on semi complete bornological vector...

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

KARRAR

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

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-Cauchy Net

11

3.2 Semi-complete bornological vector space 12

Abstract

The definition of semi-convergence of nets, semi-cauchy nets in

convex bornological vector space and semi-complete bornological vector

space and the relationship among these concepts have been studied in

this paper. Before that We study the concept of semi- bounded set in

Bornological spaces, semi bounded linear map and study some of their

properties.

introduction

The concept of bornological convergence had been studied in [6]

.The convergent net in convex bornological vector space and its results

studied after that .In chapter one we study bornological space, sub space,

bornological vector space, convex bornological vector space, sparated

bornological vector space…etc. Semi bounded sets were introduced and

studied in chapter two, semi bounded sets have been used to define and

study many new bornological proprieties. In chapter three We define

semi- cauchy net in convex bornological vector space in the context of

this study and then semi-complete bornological convex vector has been

investigated in section four. space some results.

Chapter One

1

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 𝑋 is a family 𝛽 of subset 𝑋 satisfying the following

axioms :

(i) 𝛽 𝑖𝑠 𝑎 𝑐𝑜𝑛𝑣𝑒𝑟𝑖𝑛𝑔 𝑋, 𝑖. 𝑒. 𝑋 = ⋃ 𝛽𝐵∈𝛽 ;

(ii) 𝛽 is hereditary under inclusion i.e. if 𝐴 ∈ 𝛽 and 𝐵 is a subset of 𝑋 contained

in 𝐴, then 𝐵 ∈ 𝛽

(iii) 𝛽 𝑖𝑠 𝑠𝑡𝑎𝑏𝑙𝑒 𝑢𝑛𝑑𝑒𝑟 𝑓𝑖𝑛𝑖𝑡𝑒 𝑢𝑛𝑖𝑜𝑛.

A pair (𝑋, 𝛽) consisting of a sat 𝑋 and a bornology 𝛽 on 𝑋 is called a

bornological space, and the elements of 𝛽 are called the bounded subset of 𝑋.

Example (1.1.2)[1]:

Let 𝑋 = {𝑎, 𝑏, 𝑐}, 𝛽 = {∅, {𝑎}, {𝑏}, {𝑐}, {𝑎, 𝑏}, {𝑎, 𝑐}, {𝑏, 𝑐}, 𝑋}

(i) since 𝑋 ∈ 𝛽, then 𝑋 is converging itself, we have 𝛽 is converging of 𝑋

(ii) since 𝛽 is the set of all subset of 𝑋 i.e.𝛽 = 𝑝(𝑋)and 𝛽 hereditary under

inclusion i.e. 𝐴 ∈ 𝛽, 𝐵 ⊂ 𝐴,then 𝐵 ∈ 𝛽

(iii) since 𝛽 = 𝑝(𝑋)then ⋃ 𝐴𝑖 ∈ 𝛽 𝑛𝑖=1 for all 𝐴𝑖 ∈ 𝛽 𝑖 = 1,2, … , 𝑛

Example (1.1.3)[6]:

Let 𝑅 be a field with absolute value. The collection

𝛽={A≤ R:A is a bounded subset of R in the usual sense for the absolute value}.

Then 𝛽 is a bornology on 𝑅 called the Canonical bornology of 𝑅.

Example (1.1.4)[6]:

Let 𝐸 be a vector space over 𝐾 (we shall assume that 𝐾 is either 𝑅 or 𝐶).

Let 𝑝 be a semi-normed 𝑝 if 𝑝(𝐴) is bounded subset of 𝑅 in the sense of

(Example (1.1.3). The subset of 𝐸 which are bounded for the semi-normed 𝑝

2

form a bornology on 𝐸 called the canonical bornology on the semi-normed

space (𝐸, 𝑃)

(i) since ∀𝑥 ∈ 𝐸, 𝑝(𝑥) ∈ 𝑅, then ∃𝑟 > 0 such that 𝑝(𝑋) ≤ 𝑟 implies ∀𝑥 ∈

𝐸, {𝑋}is a bounded subset for the semi-normed, i.e. {𝑥} ∈ 𝛽, then β is a covering

of 𝐸

(ii) 𝑖𝑓 𝐴 ∈ 𝛽 𝑎𝑛𝑑 𝐵 ⊆ 𝐴, then 𝑝(𝐴)is bounded subset of 𝑅 in the sense for the

absolute value since 𝐵 ⊆ 𝐴 then 𝑝(𝐵) ⊆ 𝑝(𝐴), then 𝑝(𝐵)is a bounded of

𝑅(every subset of a bounded set is bounded)

(iii) 𝑖𝑓 𝐴1, … , 𝐴𝑛 ∈ 𝛽, then 𝑝(𝐴1), … , 𝑝(𝐴𝑛)are bounded subsets of 𝑅.

