Preservation of Closure in a Locally Convex Space. I by I
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Sisältö Part I: Topological Vector Spaces 4 1. General Topological
Sisalt¨ o¨ Part I: Topological vector spaces 4 1. General topological vector spaces 4 1.1. Vector space topologies 4 1.2. Neighbourhoods and filters 4 1.3. Finite dimensional topological vector spaces 9 2. Locally convex spaces 10 2.1. Seminorms and semiballs 10 2.2. Continuous linear mappings in locally convex spaces 13 3. Generalizations of theorems known from normed spaces 15 3.1. Separation theorems 15 Hahn and Banach theorem 17 Important consequences 18 Hahn-Banach separation theorem 19 4. Metrizability and completeness (Fr`echet spaces) 19 4.1. Baire's category theorem 19 4.2. Complete topological vector spaces 21 4.3. Metrizable locally convex spaces 22 4.4. Fr´echet spaces 23 4.5. Corollaries of Baire 24 5. Constructions of spaces from each other 25 5.1. Introduction 25 5.2. Subspaces 25 5.3. Factor spaces 26 5.4. Product spaces 27 5.5. Direct sums 27 5.6. The completion 27 5.7. Projektive limits 28 5.8. Inductive locally convex limits 28 5.9. Direct inductive limits 29 6. Bounded sets 29 6.1. Bounded sets 29 6.2. Bounded linear mappings 31 7. Duals, dual pairs and dual topologies 31 7.1. Duals 31 7.2. Dual pairs 32 7.3. Weak topologies in dualities 33 7.4. Polars 36 7.5. Compatible topologies 39 7.6. Polar topologies 40 PART II Distributions 42 1. The idea of distributions 42 1.1. Schwartz test functions and distributions 42 2. Schwartz's test function space 43 1 2.1. The spaces C (Ω) and DK 44 1 2 2.2. -
Some Topics on the Theory of Cones
POSITIVITY 2013 22-26 July, Leiden, The Netherlands Jointly organized by Leiden University and Delft University of Technology. Some topics on the theory of cones by Ioannis A. Polyrakis Department of Mathematics National Technical University of Athens Let X be a normed space. A convex subset P ⊆ X is a cone in λP = P for any λ ≥ 0. If moreover P \ (−P ) = f0g, the cone P is pointed (or proper). Denote X0 is the algebraic and X∗ topological dual of X. A convex subset B of P is a base for P if a strictly positive linear functional f of X exists such that B = fx 2 P j f(x) = 1g: Then we say that B is defined by f and is denoted by Bf . Theorem 1. The base Bf of P defined by f is bounded if and only if f is uniformly monotonic (i.e f(x) ≥ akxk for each x 2 P , where a > 0 is a real constant). Theorem 2. If f 2 X∗ is strictly positive we have: The base Bf is bounded if and only if f is an interior point of P 0. 1 Unbounded, convex subsets of cones Suppose that hX; Y i is a dual system X; Y ordered normed spaces. For any cone P of X 0 f 2 h i ≥ 2 g PY = y Y : x; y 0 for each x P ; is the dual cone of P in Y . If dual cone of X+ in Y is Y+ and the dual cone of Y+ in X is X+, hX; Y i is an ordered dual system. -
Reflexive Cones
Reflexive cones∗ E. Casini† E. Miglierina‡ I.A. Polyrakis§ F. Xanthos¶ November 10, 2018 Abstract Reflexive cones in Banach spaces are cones with weakly compact in- tersection with the unit ball. In this paper we study the structure of this class of cones. We investigate the relations between the notion of reflexive cones and the properties of their bases. This allows us to prove a characterization of reflexive cones in term of the absence of a subcone isomorphic to the positive cone of ℓ1. Moreover, the properties of some specific classes of reflexive cones are investigated. Namely, we consider the reflexive cones such that the intersection with the unit ball is norm compact, those generated by a Schauder basis and the reflexive cones re- garded as ordering cones in Banach spaces. Finally, it is worth to point out that a characterization of reflexive spaces and also of the Schur spaces by the properties of reflexive cones is given. Keywords Cones, base for a cone, vector lattices, ordered Banach spaces, geometry of cones, weakly compact sets, reflexivity, positive Schauder bases. Mathematics Subject Classification (2010) 46B10, 46B20, 46B40, 46B42 1 Introduction The study of cones is central in many fields of pure and applied mathematics. In Functional Analysis, the theory of partially ordered spaces and Riesz spaces arXiv:1201.4927v2 [math.FA] 28 May 2012 ∗The last two authors of this research have been co-financed by the European Union (Euro- pean Social Fund - ESF)and Greek national funds through the Operational Program "Educa- tion and Lifelong Learning" of the National Strategic Reference Framework (NSRF) - Research Funding Program: Heracleitus II. -
