Statistically Convergent Double Sequence Spaces in 2 Normed Spaces Defined by Orlicz Function

doi:10.4236/am.2011.24048 Published Online April 2011 (http://www.SciRP.org/journal/am)

Statistically Convergent Double Sequence Spaces in

2-Normed Spaces Defined by Orlicz Function

Vakeel A. Khan, Sabiha Tabassum

Department of Mathematics, Aligarh Muslim University, Aligarh, India E-mail: [email protected], [email protected]

Received November 26, 2010; revised January 15, 2011; accepted January 18, 2011

Abstract

The concept of statistical convergence was introduced by Stinhauss [1] in 1951. In this paper, we study con- vergence of double sequence spaces in 2-normed spaces and obtained a criteria for double sequences in 2-

normed spaces to be statistically Cauchy sequence in 2-normed spaces.1

Keywords:Double Sequence Spaces, Natural Density, Statistical Convergence, 2-Norm, Orlicz Function

1. Introduction

In order to extend the notion of convergence of sequen- ces, statistical convergence was introduced by Fast [2] and Schoenberg [3] independently. Later on it was fur-ther investigated by Fridy and Orhan [4]. The idea de-pends on the notion of density of subset of .

The concept of 2-normed spaces was initially intro-duced by G ahler [5-7] in the mid of 1960’s. Since then, many researchers have studied this concept and ob- tained various results, see for instance [8].

Let X be a real vector space of dimension d , where 2  d . A 2-norm on X is a function

.,. :XX R which satisfies the following four conditions:

1) x x1, 2 = 0 if and only if x x1, 2 are linearly

de-pendent;

2) x x1, 2 = x x2, 1 :

3) x x1, 2 = x x1, 2 , for any R:

4) xx x, 2  x x, 2  x x, 2

The pair



X, .,.



is then called a 2-normed space (see [9]).

Example 1.1. A standard example of a 2-normed space is R2 equipped with the following 2-norm

, :=

x y the area of the triangle having vertices 0, , .x y

Example 1.2. Let Y be a space of all bounded real-valued functions on R. For f g, in Y, define

, = 0

f g , if f, g are linearly dependent,

   

, =sup ,

t R

f g f t g t 

 if f, g are linearly independent. Then .,. is a 2-norm on Y.

We recall some facts connecting with statistical con-vergence. If K is subset of positive integers , then

n

K denotes the set {kK k: n}. The natural density of K is given by

 

=lim| n|,

n

K K

n 

 where Kn deno- tes the number of elements in Kn, provided this limit exists. Finite subsets have natural density zero and

 

Kc = 1

 

K

   where Kc =\K, that is the comp- lement of K. If K1K2 and K1 and K2 have

natu-ral densities then 

 

K1  

 

K2 . Moreover, if

 

K1 =

 

K2 = 1,

  then 



K1K2



= 1 (see [10]).

A real number sequence x=

 

xj is statistically con- vergent to L provided that for every > 0 the set



n:|xj L| 



has natural density zero. The sequ- ence x=

 

xj is statustically Cauchy sequence if for each > 0 there is positive integer N =N

 

 such that 





n:|xjxN

 







= 0 (see [11]).

If x=

 

xj is a sequence that satisfies some property P for all n except a set of natural density zero, then we say that

 

xj satisfies some property P for “almost all n”.

An Orlicz Function is a function M: 0,



 





0,



which is continuous, nondecreasing and convex with

 

0 = 0

M , M x

 

> 0 for x> 0 and M x

 

 , as x .

If convexity of M is replaced by M x



y



M x

 

 

M y

bounded. For example, M x

 

=xp



0 <p1



is un- bounded and

 

=

1 x M x

x is bounded.

Lindesstrauss and Tzafriri [13] used the idea of Orlicz sequence space;

=1

:= : k < , for some > 0

M

k

x

l x w M 

       _{ }          





which is Banach space with the norm

=1

| | = inf > 0 : k 1 .

M

k

x

x  M

             





The space lM is closely related to the space lp, which is an Orlicz sequence space with M x

 

=xp for

1 p<.

An Orlicz function M satisfies the  2 condition 2

(M  for short) if there exist constant K2 and

0> 0

u such that

 

2

 

M u KM u

whenever |u|u0.