⋃ 𝑝(𝐴𝑖) 𝑛𝑖 is a bounded subset of 𝑅 by (iv); proposition 1,2.3) i.e. 𝑝(⋃ 𝐴𝑖) 𝑛

𝑖 is

bounded subset of 𝑅. Then ⋃ 𝐴𝑖 𝑛𝑖 is a bounded subset of semi-normed 𝑝,

i.e. ⋃ 𝐴𝑖 𝑛𝑖 ∈ 𝛽, then 𝛽 is abonology on 𝐸.

Definition (1.1.5)[2]:

A base of abornology 𝛽 on 𝑋 is any subfamily 𝛽° of 𝛽 such that every

element of 𝛽 is contained in an element of 𝛽°

Example (1.1.6):

If 𝑋 = {1,2,3} , 𝛽 = {∅, {1}, {2}, {1,2}, {1,3}, {2,3}, 𝑋}

Let 𝛽° = {𝑋} be a base for 𝛽. It is clear that each element of 𝛽 is contained in an

element of 𝛽°

Proposition (1.1.7)[2]:

Let (𝑋, 𝛽) be a bornological space and 𝛽° is the base of 𝛽, then the

following hold :

(𝑖)𝑋 = ⋃ 𝛽°

𝛽°∈𝛽

(𝑖𝑖) A finite union of elements of 𝛽° is contained in an element of 𝛽°.

Conversely; if 𝑋 ≠ ∅ and 𝛽° is a family of subset of 𝑋 satisfying : (i) and (ii)

then ∃ abornology 𝛽 𝑜𝑛 𝑋 such that 𝛽° is base for 𝛽.

3

Proof :-Since 𝛽° forms a base of abornology 𝛽 an a set 𝑋, then every element of

𝛽 is contained in an element of 𝛽° 𝑖. 𝑒. ∀𝐵 ∈ 𝛽, ∃𝛽° ∈ 𝛽 such that 𝐵 since 𝛽 is

covering of 𝑋.

𝑖. 𝑒. 𝑋 = ⋃ 𝐵

𝐵∈𝛽

, 𝑡ℎ𝑒𝑛 𝑋 = ⋃ 𝛽°

𝛽°∈𝛽

𝑖. 𝑒. 𝛽° 𝑐𝑜𝑛𝑣𝑒𝑟𝑠 𝑋, 𝑤ℎ𝑒𝑛𝑐𝑒 (𝑖)

(𝑖𝑖)𝐿𝑒𝑡 𝐵𝑖 ∈ 𝛽° ∀𝑖 = 1,2, … , 𝑛

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

Since 𝛽° ⊆ 𝛽 𝑡ℎ𝑒𝑛 𝐵𝑖 ∈ 𝛽 ∀ 𝑖 =

1,2, … , 𝑛 (𝑏𝑦 𝑑𝑒𝑓𝑖𝑛𝑖𝑜𝑛 (1.1.1)) 𝑡ℎ𝑒𝑛 ⋃ 𝐵𝑖 ∈ 𝛽𝑛𝑖=1 , 𝑏𝑦 𝑑𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛(1.1.5)

We have ⋃ 𝐵𝑖𝑛𝑖 is contained in an element of 𝛽°

Conversely; let 𝛽 = {𝐵 ⊆ 𝑋: ∃𝐵° ∈ 𝛽°, 𝐵 ⊆ 𝛽°}

Then to show that 𝛽 is a bornology on 𝑋 we must satisfy the following

conditions :

(𝑖)∀𝐵° ∈ 𝛽° → 𝐵∘ ⊆ 𝑋 → 𝐵° ∈ 𝛽° → 𝐵° ∈ 𝛽 (𝑏𝑦 𝑑𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 (1.1.1))

Then 𝐵° ∈ 𝛽

Since 𝛽° covers 𝑋, then 𝛽 is covering to 𝑋

(𝑖𝑖)𝐼𝑓 𝐵 ∈ 𝛽, 𝐴 ⊆ 𝑋, 𝐴 ⊆ 𝐵 , 𝑠𝑖𝑛𝑐𝑒 𝐴 ⊆ 𝐵 ⊆ 𝐵°

For some 𝐵° ∈ 𝛽°, 𝑡ℎ𝑒𝑛 𝐴 ⊆ 𝛽°, 𝑖. 𝑒. 𝐴 𝑖𝑠 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑒𝑑 𝑖𝑛 𝑎𝑛 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝑜𝑓 𝛽°

(by definition (1.1.1), 𝐴 ∈ 𝛽.

