Republic of Iraq Ministry of Higher Education & Scientific Research AL-Qadisiyah University College of Computer Science and Department of Mathematics

On Semi –Complete Bornological . 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

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وأخص بالتقدير والشكر

الى

االستاذه م فاطمه كامل مجيد

The Contents

Subject Page

Chapter One

1.1 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- in Bornological spaces, semi bounded 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 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 𝑖푠 푎 푐표푛푣푒푟𝑖푛푔 푋, 𝑖. 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 𝑖푠 푠푡푎푏푙푒 푢푛푑푒푟 푓𝑖푛𝑖푡푒 푢푛𝑖표푛.

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 𝑖 = 1,2, … , 푛 Example (1.1.3)[6]:

Let 푅 be a 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

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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) 𝑖푓 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) 𝑖푓 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 :

(𝑖)푋 = ⋃ 훽°

훽°∈훽

(𝑖𝑖) 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.

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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° 𝑖. 푒. ∀퐵 ∈ 훽, ∃훽° ∈ 훽 such that 퐵 since 훽 is covering of 푋.

𝑖. 푒. 푋 = ⋃ 퐵 , 푡ℎ푒푛 푋 = ⋃ 훽° 𝑖. 푒. 훽° 푐표푛푣푒푟푠 푋, 푤ℎ푒푛푐푒 (𝑖)

퐵∈훽 훽°∈훽

(𝑖𝑖)퐿푒푡 퐵푖 ∈ 훽° ∀𝑖 = 1,2, … , 푛

To prove that the union ⋃훽°∈훽 훽° is contained in as element of 훽°

Since 훽° ⊆ 훽 푡ℎ푒푛 퐵푖 ∈ 훽 ∀ 𝑖 = 푛 1,2, … , 푛 (푏푦 푑푒푓𝑖푛𝑖표푛 (1.1.1)) 푡ℎ푒푛 ⋃푖=1 퐵푖 ∈ 훽 , 푏푦 푑푒푓𝑖푛𝑖푡𝑖표푛(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 :

(𝑖)∀퐵° ∈ 훽° → 퐵∘ ⊆ 푋 → 퐵° ∈ 훽° → 퐵° ∈ 훽 (푏푦 푑푒푓𝑖푛𝑖푡𝑖표푛 (1.1.1))

Then 퐵° ∈ 훽

Since 훽° covers 푋, then 훽 is covering to 푋

(𝑖𝑖)퐼푓 퐵 ∈ 훽, 퐴 ⊆ 푋, 퐴 ⊆ 퐵 , 푠𝑖푛푐푒 퐴 ⊆ 퐵 ⊆ 퐵°

For some 퐵° ∈ 훽°, 푡ℎ푒푛 퐴 ⊆ 훽°, 𝑖. 푒. 퐴 𝑖푠 푐표푛푡푎𝑖푛푒푑 𝑖푛 푎푛 푒푙푒푚푒푛푡 표푓 훽° (by definition (1.1.1), 퐴 ∈ 훽.

(𝑖𝑖𝑖)𝑖푓 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

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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 = ∅

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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 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 , let 퐸 be a topological vector space. The collection

훽 = {퐴 ⊆ 퐸: 퐴 𝑖푠 푏표푢푛푑푒푑 푠푢푏푠푒푡 표푓 푎 푡표푝표푙표푔𝑖푐푎푙 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. Since every neighborhood of zero is absorbent, 훽 is a

5

converting of 퐸. 훽 is obviously hereditary and we shall see that it is also stable under vector addition.

Let퐴1, 퐴2 ∈ 훽 and훽° , there exists 퐵́° such that 퐵° + 퐵° ⊆ 퐵°(proposition (1.2.3)).

Since 퐴1 and 퐴2 are bounded in 퐸, there exists positive scalars 휆 푎푛푑 휇 such that

퐴1 ⊆ 휆퐵́° and 퐴2 ⊆ 휇퐵́°. With 훼 = max(휆, 휇) we have

퐴1 + 퐴2 ⊆ 휆퐵́° + 휇퐵́° ⊂ 훼퐵́° + 훼퐵́° ⊂ 훼(퐵́° + 퐵́°) ⊂ 퐵°

Finally since 훽° is stable under the formation of circled hulls (resp. under homothetic transformation). Then so is 훽, we conclude that 훽 is a vector bornology on 퐸. If 퐸 is locally convex, then clearly 훽 is a convex bornology. Moreover, since every topological vector space has a base of closed neighborhood of 0, the closure of each bounded subset of 퐸 is again bounded.

