Vol 7, No 1 (2016)

(1)

Available online throug

ISSN 2229 – 5046

International Journal of Mathematical Archive- 7(1), Jan. – 2016 116

FINITE DIMENSIONAL FUZZY ANTI 2- NORMED LINEAR SPACE

PARIJAT SINHA

1

**_{, DIVYA MISHRA*}**

2

_{AND GHANSHYAM LAL}

2

1

Department of Mathematics, V. S. S. D. College, Kanpur, India.

2

_{Department of Mathematics, M. G. C. G. University, Satna, India.}

(Received On: 17-01-16; Revised & Accepted On: 31-01-16)

ABSTRACT

In this paper we have generalized fuzzy anti 2-norm by introducing t-conorm in the earlier definition. The Riesz lemma

and a few properties of finite dimensional fuzzy anti 2-normed linear space has been established with respect to t-conorm◊.

Keywords: Fuzzy anti 2- norm,

α

-2-norm, Riesz lemma.

INTRODUCTION

The concept of fuzzy set was introduced by Zadeh [11] in 1965 and thereafter several authors applied it different branches of pure and applied mathematics. The concept of fuzzy norm was introduced by Katsaras [9] in 1984. In 1992 Felbin [8] introduced the concept of fuzzy normed linear space. A satisfactory theory of 2-norm on a linear space has been introduced and developed by Gähler [6]. Jebril and Samanta [7] gave the definition of fuzzy anti-normed linear space. In 2011, B. Surender Reddy [1] introduced the idea of fuzzy anti 2-normed linear space.

In the present paper we have modified the definition of fuzzy anti 2- normed linear space. The Riesz lemma and important properties of finite dimensional fuzzy anti 2- normed linear space has been established with respect to t-conorm◊.

PRELIMINARIES

Definition 2.1[10]: A binary operation ◊:[0,1]×[0,1]→[0,1] is a t- conorm if ◊satisfies the following condition: (i) ◊ is commutative and associative,

(ii) a◊ 0=a, ∀a∈[0,1],

(iii) a◊b≤c◊d, whenever a≤c,b≤d and a,b,c,d∈[0,1].

Example: (i) a◊b = a+b-ab (ii) a◊b = max {a, b} (iii) a◊b = {a+b, 1}

Definition 2.2[1]: Let X be a linear space over a real fieldF. A fuzzy subset N∗ of X×X×

R

is called a fuzzy anti 2-norm on X if and only if it satisfies,

(Fa2-N1) for all t∈R with t≤0, N∗

(

x₁,x₂,t

)

=1

(Fa2-N2) for all t∈R with t>0, N∗

(

x₁,x₂,t

)

=0 if and only if x₁ and x₂ are linearly dependent.

(Fa2-N3) N∗

(

x₁,x₂,t

)

is invariant under any permutation.

(Fa2-N4) for all t∈R with t>0,

(

)

c c F c

t x x N t cx x

N _ ≠ ∈

  

   = ∗

∗

, 0 if , , ,

, ₂ ₁ ₂

1

(Fa2-N5) for s,t∈R_witht >0 all N∗

(

x₁,x₂+x₂′,s+t

)

≤max

{

N∗

(

x₁,x₂,s

) (

,N∗ x₁,x₂′,t

)

}

(Fa2-N6) N∗

(

x₁,x₂,t

)

is non-increasing function of t∈R and lim

(

₁, ₂,

)

0.

t =

∗ ∞

→ N x x t

Corresponding Author: Divya Mishra*

2

(2)

Then

(

X,N∗

)

is called a fuzzy anti 2-normed linear space. The following condition of fuzzy anti 2-norm N∗will be required later on,

(Fa2-N7) for t∈Rwith t>0,N∗

(

x₁,x₂,t

)

<1, ∀t>0⇒

x

1and

x

2 are linearly dependent.

Definition 2.3[1]: Let

(

X,N∗

)

be a fuzzy anti 2- normed linear space. A sequence

{ }

x_n in X is said to be convergent to x∈X _ift>0, 0<r<1, ∃ an integer n₀∈Nsuch that N∗

(

x_n−x,x₀,t

)

<r, ∀n≥n₀.

(

X,N∗

)

be a fuzzy anti 2- normed linear space. A sequence

{ }

x_n in X is said to be cauchy sequence if t>0, 0<r<1, ∃an integer n₀∈Nsuch thatN∗

(

x_n₊_p−x_n,x₀,t

)

<r, for all n≥n₀. p=1,2,3...