(𝑖𝑖𝑖)𝑖𝑓 𝐵1, 𝐵2 ∈ 𝛽 𝑡ℎ𝑒𝑛 ∃𝐵°, 𝐵° ∈ 𝛽° such that

𝐵1 ⊆ 𝐵°(by condition (i)) and 𝐵2 ⊆ 𝐵°

Then 𝐵1 ∪ 𝐵2 ⊆ 𝐵° ∪ 𝐵° for some 𝐵∘

” ∈ 𝛽° (by condition (ii))

then (by the definition (1.1.1))𝐵1 ∪ 𝐵2 ∈ 𝛽

4

Then 𝛽 is a bornology on 𝑋, and (by definition (1.1.5)) 𝛽° is a base for 𝛽, we say

𝛽 is a bornology generated by a base

Definition (1.1.8)[6]:-

A bornology an a set 𝑋 is said a bornology with countable base if and only

if 𝛽 possesses a base consisting of an increasing sequence of bounded sets (an

element of bornology).

1.2 Subspaces Of Bornological Space


Let (𝑋, 𝐵) be a bornological space and let 𝑌 ⊆ 𝑋. The bornology for 𝑌 is

the collection

𝛽𝛾 = {𝐺 ∩ 𝑌: 𝐺 ∈ 𝛽}. The bornological space (𝑌, 𝛽𝛾) is called a subspace of

(𝑋, 𝛽),and 𝛽𝛾 the relative bornology on 𝑌.

Example (1.2.2):

Let 𝑋 = {1,2,3}, 𝛽 = {{∅}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}}, and the base

for the bornology is

𝛽° = {{1,2}, {1,3}, 𝑋}, and let 𝑌 = {1,2}. Then we have

∅ ∩ 𝑌 = ∅

{1,2} ∩ 𝑌 = 𝑌

{1} ∩ 𝑌 = {1}

{1,3} ∩ 𝑌 = {1}

{2} ∩ 𝑌 = {2}

{2,3} ∩ 𝑌 = {2}

{3} ∩ 𝑌 = ∅

5

𝑋 ∩ 𝑌 = 𝑌

Then the subspace 𝛽𝛾 = {∅, {1}, {2}, 𝑌}, and a base of the bornological subspace

is 𝛽𝛾 = {{1}, 𝑌}.


Let 𝐸 be a vector space over the field 𝐾 (the real or complex field). A

bornology 𝛽 on 𝐸 is said to be a bornology compatible with a vector space of 𝐸

or to be a vector bornology on 𝐸, if 𝛽 is stable under vector addition homothetic

transformations and the formation of circled hulls, in other words, if the sets

𝐴 + 𝐵, 𝜆𝐴, ⋃ 𝛼𝐴|𝛼|≤1 belongs to 𝛽 whenever 𝐴 and 𝐵 belong to 𝛽 and 𝜆 ∈ 𝐾.


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 (𝐸, 𝛽) is one where {0} is the only

bounded vector subspace of 𝐸.

Example (1.2.6)[6]:-

The Von-Neumann bornology of topological vector space, let 𝐸 be a

topological vector space. The collection

𝛽 = {𝐴 ⊆ 𝐸: 𝐴 𝑖𝑠 𝑏𝑜𝑢𝑛𝑑𝑒𝑑 𝑠𝑢𝑏𝑠𝑒𝑡 𝑜𝑓 𝑎 𝑡𝑜𝑝𝑜𝑙𝑜𝑔𝑖𝑐𝑎𝑙 𝑣𝑒𝑐𝑡𝑜𝑟 𝑠𝑝𝑎𝑐𝑒 𝐸} forms a

vector bornology on 𝐸

Called the Von-Neumann bornology of 𝐸. Let us verify that 𝛽 is indeed a vector

bornology on 𝐸, if 𝛽° is a base of circled neighborhoods of zero in 𝐸, it is clear

that a subset 𝐴 of 𝐸 is bounded if and only if for every 𝐵 ∈ 𝛽° there exists 𝜆 > 0

6

such that 𝐴 ⊆ 𝜆𝐵. Since every neighborhood of zero is absorbent, 𝛽 is a

converting of 𝐸. 𝛽 is obviously hereditary and we shall see that it is also stable

under vector addition.

