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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. -
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. -
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 ::::::::::::::::::::::::: -
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. -
COUNTABLE-CODIMENSIONAL SUBSPACES of LOCALLY CONVEX SPACES by J
COUNTABLE-CODIMENSIONAL SUBSPACES OF LOCALLY CONVEX SPACES by J. H. WEBB (Received 9th December 1971) A barrel in a locally convex Hausdorff space E[x] is a closed absolutely convex absorbent set. A a-barrel is a barrel which is expressible as a countable intersection of closed absolutely convex neighbourhoods. A space is said to be barrelled {countably barrelled) if every barrel (a-barrel) is a neighbourhood, and quasi-barrelled (countably quasi-barrelled) if every bornivorous barrel (ff-barrel) is a neighbourhood. The study of countably barrelled and countably quasi-barrelled spaces was initiated by Husain (2). It has recently been shown that a subspace of countable codimension of a barrelled space is barrelled ((4), (6)), and that a subspace of finite codimension of a quasi-barrelled space is quasi-barrelled (5). It is the object of this paper to show how these results may be extended to countably barrelled and countably quasi-barrelled spaces. It is known that these properties are not preserved under passage to arbitrary closed subspaces (3). Theorem 6 shows that a subspace of countable codimension of a countably barrelled space is countably barrelled. Let {£„} be an expanding sequence of subspaces of E whose union is E. 00 Then £' s f] E'n, and the reverse inclusion holds as well (1) if either (i) E'\o(E', E)~\ is sequentially complete or (ii) E'[fi(E', £)] is sequentially complete, and every bounded subset of E is contained in some En. Theorem 1. Let E[%\ be a locally convex space with r = n(E, E') (the Mackey CO topology). -
Bornological Topology Space Separation Axioms a Research Submitted by Deyar
Republic of Iraq Ministry of Higher Education & Scientific Research AL-Qadisiyah University College of Computer Science and Mathematics Department of Mathematics Bornological Topology Space Separation Axioms A Research Submitted by Deyar To the Council of the department of Mathematics ∕ College of Education, University of AL-Qadisiyah as a Partial Fulfilment of the Requirements for the Bachelor Degree in Mathematics Supervised by Fatma Kamel Majeed A. D. 2019 A.H. 1440 Abstract we study Bornological Topology Separation Axioms like bornological topology , bornological topology , bornological topology , bornological topology , bornological topology and the main propositions and theorems about this concept. introduction For the first time in (1977), H. Hogbe–NIend [1] introduced the Concept of Bornology on a set and study Bornological Construction. In chapter one study Bornology on a set , Bornological subspace, convex Bornological space, Bornological vector space and Bornivorous set. Bornological topology space were first introduced and investigated in [4], we introduce in chapter two Bornological topology space and we study Bornological topology continuous and bornological topology homeomorphism. Bornological topology open map, bornological topology separation axioms studied in chapter three like bornological topology , bornological topology , bornological topology And bornological topology Bornological topology and main properties have been studied. The Contents Subject Page Chapter One 1.1 Bornological Space 1 1. 2 Bornivorous Set 4 Chapter Two 2.1 Bornological Topological Space 6 2.2 Bornological Topology Continuous 8 Chapter three 3.1 Bornological topology And Bornological 9 topology 3.2 Bornological topology , Bornological topology 10 And Bornological topology Chapter One 1.1 Bornological space In this section, we introduce some definitions, bornological space, bornological vector space, convex bornological vector space, separated bornological vector space, bounded map and some examples . -
Tightness and Distinguished Fréchet Spaces ✩