Definition 2.5[3]: A subset A of a fuzzy anti 2-normed linear space

(

X,N∗

)

is said to be bounded iff ∃ t > 0,

( )

st N

(

x y t

)

r x y A r∈ 0,1 ., ∗ , , < , ∀ , ∈ .

( )

X,N∗ bea fuzzy anti 2- normed linear space. A subset B of X is said to be closed if any sequence

{ }

x_n in B converges to x∈B that islim ∗

(

− , ,

)

=0, ∀ >0

∞

→ N xn x y t t

n ⇒x,y∈B.

Definition 2.7[1]: A subset

A

of a fuzzy anti 2-normed linear space

(

X,N∗

)

is said to be compact if any sequence

{ }

x_n

in

A

has a subsequence converging to an element of

A

.

3. FUZZY ANTI 2- NORMED LINEAR SPACE

In this section we have modified the definition of fuzzy anti 2- norm with respect to a t-conorm ◊and deduced some important results.

Definition 3.1: Let X be a linear space over a real fieldF. A fuzzy subset N∗ of X×X×R is called a fuzzy anti 2-norm on X if and only if it satisfies,

(Fa2-N1) for all t∈R with t≤0,N∗(x₁,x₂,t)=1

(Fa2-N2) for all t∈R with

t

>

0 ,

N

∗

(

x

₁

,

x

₂

,

t

)

=

0

if and only if x₁and x₂ are linearly dependent.

(Fa2-N3) N∗(x₁,x₂,t) is invariant under any permutation of x₁ and x₂.

(Fa2-N4) for all t∈R with t>0

, , , )

, ,

( ₁ ₂ ₁ ₂ _

  

  

= ∗

∗

c t x x N t cx x

N if c≠0,c∈F

(Fa2-N5) for s,t∈R_witht>0_all

N

∗

(

x

₁

,

x

₂

+

x

′

₂

,

s

+

t

)

≤

{

N

∗

(

x

₁

,

x

₂

,

s

) (

◊

N

∗

x

₁

,

x

₂

′

,

t

)

}

(Fa2-N6) N∗

(

x₁,x₂,t

)

is non-increasing function of t∈R and lim

(

₁, ₂,

)

0

t =

∗ ∞

→ N x x t

We further assume that for a fuzzy anti 2- normed linear space

(

X,N∗

)

,

(Fa2-N7) for all t∈R_witht>0,N∗

(

x₁,x₂,t

)

<1, ∀t>0⇒

x

₁and

x

₂ are linearly dependent.

(Fa2-N8) N∗

(

x₁,x₂,.

)

is a continuous function on R and strictly decreasing on the subset

{

t:0<N∗

(

x₁,x₂,t

)

<1

}

of R.

(Fa2-N9) a◊a=a , ∀a∈[0,1] .

Remark 3.1: Let N∗ be a fuzzy anti 2- norm onX then N∗

(

x₁,x₂,t

)

is non-increasing with respect to t for each

. , 2 1 x X

x ∈

Proof: Let t<s. Then k=s−t>0_{, we have}

(

₁, ₂,

)

= ∗

(

₁, ₂,

)

◊0

∗

t x x N t x x

N (by property of t-conorm)

N

(

x1,x2,t

)

N

(

0,0,k

)

N

(

x1,x2,t k

)

N

(

x1,x2,s

)

.

∗ ∗

∗

∗ _◊ _≥ ₊ ₌

=

(3)

Example 3.1: Let

(

X,.,.

)

be 2-normed linear space and define a◊b=a+b−ab. Define N∗:X×X×R→[0,1] by

(

)

   

≤ > =

∗

2 1

2 1 2

1

, t if , 1

, t if , 0 , ,

x x

x x t

x x N

Then N∗_{is a fuzzy anti 2- norm on}X with respect to the t- conorm ◊ and

(

X,N∗

)

is a fuzzy anti 2- normed linear space with respect to the t- conorm ◊.

Solution:

(i) ∀x1,x2∈X×X and∀t∈R,t≤0 we have ( 1, 2, )=1.