On Bornivorous Set
On Bornivorous Set By Fatima Kamil Majeed Al-Basri University of Al-Qadisiyah College Of Education Department of Mathematics E-mail:[email protected] Abstract :In this paper, we introduce the concept of the bornivorous set and its properties to construct bornological topological space .Also, we introduce and study the properties related to this concepts like bornological base, bornological subbase , bornological closure set, bornological interior set, bornological frontier set and bornological subspace . Key words : bornivorous set , bornological topological space,b-open set 1.Introduction- The space of entire functions over the complex field C was introduced by Patwardhan who defined a metric on this space by introducing a real-valued map on it[6]. In(1971), H.Hogbe- Nlend introduced the concepts of bornology on a set [3].Many workers such as Dierolf and Domanski, Jan Haluska and others had studied various bornological properties[2]. In this paper at the second section ,bornivorous set has been introduced with some related concepts. While in the third section a new space “Bornological topological space“ has been defined and created in the base of bornivorous set . The bornological topological space also has been explored and its properties .The study also extended to the concepts of the bornological base and bornological subbase of bornological topological space .In the last section a new concepts like bornological closure set, bornological drived set, bornological dense set, bornological interior set, bornological exterior set, bornological frontier set and bornological topological subspace, have been studied with supplementary properties and results which related to them. 1 Definition1.1[3] Let A and B be two subsets of a vector space E. -
NONARCHIMEDEAN COALGEBRAS and COADMISSIBLE MODULES 2 of Y
NONARCHIMEDEAN COALGEBRAS AND COADMISSIBLE MODULES ANTON LYUBININ Abstract. We show that basic notions of locally analytic representation the- ory can be reformulated in the language of topological coalgebras (Hopf alge- bras) and comodules. We introduce the notion of admissible comodule and show that it corresponds to the notion of admissible representation in the case of compact p-adic group. Contents Introduction 1 1. Banach coalgebras 4 1.1. Banach -Coalgebras 5 ̂ 1.2. Constructions⊗ in the category of Banach -coalgebras 6 ̂ 1.3. Banach -bialgebras and Hopf -algebras⊗ 8 ̂ ̂ 1.4. Constructions⊗ in the category of⊗ Banach -bialgebras and Hopf ̂ -algebras. ⊗ 9 ̂ 2. Banach comodules⊗ 9 2.1. Basic definitions 9 2.2. Constructions in the category of Banach -comodules 10 ̂ 2.3. Induction ⊗ 11 2.4. Rational -modules 14 ̂ 2.5. Tensor identities⊗ 15 3. Locally convex -coalgebras 16 ̂ Preliminaries ⊗ 16 3.1. Topological Coalgebras 18 3.2. Topological Bialgebras and Hopf algebras. 20 4. modules and comodules 21 arXiv:1410.3731v2 [math.RA] 26 Jul 2017 4.1. Definitions 21 4.2. Rationality 22 4.3. Quotients, subobjects and simplicity 22 4.4. Cotensor product 23 5. Admissibility 24 Appendix 28 References 29 Introduction The study of p-adic locally analytic representation theory of p-adic groups seems to start in 1980s, with the first examples of such representations studied in the works 1 NONARCHIMEDEAN COALGEBRAS AND COADMISSIBLE MODULES 2 of Y. Morita [M1, M2, M3] (and A. Robert, around the same time), who considered locally analytic principal series representations for p-adic SL2. -