Definition (1.2.6)[2]:-

Let (푋푖, 훽푖)푖∈퐼 be a family of bornological vector space indexed by a non- empty set 퐼 and let 푋 = ∏푖∈퐼 푋푖 be the product of the sets 푋푖. For every 𝑖 ∈

퐼 , 푙푒푡 푝푖: 푋 → 푋푖 be the canonical projection the product bornological 푋 is the initial bornology on 푋 for the maps 푝푖. The set endowed with the product bornology is called bornological product of the space (푋푖, 훽푖).

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Chapter Two

2-1 Semi- Bounded Sets In this section we give the concept of semi- bounded sets and prove some results on semi bounded sets in Bornological space. Definition (2.1.1)[2]:- A subset 퐴 of a Bornological space 푋 is said to be a semi-bounded (written s- bounded) if and only if there exists a bounded subset Β of 푋 such that Β ⊆ 퐴 ⊆ Β̅ where Β̅ is the set of all upper and lower bounds of Β which continues Β. 푆Β(X) will denoted the class of all s- bounded subsets of 푋. Remark (2.1.2)[2]:- In any Bornological space, every bounded set is s- bounded. But the converse is not true in general. Example(2.1.3)[2]:- Let 푅 the real line with the canonical Bornology, and let Β = [a, b] is bounded set in 푅, let 퐴 = [푎, ∞),then 퐴 is s- bounded in 푅, but is not bounded in 푅. Remark(2.1.4)[2]:- In any Bornological space 푋

(i) If {퐴훼}훼휖Λ is a collection of s- bounded subset of 푋, then ∩훼휖Λ 퐴훼 need not be s- bounded in 푋. (ii) The union of two s-bounded sets in 푋 is s- bounded in general.

Proof:- (ii) Let 퐴1, 퐴2 is s-bounded, then there exists a bounded set Β1, Β2 in 푋 such that Β1 ⊆ A1 ⊆ Β̅̅1̅ and Β2 ⊆ A2 ⊆ Β̅̅2̅ , where Β̅̅1̅, Β̅̅2̅ is the sets of all upper and lower bounds which contains Β1, Β2 in 푋. Thus Β1 ∪ Β2 ⊆ A1 ∪ A2 ⊆ Β̅̅1̅̅̅∪̅̅Β̅̅2̅, where Β1 ∪ Β2 is a bounded set in 푋 and Β̅̅1̅̅̅∪̅̅Β̅̅2̅ is the set of all lower bounds of Β1 ∪ Β2, then A1 ∪ A2 is s- bounded set in 푋. Example (2.1.5)[2]:- Let 푅 be the real line with canonical Bornology, then 퐴 = [푎, 푏), Β = [b, 픞) are s-bounded sets in 푅 and [푎, 푏) ∪ [푏, 푐) = [푎, 푐) is s- bounded in 푅. Theorem (2.1.6)[2]:- Let 퐴 an s-bounded set in the Bornological space 푋 and 퐴 ⊆ 퐻 ⊆ 퐴̅, where A̅ is the set of upper and lower bounded of 퐴. Then 퐻 is s- bounded, where 퐻 ⊆ 푋.

7

Proof:- Since 퐴 is s-bounded in 푋 then there exists an bounded set Β such that Β ⊆ 퐴 ⊆ B̅, n Β ⊆ 퐻. But A̅ ⊆ Β̅ (since ⊆ A̅ ⊆ Β̅ ). Hence Β ⊆ 퐻 ⊆ B̅ and 퐻 is s-bounded. Remark (2.1.7)[2]:- If (푋, Β) is a Bornological space, and 푌 is a subspace of 푋, Β ⊆ 푌, then Β̅̅̅푌̅ = A̅ ∩ 푌 is the set of all upper and lower bounds of 퐴. Theorem (2.1.8)[2]:- Let 퐴 ⊆ 푌 ⊆ 푋 where 푋 is a Bornological space and 푌 is a subspace, then 퐴휖푆Β(푋) if and only if 퐴휖푆Β(Y). Proof:- Let 퐴휖푆Β(푋) , then there exists a bounded set Β in 푋 such that