∗ _x _x _t

N

(ii) ∀t∈R,t≤0 if x₁,x₂ are linearly dependent then x₁,x₂ =0 so N∗(x₁,x₂,t)=0. Again if N∗(x₁,x₂,t)=0 with

⇒ = ⇒

∈ > ∀ < ⇒

>0 x₁,x₂ t, t( 0) R x₁,x₂ 0

t x₁,x₂ are linearly dependent.

(iii) It is obvious thatN∗(x₁,x₂,t) is invariant under any permutation.

(iv) If ( ₁, ₂, ) 0 ₁, ₂ ₁, ₂ x₁,x₂ c

t x x c t cx x t t

cx x

N∗ = ⇔ > ⇔ > ⇔ >

1, 2, =0

  

  

⇔ ∗

c t x x N

. 1 , , ,

, ,

1 ) , ,

( 1 2 1 2 1 2 1 2 1 2 =

  

   ⇔ ≤

⇔ ≤

⇔

= ∗

∗

c t x x N x x c t x x c t cx x t t

cx x N

(v) N∗(x1,x2,s)◊N∗(x1,x2′,t)=N∗(x1,x2,s)+N∗(x1,x2′,t)−N∗(x1,x2,s).N∗(x1,x′2,t). If s> x₁,x₂ and t> x₁,x′₂ so s+t> x₁,x₂ + x₁,x′₂ then N∗(x₁,x₂+x′₂,s+t)=0 and N∗(x₁,x₂,s)◊N∗(x₁,x′₂,t)=0+0−0=0.

So N∗(x₁,x₂+x₂′,s+t)=N∗(x₁,x₂,s)◊N∗(x₁,x′₂,t).

If s> x₁,x₂ and t≤ x₁,x₂′ then N∗(x₁,x₂,s)◊N∗(x₁,x′₂,t)=0+1−0=1.

If s≤ x₁,x₂ and then N∗(x₁,x₂,s)◊N∗(x₁,x′₂,t)=1+0−0=1.

If s≤ x₁,x₂ and t≤ x₁,x₂′ then N∗(x₁,x₂,s)◊N∗(x₁,x′₂,t)=1+1−1=1. Then in all the above three cases,

). , , ( 1 ) , , ( ) , ,

(x₁ x₂ s N x₁ x₂ t N x₁ x₂ x₂ s t

N∗ ◊ ∗ ′ = ≥ ∗ + ′ +

Thus N∗(x1,x2+x′2,s+t)≤N∗(x1,x2,s)◊N∗(x1,x′2,t).

(vi) From the definition if t> x₁,x₂ , then lim

(

1, 2,

)

0

t =

∗ ∞

→ N x x t . Thus

(

)

∗

N

X, is a fuzzy anti 2- normed linear space with respect to the t- conorm ◊.

(

X,.,.

)

be 2- normed linear space and define a◊b= min {a+b, 1}. Define N∗:X×X×R→[0,1] by

(

)

      

≤

> ≤

+ >

=

∗

0 t if , 1

0 , , t if , , ,

, t if , 0

,

, ₁ ₂

2 1

2

1 x x t

x x t

x x

t x x N

Then N∗is a fuzzy anti 2- norm on X with respect to the t- conorm ◊ and

(

X,N∗

)

Solution:

(4)

(ii) If t>0 and t≤ x₁,x₂ the 2 1 2 1 2 1 , , ) , , ( x x t x x t x x N + =

∗ _if

2 1,x

x are linearly dependent so x₁,x₂ =0therefore

0 ) , , ( 1 2 =

∗ _x _x _t

N .

Conversely, N∗(x₁,x₂,t)=0 then t> x₁,x₂ ,∀t ⇒ x₁,x₂ =0 , so x₁,x₂are linearly dependent.

(iii) It is obvious that N∗(x₁,x₂,t) is invariant under any permutation of x₁ andx₂.

(iv) If ( ₁, ₂, ) 0 ₁, ₂ ₁, ₂ x₁,x₂ c t x x c t cx x t t cx x

N∗ = ⇔ > ⇔ > ⇔ >

0 , , 2 1 =

      ⇔ ∗ c t x x N

If ₁ ₂ ₁ ₂

2 1

2 1 2

1 , ,

, , ) , ,

( x x

c t cx x t cx x t cx x t cx x

N ⇔ ≤ ⇔ ≤

+ = ∗ 2 1 2 1 2 1 2 1 2 1 , , , , , , cx x t cx x x x c t x x c t x x N + = + =         ⇔ ∗ (v)

N (x1,x2,s)◊N (x1,x′2,t)=min

{

N (x1,x2,s)+N (x1,x2′,t), 1

}

.