Arxiv:Math/0306201V1 [Math.QA] 12 Jun 2003 Ftealgebra the of .Introduction 1
Big q-Laguerre and q-Meixner polynomials and representations of the algebra Uq(su1,1) M. N. Atakishiyev, N. M. Atakishiyev, and A. U. Klimyk Instituto de Matem´aticas, UNAM, CP 62210 Cuernavaca, Morelos, M´exico E-mail: [email protected] and [email protected] Abstract Diagonalization of a certain operator in irreducible representations of the positive discrete series of the quantum algebra Uq(su1,1) is studied. Spectrum and eigenfunctions of this operator are found in an explicit form. These eigenfunctions, when normalized, constitute an orthonormal basis in the representation space. The initial Uq(su1,1)-basis and the basis of eigenfunctions are interrelated by a matrix with entries, expressed in terms of big q-Laguerre polynomials. The unitarity of this connection matrix leads to an orthogonal system of functions, which are dual with respect to big q-Laguerre polynomials. This system of functions consists of two separate sets of functions, which can be expressed in terms of q-Meixner polynomials Mn(x; b,c; q) either with positive or negative values of the parameter b. The orthogonality property of these two sets of functions follows directly from the unitarity of the connection matrix. As a consequence, one obtains an orthogonality relation for the q-Meixner polynomials Mn(x; b,c; q) with b < 0. A biorthogonal system of functions (with respect to the scalar product in the representation space) is also derived. PACS numbers: 02.20.Uw, 02.30.Gp, 03.65.Fd 1. Introduction The significance of representations of Lie groups and Lie algebras for studying orthogonal polynomials and special functions is well known. -
Subspaces and Quotients of Topological and Ordered Vector Spaces
Zoran Kadelburg Stojan Radenovi´c SUBSPACES AND QUOTIENTS OF TOPOLOGICAL AND ORDERED VECTOR SPACES Novi Sad, 1997. CONTENTS INTRODUCTION::::::::::::::::::::::::::::::::::::::::::::::::::::: 1 I: TOPOLOGICAL VECTOR SPACES::::::::::::::::::::::::::::::: 3 1.1. Some properties of subsets of vector spaces ::::::::::::::::::::::: 3 1.2. Topological vector spaces::::::::::::::::::::::::::::::::::::::::: 6 1.3. Locally convex spaces :::::::::::::::::::::::::::::::::::::::::::: 12 1.4. Inductive and projective topologies ::::::::::::::::::::::::::::::: 15 1.5. Topologies of uniform convergence. The Banach-Steinhaus theorem 21 1.6. Duality theory ::::::::::::::::::::::::::::::::::::::::::::::::::: 28 II: SUBSPACES AND QUOTIENTS OF TOPOLOGICAL VECTOR SPACES ::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 39 2.1. Subspaces of lcs’s belonging to the basic classes ::::::::::::::::::: 39 2.2. Subspaces of lcs’s from some other classes :::::::::::::::::::::::: 47 2.3. Subspaces of topological vector spaces :::::::::::::::::::::::::::: 56 2.4. Three-space-problem for topological vector spaces::::::::::::::::: 60 2.5. Three-space-problem in Fr´echet spaces:::::::::::::::::::::::::::: 65 III: ORDERED TOPOLOGICAL VECTOR SPACES :::::::::::::::: 72 3.1. Basics of the theory of Riesz spaces::::::::::::::::::::::::::::::: 72 3.2. Topological vector Riesz spaces ::::::::::::::::::::::::::::::::::: 79 3.3. The basic classes of locally convex Riesz spaces ::::::::::::::::::: 82 3.4. l-ideals of topological vector Riesz spaces ::::::::::::::::::::::::: -
Dual-System Theories of Decision Making: Analytic Approaches and Empirical Tests
