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On Semi –Complete Bornological Vector Space. a Research Submitted by Abbas Shakir Mansor

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Republic of Iraq Ministry of Higher Education & Scientific Research AL-Qadisiyah University College of Computer Science and Mathematics Department of Mathematics On Semi –Complete Bornological Vector Space. A Research Submitted by Abbas Shakir Mansor 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. 2018 A.H. 1439 كلمة شكر ﻻبد لنا ونحن نخطو خطواتنا اﻷخيرة في الحياة الجامعية من وقفة نعود إلى أعوام قضيناها في رحاب الجامعة مع أساتذتنا الكرام الذين قدموا لنا الكثير باذلين بذلك جهودا كبيرة في بناء جيل الغد لتبعث اﻷمة من جديد ... وقبل أن نمضي تقدم أسمى آيات الشكر واﻻمتنان والتقدير والمحبة إلى الذين حملوا أقدس رسالة في الحياة ... إلى الذين مهدوا لنا طريق العلم والمعرفة ... إلى جميع أساتذتنا اﻷفاضل....... "كن عالما .. فإن لم تستطع فكن متعلما ، فإن لم تستطع فأحب العلماء ،فإن لم تستطع فﻻ تبغضهم" وأخص بالتقدير والشكر الى اﻻستاذه م فاطمه كامل مجيد The Contents Subject Page Chapter One 1.1 Bornological Space 1 1.2 Subspace of Bornological Space 4 Chapter Two 2.1 Semi- Bounded Sets 7 2.2 Semi- Bounded Linear Maps 8 Chapter three 3.1 Bornological semi-convergence 12 Abstract We introduce the concept of semi- bounded set in Bornological spaces, semi bounded linear map and study some of their basic properties. We study the definition of semi-convergence of nets, some theorems of these concept and some results. introduction For the first time in (1977), H. Hogbe–NIend [6] 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 and Bornological vector space . Semi bounded sets were first introduced and investigated in [2], we introduce in chapter two semi bounded sets have been used to define and study many new bornological proprietiesi. The convergent net in convex bornological vector space and its results studied [1]. A semi- convergent net in convex bornological vector space and some properties have been studied in chapter three. Chapter One 1.1 Bornological Space In this section we introduce the basic definitions, notions and the theories of bornological vector spaces and construction of bornological vector space. Definition (1.1.1)[6]: A bronology on a set 푋 is a family 훽 of subset 푋 satisfying the following axioms : (i) 훽 푠 푎 푐표푛푣푒푟푛푔 푋, . 푒. 푋 = ⋃퐵∈훽 훽 ; (ii) 훽 is hereditary under inclusion i.e. if 퐴 ∈ 훽 and 퐵 is a subset of 푋 contained in 퐴, then 퐵 ∈ 훽 (iii) 훽 푠 푠푡푎푏푙푒 푢푛푑푒푟 푓푛푡푒 푢푛표푛. A pair (푋, 훽) consisting of a sat 푋 and a bornology 훽 on 푋 is called a bornological space, and the elements of 훽 are called the bounded subset of 푋. Example (1.1.2)[1]: Let 푋 = {푎, 푏, 푐}, 훽 = {∅, {푎}, {푏}, {푐}, {푎, 푏}, {푎, 푐}, {푏, 푐}, 푋} (i) since 푋 ∈ 훽, then 푋 is converging itself, we have 훽 is converging of 푋 (ii) since 훽 is the set of all subset of 푋 i.e.