∗ ∗ ∗ ∗ If s x

x₁, ₂ ≥ and x₁,x′₂ ≥t then

2 1 2 1 2 1 2 1 2 1 2 1 , , , , ) , , ( ) , , ( x x t x x x x s x x t x x N s x x N ′ + ′ + + = ′ + ∗ ∗

(

)

(

)

1

, , , , , , , , , , 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 _≥ + ′ + ′ + ′ + ′ + ′ + = st x x s x x x x x x t x x x x x x s x x x x x x t

Since x₁,x₂ x₁,x₂′ >st.

In this case N∗(x₁,x₂,s)◊N∗(x₁,x₂′,t)=1≥N∗(x₁,x₂+x′₂,s+t).

If

s x

x₁, ₂ ≥ and x₁,x′₂ <t then either x₁,x₂+x₂′ ≥s+tor x₁,x₂+x₂′ <s+t.

Now,

, 0 1.

, ) , , ( ) , , ( 2 1 2 1 2 1 2

1 + ′ = ₊ + <

∗ ∗ x x s x x t x x N s x x N

Hence .

, , ) , , ( ) , , ( 2 1 2 1 2 1 2 1 x x s x x t x x N s x x N + = ′ ◊ ∗ ∗

If x₁,x₂+x′₂ ≥s+t then consider

) , , ( ) , , ( ) , ,

(x1 x2 x2 s t N x1 x2 s N x1 x2 t

N∗ + ′ + − ∗ ◊ ∗ ′

2 1 2 1 2 2 1 2 2 1 , , , , x x s x x x x x t s x x x + − ′ + + + ′ + = 1 2 2 1 2 1 2 1 2 1 2 1 , , , , , , x x s x x x x x x t s x x x x + − ′ + + + ′ + ≤

(

1 2 1 2

)(

1 2

)

2 1 2 1 , , , , , x x s x x x x t s x x t x x s + ′ + + + − ′ =

(

1 2 1 2

)(

1 2

)

2 1 , , , , x x s x x x x t s x x t st + ′ + + + −

< , Since x₁,x₂′ <t,

≤0, Since s≤ x1,x2 so st<t x1,x2 . So, N∗(x₁,x₂+x₂′,s+t)<N∗(x₁,x₂,s)◊N∗(x₁,x′₂,t).

If x₁,x₂+x′₂ <s+t then

) , , ( ) , , ( , , 0 ) , ,

( 1 2 1 2

2 1 2

2

1 N x x s N x x t

x x s x x t s x x x

N = ◊ ′

(5)

If x1,x2 <s and x1,x′2 ≥t then in the similar way we can show that )

, , ( ) , , ( ) , ,

(x₁ x₂ x₂ s t N x₁ x₂ s N x₁ x₂ t

N∗ + ′ + ≤ ∗ ◊ ∗ ′ .

If x₁,x₂ <s and x₁,x₂′ ≥t then N∗(x₁,x₂,s)+N∗(x₁,x′₂,t)=0+0<1. Therefore, N∗(x1,x2,s)◊N∗(x1,x2′,t)=0.

Also x₁,x₂+x₂′ ≤ x₁,x₂ + x₁,x₂′ <s+t and N∗(x₁,x₂+x′₂,s+t)=0.

So N∗(x₁,x₂+x₂′,s+t)=N∗(x₁,x₂,s)◊N∗(x₁,x′₂,t).

So N∗(x₁,x₂+x′₂,s+t)≤N∗(x₁,x₂,s)◊N∗(x₁,x′₂,t)

(vi) If t> x1,x2 then from the definition lim

(

1, 2,

)

=0

∗ ∞

→ N x x t

t . If x1,x2 are not independent and t≤ x1,x2 then

(

)

0

, , lim , , lim

2 1

2 1 2

1 = _→_∞ ₊ =

∗ ∞

→ t x x

x x t

x x N

t t

.

If x1,x2 are linearly dependent and t≤ x1,x2 then lim

(

1, 2,

)

=0.

∗ ∞

→ N x x t

t

Hence lim

(

1, 2,

)

=0.

∗ ∞

→ N x x t

t x x X X

× ∈ ∀₁, ₁ .