Β ⊆ 퐴 ⊆ B̅, where Β̅̅̅푋̅ is the set of all upper andlower bounds which contains Β in 푋. Now Β ⊆ 푌 (since Β ⊆ 퐴 ⊆ 푌), and thus

Β ⊆ Β ∩ 푌 ⊆ 퐴 ∩ 푌 ⊆ 푌 ∩ Β̅̅̅푋̅ 표푟 Β ⊆ 퐴 ⊆ Β̅̅̅푌̅ (by remark (2.1.7)) Β̅̅̅푌̅ = 풀 ∩ Β̅̅̅푋̅ is the set of all upper and lower bounds of Β ∩ Y in 푌, and Β = Β ∩ Y is bounded in 푌, then 퐴휖푆Β(푌). Conversely,

Let 퐴휖푆Β(푌), then there exists a bounded set Β푌 in 푌 such that Β푌 ⊆ 퐴 ⊆ Β̅̅̅푌̅ where Β̅̅̅푌̅ is the set of all upper and lower bounds of Β푌 in 푌. Β푌 in 푋, then 퐴휖푆Β(푋). Example(2.1.9)[2]:- Let 푅 be the real line with canonical Bornology and푌 = {0} and 퐴 = {0}. Then 퐴 is bounded in 푌 hence 퐴휖푆Β(푌) and (by theorem 2.1.8) 푆Β(푋) . Definition (2.1.10)[2]:- If (푋, 퐵) is a Bornological space and 퐴 ⊆ 푋. Then 퐴 is semi unbounded if and only if 푋 − 퐴 is semi bounded and (written s- unbounded) Remark (2.1.11)[2]:- In any Bornological space, every Bornological unbounded set is s- unbounded and the converse is true if 푋 is finite (since the Bornological space is hereditary under inclusion) Remark (2.1.12)[2]:- In any Bornological space (i) The union of any family of s-unbounded sets need not be s-unbounded (ii) The intersection of two s-unbounded sets is s-unbounded. (iii) If 푋 is a finite set, then the only Bornology on 푋 is 푝(푋) and so 훣 = SΒ(푋) also the collection of s-unbounded set in 푋 is 훣 itself.

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2.2 Semi- Bounded Linear Maps In this section we defined semi- bounded linear maps and prove some results on semi bounded linear map in Bornological space Definition (2.2.1)[2]:- A map 푓 from a Bornological space 푋 into a Bornological space 푌 is said to be semi- bounded map if the image 푓 of every bounded subset of 푋 is semi bounded in 푌. Remark (2.2.2)[2]:- (i) Obviously the identity map of any Bornological space is semi- bounded by (2.1.2)

(ii) Let Β1 and Β2 be two Bornologies on 푋 such that Β1 is finer than Β2 (i. e Β1 ⊂ Β2) (or Β2 is coarser than Β1) if the inclusion map (푋, Β1) → (푋, Β2) is semi- bounded. Definition (2.2.3)[2]:- Let 퐸 and 퐹 be two Bornological vector spaces. A semi- bounded linear map of 퐸 into 퐹 is that map which is bouth linear and semi bounded at the same time. Definition (2.2.4)[2]:- A semi bounded linear functional on a Bornological vector space 퐸 is a semi bounded linear map of 퐸 into the scalar field 퐾 (푅 or 퐶). Remark (2.2.5)[2]:- Every bounded map is s- bounded by (2.1.2), and the converse is not true. Example (2.2.6)[2]:- Let 푅 be the real line with the canonical Bornology, then the map 1 푓: 푅 \{0} → 푅 such that 푓(푥) = ∀ 푥휖푅\{0} is s- bounded map but it is 푥 not bounded map since, let 퐹 = (0,1]is bounded set in 푥휖푅\{0}, then the image of this bounded set under 푓 is [1, ∞) is s- bounded in 푅 but it is not bounded in 푅. Definition (2.2.7)[2]:- Let 푋 and 푌 be Bornological spaces. A map 푓: 푋 → 푌 is said to be a 푠∗- bounded map if and only if image of every s- bounded subset of 푋 under 푓 is s- bounded in 푌.