Let𝐴1, 𝐴2 ∈ 𝛽 and𝛽° , there exists 𝐵° such that 𝐵° + 𝐵° ⊆ 𝐵°(proposition (1.2.3)).

Since 𝐴1 and 𝐴2 are bounded in 𝐸, there exists positive scalars 𝜆 𝑎𝑛𝑑 𝜇 such that

𝐴1 ⊆ 𝜆𝐵° and 𝐴2 ⊆ 𝜇𝐵°

. With 𝛼 = max(𝜆, 𝜇) we have

𝐴1 + 𝐴2 ⊆ 𝜆𝐵° + 𝜇𝐵°

⊂ 𝛼𝐵° + 𝛼𝐵°

⊂ 𝛼(𝐵° + 𝐵°

) ⊂ 𝐵°

Finally since 𝛽° is stable under the formation of circled hulls (resp. under

homothetic transformation). Then so is 𝛽, we conclude that 𝛽 is a vector

bornology on 𝐸. If 𝐸 is locally convex, then clearly 𝛽 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 𝐸 is again bounded.


Let (𝑋𝑖 , 𝛽𝑖)𝑖∈𝐼 be a family of bornological vector space indexed by a non-

empty set 𝐼 and let 𝑋 = ∏ 𝑋𝑖𝑖∈𝐼 be the product of the sets 𝑋𝑖. For every 𝑖 ∈

𝐼 , 𝑙𝑒𝑡 𝑝𝑖: 𝑋 → 𝑋𝑖 be the canonical projection the product bornological 𝑋 is the

initial bornology on 𝑋 for the maps 𝑝𝑖. The set endowed with the product

bornology is called bornological product of the space (𝑋𝑖 , 𝛽𝑖).

Chapter Two

7

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.


A subset 𝐴 of a Bornological space 𝑋 is said to be a semi-bounded

(written s- bounded) if and only if there exists a bounded subset Β of 𝑋

such that Β ⊆ 𝐴 ⊆ Β where Β is the set of all upper and lower bounds of

Β which continues Β. 𝑆Β(X) will denoted the class of all s- bounded

subsets of 𝑋.

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 𝑅 the real line with the canonical Bornology, and let Β = [a, b]

is bounded set in 𝑅, let 𝐴 = [𝑎, ∞),then 𝐴 is s- bounded in 𝑅, but is not

bounded in 𝑅.

Remark(2.1.4)[2]:-

In any Bornological space 𝑋

(i) If {𝐴𝛼}𝛼𝜖Λ is a collection of s- bounded subset of 𝑋, then ∩𝛼𝜖Λ 𝐴𝛼

need not be s- bounded in 𝑋.

(ii) The union of two s-bounded sets in 𝑋 is s- bounded in general.

Proof:- (ii) Let 𝐴1, 𝐴2 is s-bounded, then there exists a bounded set

Β1, Β2 in 𝑋 such that Β1 ⊆ A1 ⊆ Β1 and Β2 ⊆ A2 ⊆ Β2

, where Β1 , Β2

is

the sets of all upper and lower bounds which contains Β1, Β2 in 𝑋. Thus

Β1 ∪ Β2 ⊆ A1 ∪ A2 ⊆ Β1 ∪ Β2 , where Β1 ∪ Β2 is a bounded set in 𝑋 and

Β1 ∪ Β2 is the set of all lower bounds of Β1 ∪ Β2, then A1 ∪ A2 is s-

bounded set in 𝑋.

Example (2.1.5)[2]:-

Let 𝑅 be the real line with canonical Bornology, then 𝐴 =

[𝑎, 𝑏), Β = [b, 𝔞) are s-bounded sets in 𝑅 and [𝑎, 𝑏) ∪ [𝑏, 𝑐) = [𝑎, 𝑐) is s-

bounded in 𝑅.

Theorem (2.1.6)[2]:-

Let 𝐴 an s-bounded set in the Bornological space 𝑋 and 𝐴 ⊆ 𝐻 ⊆ ��,

where A is the set of upper and lower bounded of 𝐴. Then 𝐻 is s-

bounded, where 𝐻 ⊆ 𝑋.