ThusN∗ is a fuzzy anti 2-norm on Xwith respect to the t-conorm ◊ and

(

X,N∗

)

is a fuzzy anti 2-normed linear space with respect to the t-conorm◊.

(

X,.,.

)

be 2-normed linear space and define a◊b= min{a+b, 1}. Define N∗:X×X×R→[0,1]

by

(

)

    

≤ > −

=

∗

2 1

2 1 2

1 2 1 2

1

, t if , 1

, t if , , 2

, ,

,

x x

x x x

x t

x x

t x x N

Then N∗ satisfies all the condition of fuzzy anti 2- norm with respect to t-conorm ◊.So N∗is a fuzzy anti 2- norm on

X with respect to the t- conorm ◊ and

(

X,N∗

)

Theorem 3.1: Let

(

X,N∗

)

be a fuzzy anti 2- normed linear space with respect to a t- conorm ◊satisfying (Fa2-N7) and (Fa2-N9). Then for anyα∈

( )

0,1the function x1,x2 _α∗ :X×X×R→

[

0,∞

)

defined as

(

)

{

0: , , 1

}

,

( )

0,1

, ₂ ₁ ₂

1 =∧ > ≤ − ∈

∗ ∗

α α

α t N x x t

x

x .

is a 2-norm on X. Then

{

.,._α∗ :α∈(0,1)

}

is an ascending family of 2-norm on a linear spaceX.

Proof:

(i) For x₁,x₂ for t≤0 , so N∗

(

x1,x2,t

)

≤1−α is not possible. So ∧

{

t>0:N∗

(

x₁,x₂,t

)

≤1−α

}

≥0, α∈

( )

0,1 ⇒ x₁,x₂ ∗_α ≥0,α∈

( )

0,1 .

(ii) It is obvious ∧

{

t>0:N∗

(

x₁,x₂,t

)

≤1−α

}

=0⇒∀t>0,N∗

(

x₁,x₂,t

)

<1 So by (Fa2-N7)

x

₁and

x

₂ are linearly dependent.

Conversely,

x

₁and

x

₂ are linearly dependent

(

)

{

0: , , 1

}

0,

( )

0,1 , 0.

> ₁ ₂ ≤ − = ∀ ∈ ⇒ ₁ ₂ =

∧

⇒ ∗ ∗

α

α x x

(6)

(iii) If c=0_{it is obvious. If}c≠0 then

(

)

{

α

}

α =∧ > ∗ ≤ −

∗

1 , , : 0

, ₂ ₁ ₂

1 cx s N x cx s

x



   

  

− ≤     

   > ∧

= ₀_: ∗ _, _, ₁ _α

2 1

c s x x N s

=∧

{

>

(

)

≤ −α

}

∗

1 , , : 0

ct N x₁ x₂ t

=∧

{

>

(

)

≤ −α

}

∗

1 , , : 0

t N x₁ x₂ t c

1, 2 .

∗

= c x x _α

(iv) x₁,x₂ ∗_α+ x₁,x′₂ ∗_α =∧

{

t>0:N∗

(

x₁,x₂,t

)

≤1−α

}

+∧

{

s>0:N∗

(

x₁,x₂′,s

)

≤1−α

}

,∀α∈

( )

0,1 ≥∧

{

t+s>0:N∗

(

x₁,x₂,t

)

≤1−

α

,N∗

(

x₁,x₂′,s

)

≤1−

α

}

So ∧

{

t+s>0:N∗

(

x₁,x₂,t

) (

◊N∗ x₁,x₂′,s

) (

≤ 1−

α

) (

◊1−

α

)

}

≥∧

{

t+s>0:N∗

(

x₁,x₂+x′₂,t+s

)

≤1−α

}

by (Fa2-N5) and (Fa2-N9)

= x₁,x₂+x′₂ ∗_α

Hence

{

.,._α∗ :α∈(0,1)

}

is a 2-norm on X.

If α ≤₁ α₂, we have

{

t>0:N∗

(

x₁,x₂,t

)

≤1−α₂

}

⊂

{

t>0:N∗

(

x₁,x₂,t

)

≤1−α₁

}

(

)

{

0: 1, 2, 1 2

}

{

0:

(

1, 2,

)

1 1

}

> ≤ −α ≥∧ > ′ ≤ −α

∧

⇒ _t _N∗ _x _x _t _t _N∗ _x _x _t

. , ,

1

2 1 2

2 1

∗ ∗

≥

⇒ x x _α x x _α

So

{

.,._α∗ :α∈(0,1)

}

is an ascending family of 2- norm on a linear spaceX. Hence proved.