Definition (2.2.8)[2]:-

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Let 푋 and 푌 be Bornological spaces. A map 푓: 푋 → 푌 is said to be a 푠∗∗- bounded map if and only if image of every s- bounded subset of 푋 under 푓 is s- bounded in 푌. Proposition (2.2.9)[2]:- Let 푋, 푌 and 푍 be Bornological spaces. A map 푓: 푋 → 푌 bounded (s- bounded) map and 푔: 푌 → 푍 be s- bounded (푠∗∗- bounded) map then 푔 ∘ 푓: 푋 → 푍 is s- bounded (푠∗∗-bounded) map. Proof:- Let 퐹 be bounded in 푋, since 푓 is bounded (s- bounded) map, then 푓(퐹) is bounded (s- bounded) in 푌. But 푔 is s- bounded (푠∗∗- bounded) map. Then 푔(푓(퐹)) is s- bounded in 푍. But 푔(푓(퐹)) = (푔 ∘ 푓)(퐹), then 푔 ∘ 푓 is s- bounded (푠∗∗- bounded)

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Chapter Three

3.Bornological semi-convergence net

In this section, we define semi-convergence of net in every convex bornological vector space, and the main propositions and theorems about this concept.

Definition (3.1.1)[1]:-

Let (푥 ) be a net in a convex bornological vector space E. We 훾 훾∈Γ 푏푠 say that (푥훾) bornologically semi-converges to 0 ( (푥훾) → 0), if there is an absolutely and semi-bounded 푆 of E and a net (휆훾) in 퐾 converging to 0, such that 푥훾 ∈ 휆훾푆, for every 훾 ∈ 훤. Then A net (푥훾) 푏푠 bornologically semi-converges to a point 푥 ∈ 퐸 and ( 푥훾 → 푥) when 푏푠 ((푥훾 − 푥) → 0).

Remark (3.1.2)[1]:-

Let 퐸 be a convex bornological vector space. A net (푥 ) in 퐸 is 훾 훾∈Γ bornologically semi- convergent to a point 푥 ∈ 퐸, if there is a decreasing 푥훾−푥 net (휆훾) of positive real numbers tending to zero such that the net ( ) 휆훾 is a semi-bounded.

Theorem (3.1.3)[1]:-

Every bornologically semi-convergent net is semi-bounded.

Proof:- Let 퐸 be a convex bornological vector space and a net (푥훾) 푏푠 bornologically semi-converges to a point 푥 ∈ 퐸. i.e. (푥훾 − 푥) → 0, there is an absolutely convex subset and semi-bounded 푆 of 퐸 and a net (휆훾) of scalars tend to 0, such that 푥훾 ∈ 푆 and (푥훾 − 푥) ∈ 휆훾푆 for every 훾 ∈ Γ.

Since (휆훾) is a net of scalars tend to 0, and 퐸 is a convex bornological vector space then 휆훾 푆 is a semi-bounded subset of 퐸 ,

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i.e. {(푥 − 푥) } ⊆ {(푥 − 푥) ⊆ 휆 푆} ( every subset of semi- 훾 훾∈훤 훾 훾∈훤 훾 bounded is semi-bounded [3]) implies (푥훾 − 푥) is a semi-bounded subset of 퐸, then (푥훾) is semi- bounded.

Theorem (3.1.4)[1]:-

Let 퐸 and 퐹 be convex bornological vector space. Then the image of a bornologically semi-convergent net under a semi-bounded linear map of E into F is a bornologically semi-convergent net.

Proof:- Let (푥훾) be a net semi- converges bornologically to a point 푥 in 푏푠 퐸, and let 푢: 퐸 ⟶ 퐹 be a semi-bounded linear map. Since (푥훾 − 푥) → 0 in 퐸, then there is an and semi-bounded 푆 of E and a net (휆훾) of scalars tending to 0, such that 푥훾 ∈ 푆 and 푥훾 − 푥 ∈ 휆훾 푆 for every 훾 ∈ 훤.

