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

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