(

X,N∗

)

be a fuzzy anti 2-normed linear space satisfying (Fa2-N7) and (Fa2-N8) Also, if

( )

{

.,._α∗ :α∈0,1

}

be ascending family of norms of X, defined by x₀,x₀′ _α∗ =∧

{

t:N∗

(

x₀,x₀′,t

)

≤1−α

}

,α∈

( )

0,1. Then forx₀,x₀′(linearly independent) ∈X, α∈

( )

0,1 ands

( )

>0 ∈R,

(

)

α

α = ⇔ ′ = −

′ ∗ ∗

1 , ,

, ₀ ₀ ₀

0 x s N x x s

x .

Proof: let x₀,x₀′ ∗_α =s thens>0. Then ∃a sequence

{ }

s_n _n,s_n >0 such that s_n→sas n→∞ and

(

x x s

)

n N

N∗ ₀, ₀′, _n ≤1−α,∀ ∈ . Therefore

(

)

α ≤ −α

  



 _′

⇒ − ≤ ′

∞ → ∗

∗ ∞

→ , , 1 , ,lim 1

lim ₀ ₀ ₀ ₀ _n

n n

n

s x x N s

x x N

(

0, ′0, 0, 0′

)

≤1− ,∀ ∈

( )

0,1

⇒ ∗ ∗ _α _α

α x x x x N

Letα∈

( )

0,1,x₀,x′₀(linearly dependent)∈X and s= x₀,x₀′ ∗_α =∧

{

t:N∗

(

x₀,x′₀,t

)

≤1−

α

}

.

Therefore N∗

(

x₀,x′₀,s

)

≤1−α (1)

If possible let N∗

(

x0,x0′,s

)

<1−α then by continuity of N∗

(

x0,x0′,.

)

ats,there exist s′<ssuch that

(

′

)

< −α

∗ _, _, ₁

0 0 x s

x

N , which is impossible since

(

)

{

′ ≤ −α

}

∧

= t:N∗ x₀,x₀,s 1

s .

Thus N∗

(

x0,x′0,s

)

≥1−α (2)

From (1) and (2) it follows that N∗

(

x₀,x₀′,s

)

=1−α. Thus

(

)

α

α= ⇒ ′ = −

′ ∗ ∗

1 , ,

, ₀ ₀ ₀

0 x s N x x s

(7)

Next if N∗

(

x₀,x₀′,s

)

=1−α,α∈

( )

0,1 then

(

)

{

t N x x s

}

s x

x0, ′0 ∗_α =∧ : ∗ 0, ′0, ≤1−α = . (4)

Hence from (3) and (4) we have for α∈

( )

0,1, x₀,x₀′(linearly independent)∈X and for

(

, ,

)

1 . ,

,

0 ₀ ₀′ = ⇔ ₀ ₀′ = −α

> _x _x _s _N∗ _x _x _s s

Hence proved.

(

X,N∗

)

_{be a fuzzy anti 2-normed linear space with respect to a t- conorm}◊satisfying (Fa2-N7), (Fa2-N8) and (Fa2-N9).

Let x₁,x₂ _α∗ =∧

{

t:N∗

(

x₁,x₂,t

)

≤1−α

}

,α∈

( )

0,1. Also, let N₁∗:X×X×R→

[ ]

0,1 be defined by

(

)

{

}

  

_∧ ₋ _≤ _≠

=

∗ ∗

otherwise

, 1

0 dependent, linearly

are if , ,

: 1 ,

, ₂ 1 2

1

t t

x x t

x x

N α α

Then N₁∗=N∗.

Proof: Let

(

x0,x′0,t0

)

∈X×X×R and α∈

( )

0,1 . To prove this we consider the following cases:

Case (i): For any

(

x₀,x′₀

)

∈X×X and t≤0,N∗

(

x₀,x′₀,t₀

)

=N₁∗

(

x₀,x′₀,t₀

)

=1.

Case (ii): Let x0,x0′

(

linearly dependent

)

,t0>0.