8

Proof:- Since 𝐴 is s-bounded in 𝑋 then there exists an bounded set Β such

that Β ⊆ 𝐴 ⊆ B, n Β ⊆ 𝐻. But A ⊆ Β (since ⊆ A ⊆ Β ). Hence Β ⊆ 𝐻 ⊆

B and 𝐻 is s-bounded.

Remark (2.1.7)[2]:-

If (𝑋, Β) is a Bornological space, and 𝑌 is a subspace of 𝑋, Β ⊆ 𝑌,

then Β𝑌 = A ∩ 𝑌 is the set of all upper and lower bounds of 𝐴.

Theorem (2.1.8)[2]:-

Let 𝐴 ⊆ 𝑌 ⊆ 𝑋 where 𝑋 is a Bornological space and 𝑌 is a

subspace, then 𝐴𝜖𝑆Β(𝑋) if and only if 𝐴𝜖𝑆Β(Y).

Proof:- Let 𝐴𝜖𝑆Β(𝑋) , then there exists a bounded set Β in 𝑋 such that

Β ⊆ 𝐴 ⊆ B, where Β𝑋 is the set of all upper andlower bounds which

contains Β in 𝑋. Now Β ⊆ 𝑌 (since Β ⊆ 𝐴 ⊆ 𝑌), and thus

Β ⊆ Β ∩ 𝑌 ⊆ 𝐴 ∩ 𝑌 ⊆ 𝑌 ∩ Β𝑋 𝑜𝑟 Β ⊆ 𝐴 ⊆ Β𝑌

(by remark (2.1.7)) Β𝑌 =

𝒀 ∩ Β𝑋 is the set of all upper and lower bounds of Β ∩ Y in 𝑌, and Β =

Β ∩ Y is bounded in 𝑌, then 𝐴𝜖𝑆Β(𝑌).

Conversely,

Let 𝐴𝜖𝑆Β(𝑌), then there exists a bounded set Β𝑌 in 𝑌 such that Β𝑌 ⊆ 𝐴 ⊆

Β𝑌 where Β𝑌

is the set of all upper and lower bounds of Β𝑌 in 𝑌. Β𝑌 in 𝑋,

then 𝐴𝜖𝑆Β(𝑋).

Example(2.1.9)[2]:-

Let 𝑅 be the real line with canonical Bornology and𝑌 = {0} and

𝐴 = {0}. Then 𝐴 is bounded in 𝑌 hence 𝐴𝜖𝑆Β(𝑌) and (by theorem 2.1.8)

𝑆Β(𝑋) .


If (𝑋, 𝐵) is a Bornological space and 𝐴 ⊆ 𝑋. Then 𝐴 is semi

unbounded if and only if 𝑋 − 𝐴 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 𝑋 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 𝑋 is a finite set, then the only Bornology on 𝑋 is 𝑝(𝑋) and so 𝛣 =

SΒ(𝑋) also the collection of s-unbounded set in 𝑋 is 𝛣 itself.

9

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


A map 𝑓 from a Bornological space 𝑋 into a Bornological space 𝑌 is

said to be semi- bounded map if the image 𝑓 of every bounded subset of

𝑋 is semi bounded in 𝑌.

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 𝑋 such that Β1 is finer than Β2

(i. e Β1 ⊂ Β2) (or Β2 is coarser than Β1) if the inclusion map (𝑋, Β1) →

(𝑋, Β2) is semi- bounded.


Let 𝐸 and 𝐹 be two Bornological vector spaces. A semi- bounded

linear map of 𝐸 into 𝐹 is that map which is bouth linear and semi

bounded at the same time.


A semi bounded linear functional on a Bornological vector space 𝐸

is a semi bounded linear map of 𝐸 into the scalar field 𝐾 (𝑅 or 𝐶).

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 𝑅 be the real line with the canonical Bornology, then the map

𝑓: 𝑅 \{0} → 𝑅 such that 𝑓(𝑥) =1

𝑥 ∀ 𝑥𝜖𝑅\{0} is s- bounded map but it is

not bounded map since, let 𝐹 = (0,1]is bounded set in 𝑥𝜖𝑅\{0}, then the

image of this bounded set under 𝑓 is [1, ∞) is s- bounded in 𝑅 but it is not

bounded in 𝑅.


Let 𝑋 and 𝑌 be Bornological spaces. A map 𝑓: 𝑋 → 𝑌 is said to be a

𝑠∗- bounded map if and only if image of every s- bounded subset of 𝑋

under 𝑓 is s- bounded in 𝑌.