훽 = 푝(푋)and 훽 hereditary under inclusion i.e. 퐴 ∈ 훽, 퐵 ⊂ 퐴,then 퐵 ∈ 훽 푛 (iii) since 훽 = 푝(푋)then ⋃푖=1 퐴푖 ∈ 훽 for all 퐴푖 ∈ 훽 = 1,2, … , 푛 Example (1.1.3)[6]: Let 푅 be a field with absolute value. The collection 훽={A≤ R:A is a bounded subset of R in the usual sense for the absolute value}. Then 훽 is a bornology on 푅 called the Canonical bornology of 푅. Example (1.1.4)[6]: Let 퐸 be a vector space over 퐾 (we shall assume that 퐾 is either 푅 or 퐶). Let 푝 be a semi-normed 푝 if 푝(퐴) is bounded subset of 푅 in the sense of (Example (1.1.3). The subset of 퐸 which are bounded for the semi-normed 푝 1 form a bornology on 퐸 called the canonical bornology on the semi-normed space (퐸, 푃) (i) since ∀푥 ∈ 퐸, 푝(푥) ∈ 푅, then ∃푟 > 0 such that 푝(푋) ≤ 푟 implies ∀푥 ∈ 퐸, {푋}is a bounded subset for the semi-normed, i.e. {푥} ∈ 훽, then β is a covering of 퐸 (ii) 푓 퐴 ∈ 훽 푎푛푑 퐵 ⊆ 퐴, then 푝(퐴)is bounded subset of 푅 in the sense for the absolute value since 퐵 ⊆ 퐴 then 푝(퐵) ⊆ 푝(퐴), then 푝(퐵)is a bounded of 푅(every subset of a bounded set is bounded) (iii) 푓 퐴1, … , 퐴푛 ∈ 훽, then 푝(퐴1), … , 푝(퐴푛)are bounded subsets of 푅. 푛 푛 ⋃푖 푝(퐴푖) is a bounded subset of 푅 by (iv); proposition 1,2.3) i.e. 푝(⋃푖 퐴푖) is 푛 bounded subset of 푅. Then ⋃푖 퐴푖 is a bounded subset of semi-normed 푝, 푛 i.e. ⋃푖 퐴푖 ∈ 훽, then 훽 is abonology on 퐸. Definition (1.1.5)[2]: A base of abornology 훽 on 푋 is any subfamily 훽° of 훽 such that every element of 훽 is contained in an element of 훽° Example (1.1.6): If 푋 = {1,2,3} , 훽 = {∅, {1}, {2}, {1,2}, {1,3}, {2,3}, 푋} Let 훽° = {푋} be a base for 훽. It is clear that each element of 훽 is contained in an element of 훽° Proposition (1.1.7)[2]: Let (푋, 훽) be a bornological space and 훽° is the base of 훽, then the following hold : ()푋 = ⋃ 훽° 훽°∈훽 () A finite union of elements of 훽° is contained in an element of 훽°. Conversely; if 푋 ≠ ∅ and 훽° is a family of subset of 푋 satisfying : (i) and (ii) then ∃ abornology 훽 표푛 푋 such that 훽° is base for 훽. 2 Proof :-Since 훽° forms a base of abornology 훽 an a set 푋, then every element of 훽 is contained in an element of 훽° . 푒. ∀퐵 ∈ 훽, ∃훽° ∈ 훽 such that 퐵 since 훽 is covering of 푋. 푒. 푋 = ⋃ 퐵 , 푡ℎ푒푛 푋 = ⋃ 훽° . 푒. 훽° 푐표푛푣푒푟푠 푋, 푤ℎ푒푛푐푒 () 퐵∈훽 훽°∈훽 ()퐿푒푡 퐵푖 ∈ 훽° ∀ = 1,2, … , 푛 To prove that the union ⋃훽°∈훽 훽° is contained in as element of 훽° Since 훽° ⊆ 훽 푡ℎ푒푛 퐵푖 ∈ 훽 ∀ = 푛 1,2, … , 푛 (푏푦 푑푒푓푛표푛 (1.1.1)) 푡ℎ푒푛 ⋃푖=1 퐵푖 ∈ 훽 , 푏푦 푑푒푓푛푡표푛(1.1.5) 푛 We have ⋃푖 퐵푖 is contained in an element of 훽° Conversely; let 훽 = {퐵 ⊆ 푋: ∃퐵° ∈ 훽°, 퐵 ⊆ 훽°} Then to show that 훽 is a bornology on 푋 we must satisfy the following conditions : ()∀퐵° ∈ 훽° → 퐵∘ ⊆ 푋 → 퐵° ∈ 훽° → 퐵° ∈ 훽 (푏푦 푑푒푓푛푡표푛 (1.1.1)) Then 퐵° ∈ 훽 Since 훽° covers 푋, then 훽 is covering to 푋 ()퐼푓 퐵 ∈ 훽, 퐴 ⊆ 푋, 퐴 ⊆ 퐵 , 푠푛푐푒 퐴 ⊆ 퐵 ⊆ 퐵° For some 퐵° ∈ 훽°, 푡ℎ푒푛 퐴 ⊆ 훽°, . 푒. 