(

, ,

)

0

Then N∗ x₀ x₀′ t₀ = also 1, 2 =0

∗

α

x

x so N₁∗

(

x₀,x₀′,t₀

)

=0

Case (iii): Let x₀,x′₀

(

inearly independent

)

,t₀>0 such that N∗

(

x₀,x₀′,t₀

)

=1. By theorem (3.2).

we haveN∗

(

x0,x0′, x0,x0′ _α∗

)

=1−α . Since

(

′

)

= > −

α

∗

1 1 , , ₀ ₀

0 x t

x

N it follows that

(

x0,x0, x0,x0

)

1 N

(

x0,x0,t0

)

N∗ ′ ′ _α∗ = −

α

< ∗ ′ and since N∗

(

x₀,x₀′,.

)

is strictly non-increasing.

So t₀< x₁,x₂ _α∗,∀

α

∈

( )

0,1. So N₁∗

(

x₀,x₀′,t₀

)

=∧

{

1−α: x₁,x₂ ∗_α ≤t₀

}

=1.

Thus N∗

(

x₀,x′₀,t₀

)

=N₁∗

(

x₀,x₀′,t₀

)

=1.

Case (iv): Let x₀,x′₀

(

linearly independent

)

,t₀>0such that N∗

(

x₀,x₀′,t₀

)

=0.

As x₀,x₀′ _α∗ =∧

{

t:N∗

(

x₀,x′₀,t

)

≤1−

α

}

,

α

∈

( )

0,1.

As N∗

(

x₀,x₀′, x₀,x′₀ _α∗

)

=1−α as N∗ is decreasing

It follows that, x₀,x₀′ _α∗ <t₀,∀

α

∈

( )

0,1 , by (Fa2-N6). Therefore,

(

, ,

)

{

1 : ,

}

0

, ₂ ₀ ₁ ₀ ₀ ₀ ₀ ₀ ₀

1 < ⇒ ′ =∧ − ′ ≤ =

∗ ∗

∗

t x x t

x x N t x

x _α

α

_α ,

Thus N∗

(

x₀,x₀′,t₀

)

=N₁∗

(

x₀,x′₀,t₀

)

=0.

Case (v): Let x₀,x₀′

(

linearly independent

)

,t₀>0, s.t., 0<N∗

(

x₀,x₀′,t₀

)

<1.

Let, N∗

(

x₀,x′₀,t₀

)

=1−

β

, as x₀,x₀′ ∗_β =∧

{

t:N∗(x₀,x₀′,t)≤1−

β

}

(8)

So N₁∗

(

x₀,x′₀,t₀

)

≤1−

β

. Therefore,

N₁∗

(

x₀,x₀′,t₀

)

≤N∗

(

x₀,x′₀,t₀

)

(1)

As N∗

(

x₀,x′₀,t₀

)

=1−

β

⇔ x₁,x₂∗_β =t₀.

If β<α<1 and let x₀,x₀′ ∗_β =t₁then

N

∗

(

x

₀

,

x

′

₀

,

t

₁

)

=

1 −

α

<

1 −

β

=

N

∗

(

x

₀

,

x

′

₀

,

t

₀

)

As N∗

(

x₀,x₀′,.

)

is monotonically decreasing so t₀<t₁ since x₀,x′₀ ∗_α =t₁>t₀.

So

N

₁∗

(

x

₀

,

x

′

₀

,

t

₀

)

>

1 −

β

=

N

∗

(

x

₀

,

x

₀

′

,

t

₀

)

(2)

So from (1) and (2) we have N₁∗

(

x₀,x₀′,t₀

)

=N∗

(

x₀,x₀′,t₀

)

. Hence proved.

Lemma 3.4: Let

(

X,N∗

)

_{be a fuzzy anti 2-normed linear space with respect to t- conorm} ◊ satisfying (Fa2-N7), (Fa2-N8) and (Fa2-N9), every sequence is convergent if and only if it is convergent with respect to its corresponding

α-2-norms, α∈(0,1).

Proof: Let

(

X,N∗

)

_{be a fuzzy anti 2-normed linear space satisfying (Fa2-N7), (Fa2-N8), (Fa2-N9) and}

{ }

xn be a

sequence inX such thatxn →x asn→∞.

(

, ,

)

0, 0. lim ∗ − 0 = ∀ >

∞

→ N xn x x t t

n

Let0<α<1. So, limN

(

xn x,x0,t

)

0 1 n0(t)

n − = < − ⇒∃

∗

→∞ α such that

(

x x,x₀,t

)

1

α

n n₀(t,

α

).