Then 푢(푥훾 − 푥) ∈ 푢(휆훾푆) for every 훾 ∈ 훤. Since 푢 is a linear map and

(휆훾) is a net of scalars, then 푢(푥훾) − 푢(푥) ∈ 휆훾푢(푆), since 푢 is a semi- bounded map, we have 푢(푆) is semi-bounded and disk (if absolutely convex set then it is disked [6]) when 푆 is semi-bounded and disked by 푏푠 (definition 2.1.1) we have 푢(푥훾) − 푢(푥) → 0 then 푢(푥훾) bornologically semi-converges to a point 푢(푥) in 퐹.

Theorem (3.1.5)[1]:-

Suppose that (푥 ) and (푦 ) are bornologically semi- 훾 훾∈Γ 훾 훾∈Γ convergent nets in a convex bornological vector space 퐸, (휆훾) is a 푏푠 푏푠 convergent net in 퐾 such that 푥훾 → 푥, 푦훾 → 푦 and 휆훾 ⟶ 휆 then

푏푠 (i) 푥훾 + 푦훾 → 푥 + 푦; 푏푠 (ii) 푐푥훾 → 푐푥, for any number c ∈ K; 푏푠 (iii) 휆푦푥훾 → 휆푥 . Proof:-

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푏푠 푏푠 푏푠 푏푠 (i) Since 푥훾 → 푥, 푦훾 → 푦 in 퐸 i.e. 푥훾 − 푥 → 0, 푦훾 − 푦 → 0 in 퐸, then there exist an absolutely convex and semi-bounded sets 푆1, 푆2 of 퐸 and nets

(훼훾), (훽훾) of scalars tending to 0, such that :

(푥훾 − 푥) ∈ 훼훾푆1 and (푦훾 − 푦) ∈ 훽훾푆2 for every 훾 ∈ Γ . Then 푥훾 − 푥 +

푦훾 − 푦 ∈ 훼훾푆1 + 훽훾푆2 = (훼훾 + 훽훾)(푆1 + 푆2) − 훼훾푆2 − 훽훾푆1.

Since 훼훾 ⟶ 0 and 훽훾 ⟶ 0.

Then ((푥훾 + 푦훾) − (푥 + 푦)) ∈ 훼훾푆1 + 훽훾푆2 ⊆ (훼훾 + 훽훾)(푆1 + 푆2)

Now, if 훼훾 ⟶ 0 and 훽훾 ⟶ 0 in 퐾 then 훼훾 + 훽훾 ⟶ 0 in 퐾 and 푆1+푆2 is semi-bounded and absolutely convex set when 푆1 and 푆2 are semi- bounded and absolutely convex sets and since 퐸 is a bornological vector space

푏푠 푏푠 implies ((푥훾 + 푦훾) − (푥 + 푦)) → 0 then (푥훾 + 푦훾) → (푥 + 푦).

푏푠 푏푠 (ii) If 푥훾 → 푥 then (푥훾 − 푥) → 0 and so there is an absolutely convex set and semi-bounded 푆 of 퐸 and a net (훼훾) of scalars tends to 0, such that

푥훾 ∈ 푆 and (푥훾 − 푥) ∈ 훼훾푆 for every 훾 ∈ 훤 푐(푥훾 − 푥) ∈ 푐훼훾푆 then 푐푥훾 − 푐푥 ∈ 훼훾(푐푆). Since 푐 ∈ 퐾 and 퐸 is a convex bornological vector space then 푐푆 is an absolutely convex set and semi-bounded of 퐸 when 푆 is an absolutely convex set and semi-bounded of 퐸 by(definition 2.1) 푏푠 푏푠 푐푥훾 − 푐푥 → 0 , then 푐푥훾 → 푐푥 .