J. Math. Anal. Appl. 324 (2006) 862–881 www.elsevier.com/locate/jmaa Tightness and distinguished Fréchet spaces ✩ J.C. Ferrando a,J.K˛akol b, M. López Pellicer c,S.A.Saxond,∗ a Centro de Investigación Operativa, Universidad M. Hernández, E-03202 Elche (Alicante), Spain b Faculty of Mathematics and Informatics, A. Mickiewicz University, 61-614 Pozna´n, Poland c Departamento de Matemática Aplicada y IMPA, Universidad Politécnica, E-46022 Valencia, Spain d Department of Mathematics, University of Florida, PO Box 11805, Gainesville, FL 32611-8105, USA Received 22 April 2005 Available online 20 January 2006 Submitted by T.D. Benavides Abstract Valdivia invented a nondistinguished Fréchet space whose weak bidual is quasi-Suslin but not K-analytic. We prove that Grothendieck/Köthe’s original nondistinguished Fréchet space serves the same purpose. In- deed, a Fréchet space is distinguished if and only if its strong dual has countable tightness, a corollary to the fact that a (DF)-space is quasibarrelled if and only if its tightness is countable. This answers a Cas- cales/Kakol/Saxon ˛ question and leads to a rich supply of (DF)-spaces whose weak duals are quasi-Suslin but not K-analytic, including the spaces Cc(κ) for κ a cardinal of uncountable cofinality. Our level of gen- erality rises above (DF)- or even dual metric spaces to Cascales/Orihuela’s class G. The small cardinals b and d invite a novel analysis of the Grothendieck/Köthe example, and are useful throughout. © 2006 Elsevier Inc. All rights reserved. Keywords: Class G; Quasibarrelled; K-analytic; Quasi-Suslin; Compact-open ✩ The research of the second author was supported by the Komitet Badan´ Naukowych (State Committee for Scientific Research), Poland, grant no. -
2-Symmetric Locally Convex Spaces
2-SYMMETRIC LOCALLYCONVEX SPACES D. E. EDMUNDS In [l] it is shown that barrelledness and quasi-barrelledness are merely the two extreme examples of a property, called 2-symmetry, which may be possessed by a locally convex Hausdorff topological vector space. The object of this note is to show how recent char- acterisations [2; 3] of barrelled and quasi-barrelled spaces may be subsumed under characterisations of 2-symmetric spaces, and to ex- hibit some properties of these spaces. First we need some definitions and simple results. 1. Let £ be a locally convex Hausdorff topological vector space (abbreviated to LCS in what follows), and let S be a class of bounded subsets of E whose union is E. Let E' denote the topological dual of E, and let E'z be the set E' endowed with the topological of uniform convergence on the members of 2. Definition 1. A subset of E is said to be X-bornivorous if it absorbs every member of 2. Definition 2. We say that E is "2-symmetric if any of the following equivalent conditions hold : (a) Every S-bornivorous barrel in £ is a neighbourhood of zero. (b) Every bounded subset of E'j¡ is equicontinuous. (c) The topology induced on E by the strong dual of E% is the original topology of E. The equivalence of these conditions was proved in [l]. If 2iG22 it is easy to see that Si-symmetry implies 22-symmetry ; the strongest restriction on E is obtained by taking for 2 the class s of all subsets of E consisting of a single point, and then S-symmetry is simply the property of being barrelled. -
181. on Nuclear Spaces with Fundamental System O F Bounded Sets
No. 8] Proc. Japan Acad., 44 (1968) 807 181. On Nuclear Spaces with Fundamental System o f Bounded Sets. II By Shunsuke FUNAKOs1 (Comm. by Kinjiro KuNUGI, M. J. A., Oct. 12, 1968) A locally convex vector space with a countable fundamental system of bounded sets has already been developed in several bibliog- raphies. Barrelled spaces and quasi-barrelled spaces with a count- able fundamental system of compact sets has been studied by J. Dieudonne [2] and by M. Mahowald and G. Gould [7] respectively. We considered, the open mapring and closed graph theorems on a nuclear dualmetric space in the previous paper [4]. Let E be a nonmed space then E is a nuclear space if and only if it is finite dimentional. It is also known that a nonmed space can only be a Montel (i.e., barrelled and perfect) space if it is finite dimen- sional. In this paper, we prove a nuclear dualmetric space which is quasi-complete is Montel space, and using this result, we consider analogous theorem to M. Mahowald and G. Gould [7], in nuclear space. For nuclear spaces and its related notion, see A. Pietsch [8] and S. Funakosi [4]. Most of the definitions and notations of the locally convex vector spaces are taken' from N. Bourbaki [1] and T. Husain [5]. Definition. Let E be a locally convex space and E' its dual. (1) I f only all countable strong bounded subset o f E' are equi- continuous, then E is called the oW-quasi-barrelled. (2) Let E be a a-quasi-barrelled space, if there exists a countable fundamental system o f bounded subset in E, then E is called the dual- metric space. -
Kov Conjecture Fails for Simple Analytical Reasons