N∗ _n− < − ∀ ≥

Now, x_n−x,x₀ _α∗ =∧

{

t>0:N∗

(

x_n−x,x₀,t

)

≤1−α

}

,

,x₀ t x

x_n− ≤

⇒ ∗_α ∀n≥n0(t,α)

Sincet>0 is arbitrary, x_n−x,x₀ ∗_α→0 asn→∞_,∀α∈(0,1).

Conversely, suppose that x_n−x,x₀ ∗_α→0 asn→∞_,∀α∈(0,1).

Then for α∈(0,1),ε>0, ∃ n0(α,ε)such that − _α <ε

∗

0

,x x

x_n ,∀n≥n0(α,ε),α∈(0,1).

Now, N∗

(

x_n−x,x₀,ε

)

=∧

{

1−α: x_n−x,x₀ ∗_α≤ε

}

(

− , ₀,

ε

)

≤1−

α

,

⇒ ∗

x x x

N _n ∀n≥n0(α,ε),α∈(0,1).

(

, ,

)

0, 0. lim − ₀ = ∀ >

⇒ ∗

∞

→ N xn x x t

n

ε

Thusx_n converges tox. Hence Proved.

Corollary 3.5: Let

(

X,N∗

)

_{be a fuzzy anti 2-normed linear space with respect to a t- conorm}◊ satisfying (Fa2-N7), (Fa2-N8) and (Fa2-N9). W⊆X is closed in

(

X,N∗

)

if and only if it is closed with respect to its corresponding α -2-norms , α∈(0,1).

Theorem 3.6 (Riesz lemma): Let W be a closed and proper subspace of a fuzzy anti 2- normed linear space

(

X

,

N

∗

)

with respect to a t- conorm ◊satisfying (Fa2-N7) (Fa2-N8) and (Fa2-N9). Then for each

ε

>0 there exist

(

)

2

,y X W

(9)

Proof: As x₁,x₂ _α∗ =∧

{

t:N∗

(

x₁,x₂,t

)

≤1−α

}

,α∈

( )

0,1 and

{

.,._α∗ :α∈(0,1)

}

is an ascending family of fuzzy α-2-norm on a linear spaceX.Now by Riesz lemma for 2- normed linear space, it follows that for any

ε

>0 there exist

(

)

2 2

1,y X W

y ∈ − such that y₁,y₂ _α∗ =1 and y₁−w,y₂−w_α∗ >1−ε ,∀w∈W

Now, from theorem (3.3), for all u(y,1)≤1−α we have

N

(

y y t

)

=∧

{

− y y ≤t

}

∗ ∗

α

α 1 2

2

1, , 1 : ,

{

1 : , 1

}

) 1 , ,

( ₁ ₂ =∧ − ₁ ₂ ≤

⇒ ∗ ∗

α

α y y y

y N

α

− ≤ ⇒N∗(y₁,y₂,1) 1

Again,

N y −w y −wt =∧

{

− y −w y −w ≤t

}

∗ ∗

α

α 1 2

2

1 , , ) 1 : ,

(

{

α ε

}

ε =∧ − − − _α ≤

− −

⇒ ∗ ∗

w y w y w

y w y

N ( ₁ , ₂ , ) 1 : ₁ , ₂

. 1 ) , ,

( ₁− ₂−

ε

≤ −

α

⇒ ∗

w y w y N

Hence proved.

(

X,N∗

)

_{be a fuzzy anti 2-normed linear space with respect to a t- conorm}◊ satisfying (Fa2-N7), (Fa2-N8) and (Fa2-N9). If the set

{

x₁,x₂:N∗

(

x₁,x₂,1

)

≤1−α

}

,α∈(0,1) is compact thenX is a space of finite dimension.

Proof: It can be easily verified that

{

1, 2:

(

1, 2,1

)

≤1−

}

=

{

1, 2: 1, 2 ≤1

}

, ∈(0,1)

∗

∗ _α _α

α x x x x x

x N x

x . By applying Riesz

lemma 3.6, it can be proved that if for some

α

∈(0,1)the set

{

x₁,x₂: x₁,x₂ ∗_α ≤1

}

is compact then X is of finite dimensional. By lemma (3.4), it follows that , for some α∈(0,1),

{

x₁,x₂:N∗

(

x₁,x₂,1

)

≤1−

α

}

is compact then X is a space of finite dimensional.

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Source of support: Nil, Conflict of interest: None Declared