Note that an Orlicz function satisfies the inequality

 

 

for all with 0 < < 1.

M x M x  

Orlicz function has been studied by V. A. Khan [14-17] and many others.

Throughout a double sequence x=

 

x_kl is a double infinite array of elements xkl for k l, .

Double sequences have been studied by V. A. Khan [18-20], Moricz and Rhoades [21] and many others.

A double sequence x=

 

xjk called statistically con- vergent to L if

 

,

1

, : , , = 0

lim jk

m n

j k x L j m k n

mn 

    

where the vertical bars indicate the number of elements in the set. (see [19])

In this case we write st2limxjk=L.

2. Definitions and Preliminaries

Let

 

xj be a sequence in 2-normed space



X, .,.



. The sequence

 

xj is said to be statistically convergent to L, if for every > 0, the set



j: xjL z, 



has natural density zero for each nonzero z in X, in other words

 

xj statistically converges to L in 2-normed space



X, .,.



if





1

: , = 0

lim j

n

j x L z n

 

for each nonzero z in X. It means that for every zX ,

, < . . . j

x L z  a a n

In this case we write

, := , . lim j

n

st x L z L z



 

Example 2.1 Let X =R2 be equiped with the 2- norm by the formula









1 2 2 1 1 2 1 2

, = , = , , = , .

x y x y x y x x x y y y

Define the

 

xj in 2-normed space



X, .,.



by

 

2

1, if = , ,

= ₁

1, otherwise. j

n n k k N

x _n n       _ _ 

and let L= 1,1

 

and z=



z z1, 2



. If z1= 0 then





= : j , =

K j x L z  

for each z in 2 | 1|

, : = , z

X j n k k



 _{  } 

 

   is a finite set,

so









1 2 2 1 : ,

= : = , 1 finite set . j

j x L z

j j k k

z           _{  } _    _{ }          Therefore,





 

1 2 2 1 1 : , 1

: = , 1 0 1

j

j x L z n

j j k k

z n               _  _         

for each z in X. Hence, 





j: x_nL z, 





= 0

for every > 0 and zX.

V. A. Khan and Sabiha Tabassum [20] defined a double sequence

 

xjk in 2-normed space



X, .,.



to be Cauchy with respect to the 2-norm if

,

, = 0 for every and , . lim jk pq

j p

x x z z X k q

   

If every Cauchy sequence in X converges to some

,

LX then X is said to be complete with respect to the 2-norm. Any complete 2-normed space is said to be 2-Banach space.

Example 2.2 Define the xi in 2-normed space



X, .,.



by

 

 

2

0, if = , , =

0, 0 otherwise.

j

j j k k N

x  

and let L= 0, 0

 

and z=



z z1, 2



. If z1= 0 then







2



: _j , 1, 4, 9,16, , ;

j x L z    j 

We have that 





j: x_j L z, 





= 0 for every

> 0

 and zX. This implies that lim j, = n

st x z





,

L z . But the sequence xj is not convergent to L. A sequence which converges statistically need not be bounded. This fact can be seen from Example [2.1] and Example [2.2].

3. Main Results

In this paper we define a double sequence

 

xjk in 2-normed space



X, .,.



to be statistically Cauchy with respect to the 2-norm if for every  > 0 and every nonzero zX there exists a number p= p

 

,z and

 

= ,

q q  z such that

 

,

1

, : , , , = 0

lim jk pq

m n

j k N N x x z j m k n

mn 

      

In this case we write st2lim xjkL z, := L z, . Theorem 3.1. Let

 

xjk be a double sequence in 2-normed space



X, .,.



and L L, X . If

2 lim jk, = ,

st  x z L z and st2lim xjk,z = L z, , then L=L.

Proof. Assume L= L,. Then LL = 0, so there exists a zX, such that LL and z are linearly in-dependent. Therefore

, = 2 , with > 0.

LL z  

Now



 



2 = ,

, , .

jk jk

jk jk

L x x L z

x L z x L z

    

   

So



 

j k, : xjkL z, <







 

j k, : xjkL z, <



. But 





 

j k, : x_jkL z, <





= 0. Contradicting the fact that x_jk L stat





.