Dual-System Theories of Decision Making: Analytic Approaches and Empirical Tests DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Aleksandr Sinayev Graduate Program in Psychology The Ohio State University 2016 Dissertation Committee: Ellen Peters, Adviser Michael DeKay Trisha Van Zandt Copyrighted by Aleksandr Sinayev 2016 Abstract Dual-system models are popular in the study of decision making. They have faced criticisms, especially for being vague and lacking specific predictions. In order to address these criticisms, three categories of dual-system models are reviewed: parallel- competitive (in which intuitive, system 1, and deliberative, system 2, processing happen at the same time and both influence the response), default-interventionist (in which system 1 executes first and then system 2 may or may not override system 1), and interactive (in which both systems process information at the same time, but they are allowed to influence each other in complex back-and-forth interactions). Tests are conducted of the former two categories. Default-interventionist dual-system models predict that individual differences in reflectiveness should be associated with less biased decision making. The Cognitive Reflection Test (CRT) is thought to measure monitoring of system 1 intuitions such that, if cognitive reflection is high enough, intuitive errors will be detected and the problem will be solved. However, CRT items also require numeric ability to be answered correctly and it is unclear how much numeric ability vs. cognitive reflection contribute to better decision making. In two studies, CRT responses were used to calculate Cognitive Reflection and numeric ability; a numeracy scale was also administered. -
Induced C* -Algebras and Landstad Duality for Twisted Coactions
transactions of the American mathematical society Volume 347, Number 8, August 1995 INDUCED C*-ALGEBRAS AND LANDSTAD DUALITY FOR TWISTED COACTIONS JOHN C. QUIGG AND IAIN RAEBURN Abstract. Suppose N is a closed normal subgroup of a locally compact group G. A coaction e: A —»M(A ® C*(N)) of N on a C*-algebra A can be inflated to a coaction S of G on A , and the crossed product A x¿ G is then isomorphic to the induced C*-algebra Ind^A xe N . We prove this and a natural generalization in which A xf N is replaced by a twisted crossed product A x G/N G ; in case G is abelian, we recover a theorem of Olesen and Pedersen. We then use this to extend the Landstad duality of the first author to twisted crossed products, and give several applications. In particular, we prove that if l-*'N-*d— G/N -> 1 is topologically trivial, but not necessarily split as a group extension, then every twisted crossed product A x G/N G is isomorphic to a crossed product of the form A x N . Introduction Let ô be a coaction of a locally compact group G on a C*-algebra B . The cocrossed product BxsG is a C*-algebra whose representations are given by the covariant representations of the cosystem (B, G, a) ; it is generated by a copy MB) of B and a copy jG(C0(G)) of C0(G) (e.g., [LPRS], [Rae3], [Rae4]). The embedding jG is C7-equivariant, and the first author has recently shown that the existence of such an equivariant embedding characterizes cocrossed products by coactions [Qui2]. -
Dual System Groups and Its Applications — Compact HIBE and More⋆
Dual System Groups and its Applications — Compact HIBE and More? Jie Chen1;?? and Hoeteck Wee2;? ? ? 1 East China Normal University, China 2 Ecole´ Normale Superieure,´ France Abstract. We introduce the notion of dual system groups. – We show how to derive compact HIBE by instantiating the dual system framework in Waters (Crypto ’09) and Lewko and Waters (TCC ’10) with dual system groups. Our construction provides a unified treatment of the prior compact HIBE schemes from static assumptions. – We show how to instantiate dual system groups under the decisional subgroup assumption in composite-order groups and the decisional linear assumption (d-LIN) in prime-order groups. Along the way, we provide new tools for simulating properties of composite-order bilinear groups in prime-order groups. In particular, we present new randomization and parameter-hiding techniques in prime-order groups. Combining the two, we obtain a number of new encryption schemes, notably – a new construction of IBE in prime-order groups with shorter parameters; – a new construction of compact HIBE in prime-order groups whose structure closely mirrors the selectively secure HIBE scheme of Boneh, Boyen and Goh (Eurocrypt ’05); – a new construction of compact spatial encryption in prime-order groups. Table of Contents 1 Introduction ........................................................................ 1 2 Preliminaries ....................................................................... 5 3 Dual System Groups ................................................................ -