퐴 푠 푐표푛푡푎푛푒푑 푛 푎푛 푒푙푒푚푒푛푡 표푓 훽° (by definition (1.1.1), 퐴 ∈ 훽. ()푓 퐵1, 퐵2 ∈ 훽 푡ℎ푒푛 ∃퐵°, 퐵̀° ∈ 훽° such that 퐵1 ⊆ 퐵°(by condition (i)) and 퐵2 ⊆ 퐵̀° ” Then 퐵1 ∪ 퐵2 ⊆ 퐵° ∪ 퐵̀° for some 퐵∘ ∈ 훽° (by condition (ii)) then (by the definition (1.1.1))퐵1 ∪ 퐵2 ∈ 훽 3 Then 훽 is a bornology on 푋, and (by definition (1.1.5)) 훽° is a base for 훽, we say 훽 is a bornology generated by a base Definition (1.1.8)[6]:- A bornology an a set 푋 is said a bornology with countable base if and only if 훽 possesses a base consisting of an increasing sequence of bounded sets (an element of bornology). 1.2 Subspaces Of Bornological Space Definition (1.2.1)[2]:- Let (푋, 퐵) be a bornological space and let 푌 ⊆ 푋. The bornology for 푌 is the collection 훽훾 = {퐺 ∩ 푌: 퐺 ∈ 훽}. The bornological space (푌, 훽훾) is called a subspace of (푋, 훽),and 훽훾 the relative bornology on 푌. Example (1.2.2): Let 푋 = {1,2,3}, 훽 = {{∅}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}}, and the base for the bornology is 훽° = {{1,2}, {1,3}, 푋}, and let 푌 = {1,2}. Then we have ∅ ∩ 푌 = ∅ {1,2} ∩ 푌 = 푌 {1} ∩ 푌 = {1} {1,3} ∩ 푌 = {1} {2} ∩ 푌 = {2} {2,3} ∩ 푌 = {2} {3} ∩ 푌 = ∅ 4 푋 ∩ 푌 = 푌 Then the subspace 훽훾 = {∅, {1}, {2}, 푌}, and a base of the bornological subspace is 훽훾 = {{1}, 푌}. Definition (1.2.3)[6]:- Let 퐸 be a vector space over the field 퐾 (the real or complex field). A bornology 훽 on 퐸 is said to be a bornology compatible with a vector space of 퐸 or to be a vector bornology on 퐸, if 훽 is stable under vector addition homothetic transformations and the formation of circled hulls, in other words, if the sets 퐴 + 퐵, 휆퐴, ⋃|훼|≤1 훼퐴 belongs to 훽 whenever 퐴 and 퐵 belong to 훽 and 휆 ∈ 퐾. Definition (1.2.4)[6]:- A convex bornological space is bornological vector space for which the disked hull of every bounded set is bounded i.e. it is stable under the formation of disked hull A separated bornogical vector space (퐸, 훽) is one where {0} is the only bounded vector subspace of 퐸. Example (1.2.5)[6]:- The Von-Neumann bornology of topological vector space, let 퐸 be a topological vector space. The collection 훽 = {퐴 ⊆ 퐸: 퐴 푠 푏표푢푛푑푒푑 푠푢푏푠푒푡 표푓 푎 푡표푝표푙표푔푐푎푙 푣푒푐푡표푟 푠푝푎푐푒 퐸} forms a vector bornology on 퐸 Called the Von-Neumann bornology of 퐸. Let us verify that 훽 is indeed a vector bornology on 퐸, if 훽° is a base of circled neighborhoods of zero in 퐸, it is clear that a subset 퐴 of 퐸 is bounded if and only if for every 퐵 ∈ 훽° there exists 휆 > 0 such that 퐴 ⊆ 휆퐵.
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  • Quasi-Barrelled Locally Convex Spaces 811

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    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.
  • Bornological Topology Space Separation Axioms a Research Submitted by Deyar

    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 .
  • 2-Symmetric Locally Convex Spaces

    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

    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.