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Let 𝑋 and 𝑌 be Bornological spaces. A map 𝑓: 𝑋 → 𝑌 is said to be a

𝑠∗∗- bounded map if and only if image of every s- bounded subset of 𝑋

under 𝑓 is s- bounded in 𝑌.

Proposition (2.2.9)[2]:-

Let 𝑋, 𝑌 and 𝑍 be Bornological spaces. A map 𝑓: 𝑋 → 𝑌 bounded (s-

bounded) map and 𝑔: 𝑌 → 𝑍 be s- bounded (𝑠∗∗- bounded) map then 𝑔 ∘

𝑓: 𝑋 → 𝑍 is s- bounded (𝑠∗∗-bounded) map.

Proof:- Let 𝐹 be bounded in 𝑋, since 𝑓 is bounded (s- bounded) map,

then 𝑓(𝐹) is bounded (s- bounded) in 𝑌. But 𝑔 is s- bounded (𝑠∗∗-

bounded) map. Then 𝑔(𝑓(𝐹)) is s- bounded in 𝑍. But 𝑔(𝑓(𝐹)) = (𝑔 ∘

𝑓)(𝐹), then 𝑔 ∘ 𝑓 is s- bounded (𝑠∗∗- bounded)

Chapter

Three

11

3.1 Bornological Semi-Cauchy Net

In this section, we introduce the definition of bornological semi-

converge nets and bornological semi- Cauchy nets for bornological vector

space and important results of these concepts.


Let (𝑥𝛾)𝛾∈Γ be a net in a convex bornological vector space E. We say

that (𝑥𝛾) bornologically semi-converges to 0 ( (𝑥𝛾)𝑏𝑠→0), if there is an

absolutely convex set and semi-bounded 𝑆 of E and a net (𝜆𝛾) in 𝐾

converging to 0, such that 𝑥𝛾 ∈ 𝜆𝛾𝑆, for every 𝛾 ∈ 𝛤. Then A net (𝑥𝛾)

bornologically semi-converges to a point 𝑥 ∈ 𝐸 and ( 𝑥𝛾𝑏𝑠→𝑥) when

((𝑥𝛾 − 𝑥)𝑏𝑠→0).

Definition (3.1.2)[1]:- Let 𝐸 be a convex bornological vector space.

(𝑥𝛾)𝛾∈𝛤 is called a bornological semi-Cauchy net in 𝐸 if there is an

absolutely convex and semi-bounded set 𝑆 ⊆ 𝐸 and a null net of postive

scalars (𝜇𝛾,𝛾′)𝛾,𝛾′∈Γ×Γ such that (𝑥𝛾 − 𝑥𝛾′) ∈ 𝜇𝛾,𝛾′𝑆 .

Proposition (3.1.3):- Every bornological semi-Cauchy net is semi-

bounded.

Proof:- Suppose (𝑥𝛾)𝛾∈𝛤 is a semi-Cauchy net in a bornological vector

space 𝐸 by definition 3.1, there is an absolutely convex set and semi-

bounded 𝑆 ⊆ 𝐸 and a null net of postive scalars (𝜇𝛾,𝛾′)𝛾,𝛾′∈Γ×Γ such that

(𝑥𝛾 − 𝑥𝛾′) ∈ 𝜇𝛾𝑆 . Since 𝑥𝛾 ∈ 𝑆 for all 𝛾 ∈ 𝛤 . Since 𝑆 is semi-bounded

subset of 𝐸, then (𝑥𝛾) is semi-bounded.

Theorem (3.1.4):- A bornological semi-convergent net in a convex

bornological vector space is a semi-Cauchy net .

Proof:- Let (𝑥𝛾) in 𝐸 bornologically semi-converges to a point 𝑥 ∈ 𝐸 by

definition 3.1, there is an absolutely convex and semi-bounded set 𝑆 ⊆ 𝐸

and a net (𝜇𝛾) of scalars tends to 0 such that (𝑥𝛾 − 𝑥) ∈ 𝜇𝛾𝑆. for every

12

𝛾 ∈ 𝛤 hence if 𝛾′ is a positive integer such that 𝛾 ≥ 𝛾′ . (𝑥𝛾′ − 𝑥) ∈ 𝜇𝛾′𝑆 .