(iii) If (푥훾) bornologically semi-converges to 푥 in 퐸, then there is an absolutely convex and semi-bounded set 푆 of 퐸 and a net (훼훾) of scalars tends to 0, such that

푥훾 ∈ 푆 and (푥훾 − 푥) ∈ 훼훾푆 for every 훾 ∈ Γ. 휆훾푥훾 − 휆푥 =

(푥훾 − 푥)(휆훾 − 휆) + 푥(휆훾 − 휆) + 휆(푥훾 − 푥). Now 휆훾 ⟶ 휆, then 휆훾 − 푏푠 푏푠 휆 ⟶ 0 and 푥훾 → 푥. If (푥훾 − 푥) → 0 then 휆훾푥훾 − 휆푥 = 휆훾푥훾 − 휆푥 =

(푥훾 − 푥)(휆훾 − 휆) ∈ 훼훾 ((휆훾 − 휆)푆) i.e. 휆훾푥훾 − 휆푥 ∈ 훼훾 ((휆훾 − 휆)푆).

Since 퐸 is a convex bornological vector space, then (휆훾 − 휆)푆 is an

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absolutely convex and semi-bounded set when 푆 is an absolutely convex 푏푠 and semi-bounded set of 퐸 by (definition 2.1) 휆훾푥훾 − 휆푥 → 0 then 휆훾푥훾 푏푠 → 휆푥.

Theorem (3.1.6)[1]:-

A convex bornological vector space 퐸 is separated if and only if every bornologically semi-convergent net in E has a unique limit.

Proof:- Necessity: Let 퐸 be a separated bornological vector space. If a net (푥훾) in 퐸 bornologically semi-converges to 푥 and 푦 . Then the net

푧훾 = 푥훾 − 푥훾 = 0 semi-converges to 푧 = 푥 − 푦.

Thus it suffices to show that the limit 푧 of the net (푧훾 = 0) must be the element 0 .

Let (휆훾) be a net of real numbers tends to 0 and let 푆 be a semi-bounded subset of 퐸 such that 푧 − 푧훾 = 푧 ∈ 휆훾푆 for every 훾 ≥ 1 .

If 푧 ≠ 0, then the line spanned by 푧 (i.e. the subspace (퐾푧) is contained in 푆,that contradicts the hypothesis that 퐸 is separated.

Sufficiency: Assume the uniqueness of limits, and suppose that there is an element 푧 ≠ 0 such that the line spanned by 푧 is semi-bounded. Then 1 we can find a semi-bounded set 푆 ⊂ 퐸 such that 푧 ∈ ( ) 푆 for every 훾 ≥ 훾

1 and hence the net (푧훾 = 푧) semi-converges to 0.

But clearly this net also semi-converges to 푧 , whence, by uniqueness of limits, 푧 = 0 we have reached a contradiction.

Theorem( 3.1.7)[1]:-

Let 퐸 be a convex bornological vector space and let (푥훾) be a net in 퐸 then the following are equivalent:

(i) The net (푥훾) bornologically semi-converges to 0; (ii) there is an absolutely convex set and semi-bounded 푆 ⊂ 퐸 and a decreasing net (훼훾) of positive real numbers, tends to 0, such that 푥훾 ∈

훼훾푆 for every 훾 ∈ 훤;

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(iii) there is an absolutely convex and semi-bounded set 푆 ⊂ 퐸 such that, given any 휀 > 0 , we can find an integer 훤(휀) for which 푥훾 ∈ 휀푆 whenever 훾 ≥ 훤(휀).

(iv) there is a semi-bounded disk 푆 ⊂ 퐸 such that (푥훾) belongs to the semi-normed . 퐸푆 and semi-converges to 0 in 퐸푆. Proof:- (i) ⟹(ii):

1 For any integer 푝 ∈ 훤 there is 훤 ∈ 훤 such that if 훾 ≥ 훤p then λγ ≤ ; 푝 p 1 hence 휆 푆 ⊂ ( ) 푆, since 푆 is disk. We may assume that the net 훤 is 훾 푝 푝 strictly increasing, and, for 훤푝 ≤ 퐾 ≤ 훤푝+1 ,

1 Let 훼 = . Then the net {훼 } satisfies the conditions of assertion (ii). 퐾 푝 퐾

Clearly (ii) ⟹ (iii).

