Applied Mathematics, 2011, 2, 398-402
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 Mat hematics, Aligarh Muslim University, Aligarh, India
Received November 26, 2010; revised January 15, 2011; accepted Ja nu ary 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
XR which satisfies the following four
conditions:
1) 12 ,=0xx if and only if 12 ,
x
x are linearly de-
pendent;
2) 12 21 ,=,
x
xxx:
3) 12 12
,= ,
x
xxx  , for any R

:
4) 22 2
,,,
x
xxxx xx


The pair

,.,.X is then called a 2-normed space
(see [9]).
Example 1.1. A standard example of a 2-normed
space is 2 R equipped wit h t he f oll owing 2-norm
,:=xy 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
,=0fg , if f, g are linearly dependent,




, =, sup
tR
f
gftgt

 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 {:}.kKkn

 The natural density
of K is given by

||
=,
lim n
n
K
Kn

 where n
K
deno-
tes the number of elements in n
K
, provided this limit
exists. Finite subsets have natural density zero and




=1
c
K
K where =\
c
K
K, that is the comp-
lement of K. If 12
K
K and 1
K
and 2
K
have natu-
ral densities then


12
.KK Moreover, if




12
==1,KK then

12
=1KK (see [10]).
A real number sequence

=
j
x
x is statistically con-
vergent to L provided that for every >0

the set


:| |
j
nxL


 has natural density zero. The sequ-
ence


=
j
x
x is statustically Cauchy sequence if for
each >0

there is positive integer


=NN

such
that





:|= 0
jN
nxx    (see [11]).
If


=
j
x
x is a sequence that satisfies some property
P for all n except a set of natural density zero, then we
say that


j
x
satisfies some property P for “almost all n”.
An Orlicz Function is a function




:0, 0,M 
which is continuous, nondecreasing and convex with


0=0M,


>0Mx for >0x and

Mx, as
x
.
If convexity of
M
is replaced by



M
xy Mx


M
y, then it is called a Modulus funtion (see Mad-
dox [12]). An Orlicz function may be bounded or un-
12000 Mathematics Subject Classification. 46E30, 46E40, 46B20.

V. A. KHAN ET AL. 399
bounded. For example,

 
=0<1
p
Mx xp is un-
bounded and

=1
x
Mx x is bounded.
Lindesstrauss and Tzafriri [13] used the idea of Orlicz
sequence space;
=1
:=:<, for some >0
k
Mk
x
lxwM







 




which is Banach space with the norm
=1
||
=inf>0:1 .
k
Mk
x
xM










The space
M
l is closely related to the space
p
l,
which is an Orlicz sequence space with

=
p
M
xx for
1<.p
An Orlicz function
M
satisfies the 2condition
2
( f )
M
or short if there exist constant 2K and
0>0u such that
 
2
M
uKMu
whenever 0
||uu.
Note that an Orlicz function satisfies the inequality
 
for all with 0<<1.Mx Mx   
Orlicz function has been studied by V. A. Khan [14-17]
and many others.
Throughout a double sequence

=kl
x
x is a double
infinite array of elements kl
x
for ,.kl
Double sequences have been studied by V. A. Khan
[18-20], Moricz and Rhoades [21] and many others.
A double sequence

=
j
k
x
x called statistically con-
vergent to L if

,
1,:, ,=0
lim jk
mn jkxLj mk n
mn

  
where the vertical bars indicate the number of elements
in the set. (see [19])
In this case we write 2lim =
jk
s
txL.
2. Definitions and Preliminaries
Let

j
x
be a sequence in 2-normed space


,.,.X.
The sequence

j
x
is said to be statistically convergent
to L, if for every >0

, the set

:,
j
jxLz


has natural density zero for each nonzero z in X, in other
words

j
x
statistically converges to L in 2-normed
space

,.,.X if

1:,=0
lim j
njx Lz n
 
for each nonzero z in X. It means that for every zX

,
,< ...
j
x
Lz aan  
In this case we write
,:=,.
lim j
n
s
tx LzLz


Example 2.1 Let 2 =
X
R be equiped with the 2-
norm by the formula

12211 21 2
,=,= ,,=,.
x
yxyxyxxxyyy
Define the


j
x
in 2-normed space

,.,.X by

2
1, if =,,
=
1
1, otherwise.
j
nnkkN
xn
n







and let


=1,1L and

12
=,zzz. If 1=0z then


=:,=
j
Kj xLz

 
for each z in 21
||
,:=,
z
Xj nkk








 is a finite set,
so



1
2
2
1
:,
=:=,1 finite set.
j
jxLz
jjkk
z










 