Theorem 3.2. Let the double sequence

 

xjk and

 

yjk in 2-normed space



X, .,.



. If

 

yjk is a con-vergent sequence such that x_jk = y_jk almost all n, then

 

xjk is statistically convergent.

Proof. Suppose 





( , )j k  N N x: _jk = y_jk





= 0 and

,

, = , lim jk

j k

y z L z

 . Then for every > 0 and zX .

 





 





, : ,

, : = .

jk

jk jk

j k N N x L z

j k N N x y



   



  

Therefore

 









 









 









, : ,

, : ,

, : = .

jk

jk

jk jk

j k N N x L z

j k N N y L z

j k N N x y





    

     



   

(3.1)

Since lim jk, = , n

y z L z

 for every zX, the set

 



j k,  N N: y_jkL z, 



contains finite number of integers. Hence, 





 

j k,  N N: y_jkL z, 





= 0. Using inequality [3.1], we get

 







j k, N N: x_jk L z, 



= 0

    

for every > 0 and zX. Consequently,

2 lim jk , = , .

st  x L z L z

Theorem 3.3. Let the double sequence

 

xjk and

 

yjk in 2-normed space



X, .,.



and L L, X and

a.

If st2lim xjk,z = L z, and st2lim yjk,z = L z, , for every nonzero zX, then

1) st2lim xjkyjk,z = LL z, , for each nonzero

zX and

2) st2lim axjk,z = aL z, , for each nonzero zX. Proof 1) Assume that st2lim xjk,z = L z, , and

2 lim jk, = ,

st  y z L z , for every nonzero zX . Then

 

K1 = 0

 and 

 

K2 = 0 where

 

 

1= 1 := , : ,

2

jk

K K  _ j k  N N x L z _

 

 

 

2 = 2 := , : ,

2

jk

K K  _ j k  N N y L z _

 

for every > 0 and zX. Let

   





= := , : jk jk ( ), .

K K  j k  N N x y  LL z 

To prove that 

 

K = 0, it is sufficient to prove that

1 2

KK K . Suppose j k0, 0K. Then





Suppose to the contrary that j k0, 0K1K2. Then

0, 0 1

j k K and j k0, 0K2. If j k0, 0K1 and

0, 0 2

j k K then

0 0 , < ₂

j k

x L z  and

0 0 , < ₂.

j k

x L z 

Then, we get





0 0 0

0 0 0

,

, , < =

2 2

j k_o j k

j k_o j k

x y L L z

x L z y L z   



  



    

which contradicts [3.2]. Hence j k0, 0K1K2, that is,

1 2

KK K .

2) Let st2lim xjk,z = L z a, ,  and a= 0 . Then

 

j k, N N: x_jk L z, = 0.

a



_ _

   

_{ }

_{ }

 

Then we have

 





 





 

, : ,

= , : ,

= , : , .

jk

jk

jk

j k N N ax aL z

j k N N a x L z

j k N N x L z a



 

   

   

 

 _{   } 

 

 

 

Hence, the right handside of above equality equals 0. Hence, st2lim axjk,z = aL z, , for every nonzero

. zX

From Theorem 1 of Fridy [11] we have

Theorem 3.4. Let

 

xjk be statistically Cauchy se-quence in a finite dimensional 2-normed space



X, .,.



. Then there exists a convergent double sequence

 

y_jk in



X, .,.



such that xjk = yjk for almost all n.

Proof. See proof of Theorem 2.9 [9].

Theorem 3.5. Let

 

x_jk be a double sequence in 2- normed space



X, .,.



The double sequence (xjk) is statistically convergent if and only if (xjk) is a statisti-cally Cauchy sequence.

Proof. Assume that st2lim xjk,z = L z, for every nonzero zX and > 0.

Then, for every zX,

, < almost all , 2

jk

x L z  n

and if p= p

 

,z and q=q

 

,z is chosen so that , < ,

2 pq

x L z  then, we have

, , ,

< almost all . 2 2

= almost all .

jk pq jk pq

x x z x L z L x z

n

n

 



    



Hence,

 

xjk is statistically Cauchy sequence.

Conversely, assume that xjk is a statistically Cauchy sequence. By Theorem 3.4, we have st2lim xjk,z =

,

L z for each zX .

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