Mappings9.7 Application to 5-Completeness9.8 On Weak Compactness9.9 References17417617818118318418618919310 Local Convergence and Schwartz Spaces10.110.210.310.410.510.610.710.810.9 -Convergence. Local ConvergenceLocal CompletenessEquicontinuous Convergence. The Topologies rf and r\Schwartz TopologiesA Universal Schwartz SpaceDiametral Dimension. Power Series SpacesQuasi-Normable LcsApplication to Continuous Function SpacesReferences19519719920120420721421621711 Barrelledness and Reflexivity11.1 Barrelled Lcs11.2 Quasi-Barrelled Lcs219222";.
12Contents11.311.411.511.611.711.811.911.10Some Permanence PropertiesSemi-Reflexive and Reflexive LcsSemi-Montel and Montel SpacesOn Frechet-Montel SpacesApplication to Continuous Function SpacesOn Uniformly Convex Banach SpacesOn Hubert SpacesReferences22322722923123323624124712 Sequential led and «0-Barrelled LcsN0-Barrelled LcsAbsorbent and Bornivorous SequencesDF-Spaces, gDF-Spaces, and df-Spaces .,Relations to Schwartz TopologiesApplication to Continuous Function SpacesReferences24925125325726326626913 Bornological and Ultrabornological onvergent and Rapidly J'-Convergent SequencesAssociated Bornological and Ultrabornological SpacesOn the Topology ß(E', E) Permanence PropertiesApplication to Continuous Function SpacesReferences27127327627928128328814 On Topological nal SequencesBases and Schauder Bases .Weak Bases. Equicontinuous BasesExamples and Additional RemarksShrinking and Boundedly Complete BasesOn Summable SequencesUnconditional and Absolute BasesOrthonormal Bases in Hubert SpacesReferences289292295299302305309315320
Contents13Part III Tensor Products and Nuclearity15 The Projective Tensor Product15.1 Generalities on Projective Tensor Products15.2 Tensor Product and Linear Mappings15.3 Linear Mappings with Values in a Dual15.4 Projective Limits and Projective Tensor Products15.5 Inductive Limits and Projective Tensor Products15.6 Some Stability Properties15.7 Projective Tensor Products with Jx (//)-spaces15.8 References32332632933133333533834116 The Injective Tensor Product16.1 8-Products and e-Tensor Products16.2 Tensor Product and Linear Mappings16.3 Projective and Inductive Limits16.4 Some Stability Properties16.5 Spaces of Summable Sequences16.6 Continuous Vector Valued Functions16.7 Holomorphic Vector Valued Functions16.8 References34334735035335736036236617 Some Classes of Operators17.1 Compact Operators17.2 Weakly Compact Operators17.3 Nuclear Operators17.4 Integral Operators17.5 The Trace for Finite Operators17.6 Some Particular Cases17.7 References36837237638038639139518 The Approximation Property18.1 Generalities18.2 Some Stability Properties18.3 The Approximation Property for Banach Spaces18.4 The Metrie Approximation Property18.5 The Approximation Property for Concrete Spaces18.6 Referencesi.397401403408410417
14Contents19 Ideals of Operators in Banach 11GeneralitiesDual, Injective, and Surjective IdealsIdeal-Quasinorms4,-SequencesAbsolutely / -Summing OperatorsFactorizationp-Nuclear Operatorsp-Approximable OperatorsStrongly Nuclear OperatorsSome Multiplication 920 Components of Ideals on Particular Spaces20.1 Compact Operators on Hubert Spaces20.2 The Schatten-von Neumann Classes20.3 Grothendieck's Inequality20.4 Applications20.5 0» and JVq on Hubert Spaces20.6 Composition of Absolutely Summing Operators20.7 Weakly Compact Operators on ( )-Spaces20.8 References45145345846246747047247621 Nuclear Locally Convex 11Locally Convex -SpacesGeneralities on Nuclear SpacesFurther Characterizations by Tensor ProductsNuclear Spaces and Choquet SimplexesOn Co-Nuclear SpacesExamples of Nuclear SpacesA Universal GeneratorStrongly Nuclear SpacesAssociated TopologiesBases in Nuclear ibliography520List of Symbols541Index543
21 Nuclear Locally Convex Spaces 21.1 Locally Convex -Spaces 478 21.2 Generalities on Nuclear Spaces 482 21.3 Further Characterizations by Tensor Products 486 21.4 Nuclear Spaces and Choquet Simplexes 489 21.5 On Co-Nuclear Spaces 491 21.6 Examples of Nuclear Spaces 496 21.7 A
For locally convex Hausdor spaces E;F we denote the completed projective tensor product of locally convex spaces E;Fby E F. De nition 2.1. A Fr echet space is a complete metrizable locally convex space. In other words, a locally convex space Xis a Fr echet space if and only if the topology
Appendix: Non-locally-convex spaces ‘pwith 0 p 1 For all [1] our purposes, topological vector spaces are locally convex, in the sense of having a basis at 0 consisting of convex opens. We prove below that a separating family of semin
Convex obj non-convex domain. Introduction Generic Problem: min Q(x); s:t: x 2F; Q(x) convex, especially: convex quadratic F nonconvex Examples: F is a mixed-integer set F is constrained in a nasty way, e.g. x 1 3 sin(x 2) 2cos(x 3) 4 Bienstock, Michalka Columbia Convex obj non-convex domain.
Solution. We prove the rst part. The intersection of two convex sets is convex. There-fore if Sis a convex set, the intersection of Swith a line is convex. Conversely, suppose the intersection of Swith any line is convex. Take any two distinct points x1 and x2 2 S. The intersection of Swith the line through x1 and x2 is convex.
What is convex projective geometry? Motivation from hyperbolic geometry Nice Properties of Hyperbolic Space Convex: Intersection with projective lines is connected. Properly Convex: Convex and closure is contained in an affine patch ()Disjoint from some projective hyperplane. Strictly Convex: Properly convex and boundary contains
3 convex-convex n [DK85] convex-convex (n;lognp lognq) [DK90] INTERSECTION DETECTION OF CONVEX POLYGONS Perhaps the most easily understood example of how the structure of geometric objects can be exploited to yield an e cient intersection test is that of detecting the intersection of two convex polygons. There are a number of solutions to this .
Appendix A:Sample Parking Garage Operations Manual: Page A-4 1.4 Parking Facility Statistics Total Capacity: 2,725 Spaces Total Compact: 464 Spaces (17%) Total Full Size: 2,222 Spaces Total Handicapped Spaces: 39 Spaces Total Reserved Spaces: 559 Spaces Total Bank of America Spaces: 2,166 Spaces Total Boulevard Reserved
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