Therefore,


1
2
2
1
1:,
1
:= ,101
j
jxLz
n
jjkk
zn






 



 



for each z in X. Hence,
 
:, =0
n
jxLz   
for every >0

and zX

.
V. A. Khan and Sabiha Tabassum [20] defined a
double sequence


j
k
x
in 2-nor med space

,.,.X to
be Cauchy with respect to the 2-norm if
,,=0for every and ,.
lim jk pq
jp xxzzX kq



If every Cauchy sequence in
X
converges to some
,LX

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
jjkkN
x





V. A. KHAN ET AL. 400
and let

=0,0L and

12
=,zzz. If 1=0z then



2
:,1,4,9,16, , ;
j
jxLzj  
We have that
 
:, =0
j
jxLz    for every
>0

and zX. This implies that ,=
lim j
n
s
txz


,Lz. But the sequence
j
x
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


j
k
x
in
2-normed space

,.,.X to be statistically Cauchy
with respect to the 2-norm if for ever y >0

and every
nonzero zX there exists a number

=,pp z

and

=,qqz  such that

,
1,:,,,=0
lim jk pq
mn jkNNxxzjmkn
mn

  
In this case we write 2lim, :=,
jk
s
tx LzLz .
Theorem 3.1. Let

j
k
x
be a double sequence in
2-normed space

,.,.X and ,LL X
. If
2lim, = , jk
s
txzLz and 2lim, =,
jk
s
txzLz

,
then =.LL

Proof. Assume =,LL


. Then =0LL

, so there
exists a zX, such that LL


and z are linearly in-
dependent. Therefore
,=2, with >0.LLz 
 
Now



2= ,
,,.
jk jk
jk jk
LxxL z
x
LzxLz

 
 
So




,: ,<,: ,<
jk jk
jkxL zjkxL z



  .
But



,:,<=0
jk
jkxL z 

 . Contradicting the
fact that

.
jk
x
Lstat


Theorem 3.2. Let the double sequence

j
k
x
and

j
k
y in 2-normed space

,.,.X. If

j
k
y is a con-
vergent sequence such that =
j
kjk
x
y almost all n, then

j
k
x
is statistically convergent.
Proof. Suppose
 
(,): ==0
jk jk
jkNN xy


and ,,=,
lim jk
jk yz Lz  . Then for every >0

and
zX.





,:,
,:=.
jk
jk jk
jkN NxLz
jkNNxy



 
Therefore








 

,:,
,:,
,:=.
jk
jk
jk jk
jkN NxLz
jkN NyLz
jkNNxy



 

 
(3.1)
Since ,=,
lim jk
nyzLz
 for every zX, the set



,:,
jk
jkN NyLz


 contains finite number
of integers. Hence,





,:,
jk
jkN NyLz


=0. Using inequality [3.1], we get





,:,=0
jk
jkN NxLz  
for every >0

and .zX

Consequently,
2lim,=, . jk
s
tx LzLz
Theorem 3.3. Let the double sequence


j
k
x
and


j
k
y in 2-normed space


,.,.X and ,LL X


and
a

.
If 2lim, = ,
jk
s
txzLz
and 2lim,=, ,
jk
s
tyzLz


for every nonzero zX

, then
1) 2lim, =,
jk jk
s
txyzLLz

  , for each nonzero
zX

and
2) 2lim, =,
jk
s
taxzaLz, for each nonzero zX

.
Proof 1) Assume that 2lim,=, ,
jk
s
txzLz and
2lim, =, jk
s
tyzLz

, for every nonzero zX

. Then


1=0K and


2=0K where
 
11
=:=,: ,
2
jk
KKjkNNx Lz








 
22
=:=,: ,
2
jk
KK jkNNyLz





 



for every >0

and zX

. Let
 


=:=,:(), .
jk jk
KKjk NNxyLLz




To prove that


=0K  , it is sufficient to prove that
12
K
KK. Suppose 00 ,jk K. Then



000,
jk jk
o
xy LLz



  (3.2)