Quasi-Barrelled Locally Convex Spaces 811
i960] quasi-barrelled locally convex spaces 811 Bibliography 1. R. E. Lane, Absolute convergence of continued fractions, Proc. Amer. Math. Soc. vol. 3 (1952) pp. 904-913. 2. R. E. Lane and H. S. Wall, Continued fractions with absolutely convergent even and odd parts, Trans. Amer. Math. Soc. vol. 67 (1949) pp. 368-380. 3. W. T. Scott and H. S. Wall, A convergence theorem for continued fractions, Trans. Amer. Math. Soc. vol. 47 (1940) pp. 155-172. 4. H. S. Wall, Analytic theory of continued fractions, New York, D. Van Nostrand Company, Inc., 1948. The University of Texas and University of Houston QUASI-BARRELLED LOCALLY CONVEX SPACES MARK MAHOWALD AND GERALD GOULD 1. Introduction and preliminary definitions. The main object of this paper is to answer some problems posed by Dieudonné in his paper Denumerability conditions in locally convex vector spaces [l]. His two main results are as follows: Proposition 1. If Eis a barrelled space on which there is a countable fundamental system of convex compact subsets, [Definition 1.2] then it is the strong dual of a Fréchet-Montel Space. Proposition 2. If E is either bornological or barrelled, and if there is a countable fundamental system of compact subsets, then E is dense in the strong dual of a Fréchet-Montel Space. Two questions raised by Dieudonné in connection with these results are: (a) If E is either bornological or barrelled then it is certainly quasi- barrelled [l, Chapter 3, §2, Example 12]. Can one substitute this weaker condition on E in Proposition 2? (b) Is there is an example of a quasi-barrelled space which is neither barrelled nor bornological? We shall show that the answer to (a) is "Yes," and that the answer to (b) is also "Yes," so that the generalization is in fact a real one. -
Nuclearity Related Properties in Operator Systems 3
NUCLEARITY RELATED PROPERTIES IN OPERATOR SYSTEMS ALI S. KAVRUK Abstract. Some recent research on the tensor products of operator systems and ensuing nuclearity properties in this setting raised many stability problems. In this paper we examine the preservation of these nuclearity properties including exactness, local liftability and the double commutant expectation property under basic algebraic operations such as quotient, duality, coproducts and tensorial products. We show that, in the finite dimensional case, exactness and the lifting property are dual pairs, that is, an operator system S is exact if and only if the dual operator system Sd has the lifting property. Moreover, the lifting property is preserved under quotients by null subspaces. Again in the finite dimensional case we prove that every operator system has the k-lifting property in the sense that whenever ϕ : S → A/I is a unital and completely positive map, where A is a C*-algebra and I is an ideal, then ϕ possess a unital k-positive lift on A, for every k. This property provides a novel proof of a classical result of Smith and Ward on the preservation of matricial numerical ranges of an operator. The Kirchberg conjecture naturally falls into this context. We show that the Kirchberg conjecture is equivalent to the statement that the five dimensional universal operator sys- tem generated by two contraction (S2) has the double commutant expectation property. In addition to this we give several equivalent statements to this conjecture regarding the preservation of various nuclearity properties under basic algebraic operations. We show that the Smith Ward problem is equivalent to the statement that every three dimensional operator system has the lifting property (or exactness).