For every integer 𝛾′ with 𝛾 ≥ 𝛾′, then 𝑥𝛾′ − 𝑥 → 0 implies 𝑥𝛾 − 𝑥𝛾′ =

𝑥𝛾 − 𝑥 − (𝑥𝛾′ − 𝑥) ∈ 𝜇𝛾𝛾′𝑆 for all 𝛾, 𝛾′ ∈ 𝛤 with 𝛾 ≥ 𝛾′, then (𝑥𝛾) is a

bornological semi-Cauchy net

Theorem (3.1.5)[1]:- Every semi-Cauchy net in a convex bornological

vector space 𝐸 which has a bornologically semi-convergent subnet , it is

bornologically semi-convergent.

Proof:- Let (𝑥𝛾) be a semi-Cauchy net in a bornological vector space 𝐸

and let (𝑥𝑘𝛾′) be a subnet of (𝑥𝛾) semi-converges to 𝑥 ∈ 𝐸, then there is

an absolutely convex set and semi-bounded 𝑆1 ⊆ 𝐸 and a net (𝜇𝛾′) of

scalars tends to 0 such that 𝑥𝑘𝛾′ − 𝑥 ∈ 𝜇𝑘𝛾′𝑆1 for every integer 𝑘𝛾′ ∈ Γ.

Since (𝑥𝛾) is a semi-Cauchy net . By (definition 3.1.1), there is an

absolutely convex set and semi-bounded 𝑆2 ⊆ 𝐸 and a null net of positive

scalars (𝜇𝛾) such that:-

𝑥𝛾 − 𝑥𝑘𝛾′ ∈ 𝜇𝑘𝛾′𝑆2

for every 𝑘𝛾′,𝛾 ∈ Γ with 𝛾 ≥ 𝑘𝛾′ let 𝑆 = 𝑆1 ∪ 𝑆2 then 𝑥𝛾 − 𝑥 − (𝑥𝑘𝛾′−

𝑥) ∈ 𝜇𝑘𝛾′𝑆 for every 𝑘𝛾′, 𝛾 ∈ Γ with 𝛾 ≥ 𝑘𝛾′ .,

Since 𝑥𝑘𝛾′ − 𝑥 → 0 then 𝑥𝛾 − 𝑥 ∈ 𝜇𝑘𝛾′𝑆 for every 𝑘𝛾′, 𝛾 ∈ Γ with 𝛾 ≥ 𝑘𝛾′

, then 𝑥𝛾 − 𝑥 ∈ 𝜇𝛾𝑆 for every 𝛾 ∈ Γ.

i.e. (𝑥𝛾) bornologically semi-converges to a point 𝑥 ∈ 𝐸 .

3.2 Semi-complete bornological vector space

In this section, we introduce the definition of semi-complete

bornological vector spaces and investigate its properties such as product,

quotient and direct sum.

Definition (3.2.1):- A separated convex bornological vector space is called

a semi-complete bornological vector space if every bornological semi-

Cauchy net in 𝐸 is semi-converges in 𝐸 .

Theorem (3.2.2):- Let 𝐸 be a separated convex bornological vector space

and let 𝐹 be a bornological subspace of 𝐸 , then:-

13

(i) If 𝐹 is semi-complete , then 𝐹 is b-closed in 𝐸 ;

(ii) If 𝐸 is semi-complete and 𝐹 is b-closed, then 𝐹 is semi-

complete .

Proof: (i) Let (𝑥𝛾) be a net in 𝐹 which bornologically semi-converges to

𝑥 ∈ 𝐸 ; (𝑥𝛾) is a semi-Cauchy net in 𝐹. Since 𝐹 is semi-complete then

(𝑥𝛾) bornologically semi-converges to 𝑦 in 𝐹 but, 𝐸 is a separated

bornological vector space, then 𝑥 = 𝑦 implies 𝑥𝛾𝑏𝑠→ 𝑥 in 𝐹, then 𝐹 is b-

closed in 𝐸 .

(ii) Let (𝑥𝛾) be a semi-Cauchy net in ; Since 𝐸 is semi-complete , then

(𝑥𝛾) bornologically semi-converge to 𝑥 in 𝐸. Since 𝐹 is b-closed, then

(𝑥𝛾) bornologically semi-converges to 𝑥 in 𝐹, then 𝐹 is semi-complete .

Theorem (3.2.3):- Let 𝐸 be a semi-complete bornological vector space

and 𝐹 be b-closed subspace of 𝐸, then the quotient 𝐸/𝐹 is semi-complete

.