To show that (iii) ⟹ (i), let , for every 훾 ∈ 훤, 휀 = 푖푛푓 {휀 > 0; 푥훾 ∈ 휀푆}, 1 and 휆 = 휀 + , 훾 훾 훾 then the net (휆훾) converges to 0 and 푥훾 ∈ 휆훾푆 for every 훾 ∈ 훤. Thus the assertion (i, ii, iii) are equivalent. Suppose that the bornology of

퐸 is convex. Clearly (iv) implies (i) with 휆훾 = 푃푠(푥훾) and 푃푆 the gauge of

푆, while (ii) implies that 푥훾 ∈ 퐸푆 and 푃푆(푥훾) ≤ 훼훾 → 0. Remark (3.1.8)[1]:-

It is clear that (xγ) for every 훾 ∈ 훤 bornologically semi-converges to

0 if and only if every subnet of (푥훾) bornologically semi- converges to 0. Theorem (3.1.9)[1]:-

A net {푥훾} in a product convex bornological vector space ∏푖∈퐼 퐸푖 bornologically semi-converges to 푦 if and only if the net {푥푖 } 훾 푖∈퐼 bornologically semi-converges to 푦푖 in convex bornological vector space 퐸푖.

푏푠 Proof:- Let 푥훾 → 푦 = (푦1, 푦2, … , 푦푛, … ) in ∏푖∈퐼 퐸푖, since 푝푖 is semi- 푏푠 bounded linear map, then by (Theorem 3.1.4) 푝푖(푥훾) → 푝푖(푦), for each

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푏푠 푏푠 푖 푖 푖 ∈ 퐼, then 푥훾 → 푦푖 in 퐸푖. Suppose that 푥훾 → 푦푖 for each 푖 ∈ 퐼, then there is an absolutely convex and semi-bounded set 푆푖 of 퐸푖 , 푖 ∈ 퐼 and a net 푖 푖 푖 (휆훾) of scalars tending to 0, such that 푥훾 ∈ 푆푖 and for each 푖 ∈ 퐼푥훾 − 푦푖 ∈ 푖 푖 휆훾 − 푆푖, 훾 ∈ 훤 since a net 휆훾 → 0, 푖 ∈ 퐼, for each 훾 ∈ 훤 then there is a 푖 diagonal net semi-converging to 0 such that when 푖 = 훾 then 휆훾 → 0.

푖 푛 Since 푥훾 − 푦푖 ∈ 휆훾퐵푖 whenever 푖 = 푛 and = 1,2, … , 푛, … .

푛 푛 If ≠ 푖 , since a disk 푆푖 and |휆훾| → 0 and if 푎 ∈ 푆푖 then 휆훾푎 ∈ 푆푖 then 푖 푛 푖 푛 1 2 푥훾 − 푦푖 ∈ 휆훾푆푖 then 푥훾 − 푦푖 ∈ 휆훾푆푖 . Therefore 푥훾 = (푥훾, 푥훾 , … ) − 푦 ∈ 푠푏 푛 휆훾 ∏푖∈퐼 푆푖 푖. 푒 푥훾 → 푦 .

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References

[1] Al-Basri, Fatma Kamil Majeed. "On Semi –Complete Bornological Vector Space." , Journal of AL-Qadisiyah for computer science and mathematics, Vol.9 No.1 Year 2017.

[2] Al-Salihi, Anwar, Nooral-Deen " Semi-bounded Set in Bornological Space, Semi-bounded map, and Semi convergence". J. of college of edu, No.3.(2009).

[3] Barreira, L. and Almeida, j. "Hausdorff Dimension in Convex Bornological Space", J Math Appl. 268,590-601, (2002).

[4] Borwein,J. and Fitzpatrick,S.; "Weak Sequential Compactness and Bornological Limit Derivatives",J. of Convex Anal. Vol. 2 , No.1/2, 59- 67,(1995)

[5] Dierolf, S. and Domanski, P.; "Bornological Space Of Null Sequences", Arch. Math. 65(1): 46-52, (1995).

[6] Hogbe Nlend, H.; "Bornologies and ", North –Holland Publishing Company Netherlands, (1977).

[7] Lechicki,A. , Leviand,S. and Spakowski,A.; "Bornological Convergence"J. Math. Anal.Appl.297,751-770,(2004).

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