Journal of Pure and Applied Algebra 216 (2012) 1700–1703 Contents lists available at SciVerse ScienceDirect Journal of Pure and Applied Algebra journal homepage: www.elsevier.com/locate/jpaa The Ra˘ıkov conjecture fails for simple analytical reasons J. Wengenroth Universität Trier, FB IV – Mathematik, 54286 Trier, Germany article info a b s t r a c t Article history: We show that a conjecture of Ra˘ıkov from category theory fails for simple reasons which Received 11 November 2011 reflect very concrete properties of, for example, partial differential operators. Received in revised form 19 December 2011 ' 2012 Elsevier B.V. All rights reserved. Available online 27 January 2012 Communicated by B. Keller MSC: 18E05; 18E10; 46A08; 46M15 1. Pre-, semi-, quasi The Ra˘ıkov conjecture states that every semi-abelian category is quasi-abelian. The occurrence of semi and quasi sounds a bit as if this conjecture would be related to the ``theory of piffles'' invented by A.K. Austin [1]. This, however, is not the case. In many concrete situations, one meets additive categories which are not abelian, for instance if one considers topological abelian groups or vector spaces instead of their naked algebraic counterparts. Nevertheless, in order to apply homological algebra one then indeed needs weaker properties than the usual invertibility of fN in the classroom diagram. f X - Y 6 ? [ fN coim f - im f In the case of topological abelian groups, fN remains bijective and hence a monomorphism and an epimorphism, the non- invertibility is only due to topological reasons. It was thus most natural to define a semi-abelian category just by this property, as was done by Palamodov [9] in his thorough investigation of homological aspects in applications of the theory of locally convex spaces (note that Ra˘ıkov [11] used the term semi-abelian in a different sense). -
Stojan Radenovi C Zoran Kadelburg
Scientiae Mathematicae Vol.1, No. 11998, 43{49 43 ON A. GROTHENDIECK'S PROBLEMS CONCERNING F AND DF{SPACES Stojan Radenovic Zoran Kadelburg For non-exp erts only Received Novemb er 7, 1996; revised March 6, 1997 Abstract. This is an exp ository article devoted to the answers to the ten op en problems from the fundamental pap er [16] of A. Grothendieck, some of whichwere given in the recent years. A signi cant contribution to the development of the theory of lo cally convex spaces lcs , as a part of functional analysis, was given in the fties by a group of French mathematicians L. Schwartz, N. Bourbaki, J. Dieudonn e, A. Grothendieck. The dominant part in their work, concerning this part of functional analysis, was devoted to the so called duality theory.Itwas esp ecially applied to the investigation of metrizable, i.e. Fr echet lcs, which are \nearest" to the normed and Banach spaces [14]. The article [16] of A. Grothendieck \on F and DF spaces" takes a sp ecial place in the theory of lcs. Apart from solving nearly all the problems of L. Schwartz and J. Dieudonn e, p osed in [14], a signi cant class of spaces|\DF-spaces"|was intro duced in it. This class contains a lot of imp ortant functional spaces|we mention just a few, following the pap er [16] itself: m 0 0 m 0 a The spaces E , E , D , D , intro duced byL.Schwartz in [28] for ap- L L p p plication in the theory of distributions; also the space C of continuous functions with compact supp ort. -
Normed Spaces Which Are Not Mackey Groups
axioms Article Normed Spaces Which Are Not Mackey Groups Saak Gabriyelyan Department of Mathematics, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva 8410501, Israel; [email protected] Abstract: It is well known that every normed (even quasibarrelled) space is a Mackey space. However, in the more general realm of locally quasi-convex abelian groups an analogous result does not hold. We give the first examples of normed spaces which are not Mackey groups. Keywords: normed space; Mackey group; locally quasi-convex; compatible group topology MSC: 46A8; 46E10; 54H11 1. Introduction Let (E, t) be a locally convex space (lcs for short). A locally convex vector topology n on E is called compatible with t if the spaces (E, t) and (E, n) have the same topological dual space. The famous Mackey–Arens Theorem states that there is a finest locally convex vector space topology m on E compatible with t. The topology m is called the Mackey topology on E = E Mackey space associated with t, and if m t, the space is called a . The most important class of Mackey spaces is the class of quasibarrelled spaces. This class is sufficiently rich and contains all metrizable locally convex spaces. In particular, every normed space is a Citation: Gabriyelyan, S. Normed Mackey space. Spaces Which Are Not Mackey ( ) Groups. Axioms 2021, 10, 217. For an abelian topological group G, t we denote by Gb the group of all continuous https://doi.org/10.3390/ characters of (G, t). Two topologies m and n on an abelian group G are said to be compatible axioms10030217 if (\G, m) = (\G, n).