Proof: Let ß0 be a base for the bornology of 𝐸. If 𝜃: 𝐸 → 𝐸/𝐹 is the

canonical map, then 𝜃(ß0) is a base for the bornology of 𝐸/𝐹, Since 𝐹 is b-

closed subspace of 𝐸, then by theorem in[6] the quotient 𝐸/𝐹 is separated

and 𝜃 is semi bounded linear map (definition 2.2.1). Thus 𝜃(𝑥𝛾) is

bornologically semi-convergent net in 𝐸/𝐹, for every bornologically semi-

convergent (𝑥𝛾) in 𝐸. Then bornologically semi-converge Cauchy net in

𝐸/𝐹. Whence 𝐸/𝐹 is semi-complete .

Theorem (3.2.4):- Every product of any family of semi-complete

bornological spaces is semi-complete .

Proof:- Let 𝐸𝑖 , 𝑖 ∈ Γ ≠ Φ be a family of semi-complete separated

bornological vector space and let 𝐸 = ∏ 𝐸𝑖𝑖∈Γ be the product of 𝐸𝑖. If (𝑥𝛾)

is a semi-Cauchy net in 𝐸. Then the canonical projection 𝑝𝑖: 𝐸 → 𝐸𝑖 of a

semi-Cauchy net is semi-Cauchy (𝑥𝛾𝑖 ) in 𝐸𝑖 for every 𝑖 ∈ Γ . Since every

𝐸𝑖 is semi-complete and it is clear that separated bornological vector space

for every 𝑖 ∈ Γ , then (𝑥𝛾𝑖 ) bornologically semi-converges to a unique point

𝑥𝑖 in 𝐸𝑖 for every 𝑖 ∈ I , and let 𝑥 = (𝑥𝑖) ∈ 𝐸, 𝑖 ∈ Γ, then the net (𝑥𝛾)

14

bornologically semi-converges to 𝑥 ∈ 𝐸 Then 𝐸 is a semi-complete

bornological vector space.

Theorem (3.2.5):- Let (𝐸𝑖 , 𝑢𝑗𝑖)𝑖∈Γ an inductive system of semi-complete

bornological vector space, i.e. 𝐸𝑖 is semi-complete for every 𝑖 ∈ Γ and let

𝐸 =𝑖∈Γ→ 𝐸𝑖. Then 𝐸 is semi-complete if and only if 𝐸 is separated.

Proof:- Since every semi-complete space is separated, only the sufficiency

needs proving. Assume, then, 𝐸 to be separated and let 𝑢𝑖 be the canonical

embedding of 𝐸𝑖 in to 𝐸. Let (𝑥𝛾𝑖 )𝛾 ∈Γ

be a semi-Cauchy net in 𝐸𝑖,

whenever 𝑖 ∈ Γ . Since 𝐸𝑖 is a semi-complete bornological vector space

then (𝑥𝛾𝑖 ) bornologically semi-converges to a point 𝑥 ∈ 𝐸𝑖; then 𝑢𝑖(𝑥𝛾

𝑖 )

bornologically semi-converges semi-Cauchy net in 𝐸 whenever 𝑖 ∈ Γ.

Since 𝐸 is separated then 𝑢𝑖(𝑥𝛾𝑖 ) has a unique limit then 𝐸 is semi-complete

.

Corollary (3.2..6):- Let (𝐸𝑖 , 𝑢𝑗𝑖)𝑖∈Γ an inductive system of semi-complete

bornological space and 𝐸 =𝑖∈Γ→ 𝐸𝑖. Hence if the maps 𝑢𝑗𝑖 are injective, then

𝐸 is semi-complete .

Proof:- the proof is complete by using theorems (3.2.5 and 3.2.6).

Corollary( 3.2.7) :- Every bornological direct sum of any family of semi-

complete bornological spaces is semi-complete .

Proof:- Let (𝐸𝑖) be a family of semi-semi-complete bornological space

and let 𝐸 = ⨁𝑖∈Γ𝐸𝑖

be their bornological direct sum. For every 𝐽 ∈ I, 𝐸𝐽 =⨁𝑖∈Γ𝐸𝑖. The space 𝐸𝐽

is bornologically isomorphic to the product ∏ 𝐸𝑖𝑖∈Γ . Whence is semi-

complete (theorem 3.2.4). If 𝐽 ⊂ 𝐽′, denote by 𝑢𝐽�� the canonical embedding

of 𝐸𝐽 into 𝐸��. Then 𝐸 is the bornological inductive limit of the spaces 𝐸𝐽

and the assertion follows from( Corollary 3.2.6) .

23

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[3] Barreira, L. and Almeida, j. "Hausdorff Dimension in Convex Bornological

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