V. A. KHAN ET AL. 401
Suppose to the contrary that 00 12 ,jk K K. Then
00 1
,jk K and 002 ,jkK. If 001 ,jk K and
00 2
,jk K then
00 ,<
2
jk
xLz

 and 00,<.
2
jk
xLz


Then, we get

000
000
,
,, <=
22
jk jk
o
jk jk
o
xy LLz
xLzyLz





 
which contradicts [3.2]. Hence 0012
,jk K K, that is,
12
K
KK.
2) Let 2lim,=, ,
jk
stxzL za and =0a.
Then

,:,=0.
jk
jkN NxLza











Then we have





,:,
=,: ,
=,: ,.
jk
jk
jk
jkN NaxaLz
jkNNa xLz
jkN NxLza



 
 


 



Hence, the right handside of above equality equals 0.
Hence, 2lim,=,,
jk
s
taxzaLz for every nonzero
.zX
From Theorem 1 of Fridy [11] we have
Theorem 3.4. Let

j
k
x
be statistically Cauchy se-
quence in a finite dimensional 2-normed space


,.,.X.
Then there exists a convergent double sequence


j
k
y
in

,.,.X such that =
j
kjk
x
y for almost all n.
Proof. See proof of Theorem 2.9 [9].
Theorem 3.5. Let

j
k
x
be a double sequence in 2-
normed space

,.,.X The double sequence ()
j
k
x
is
statistically convergent if and only if ()
j
k
x
is a statisti-
cally Cauchy sequence.
Proof. Assume that 2lim, = ,
jk
s
txzLz for every
nonzero zX and >0.

Then, for every zX,
,< almost all ,
2
jk
x
Lz n


and if


=,pp z

and


=,qq z  is chosen so that
,<,
2
pq
xLz

 then, we have
,,,
< almost all .
22
= almost all .
jk pqjkpq
x
xzxLz Lxz
n
n




Hence,


j
k
x
is statistically Cauchy sequence.
Conversely, assume that
j
k
x
is a statistically Cauchy
sequence. By Theorem 3.4, we have 2lim, =
jk
s
txz
,Lz for each zX

.
4. References
[1] H. Stinhaus, “Sur la Convergence Ordinarie et la Conver-
gence Asymptotique,” Colloqium Mathematicum, Vol. 2,
No. 1, 1951, pp. 73-74.
[2] H. Fast, “Sur la Convergence Statistique,” Colloqium
Mathematicum, Vol. 2, No. 1, 1951, pp. 241-244.
[3] I. J. Schoenberg, “The Integrability of Certain Functions
and Related Summability Methods,” American Mathema-
tical Monthly, Vol. 66, No. 5, 1959, pp. 361-375.
doi:10.2307/2308747
[4] J. A. Fridy and C. Orhan, “Statistical Limit Superior and
Limit Inferior,” Proceedings of the American Mathemati-
cal Society, Vol. 125, No. 12, 1997, pp. 3625-3631.
doi:10.1090/S0002-9939-97-04000-8
[5] S. Gähler, “2-Merische Räme und Ihre Topological Stru-
ktur,” Mathematische Nachrichten, Vol. 26, No. 1-2, 1963,
pp. 115- 148.
[6] S. Gähler, “Linear 2-Normietre Räme,” Mathematische
Nachrichten, Vol. 28, No. 1-2, 1965, pp. 1-43.
[7] S. Gähler, “Uber der Uniformisierbarkeit 2-Merische
Räme,” Mathematische Nachrichten, Vol. 28, No. 3-4,
1964, pp. 235-244.
[8] H. Gunawan and Mashadi, “On Finite Dimensional 2-
Normed Spaces,” Soochow Journal of Mathematics, Vol.
27, No. 3, 2001, pp. 631-639.
[9] M. Gurdal and S. Pehlivan, “Statistical Convergence in
2-Normed Spaces,” Southeast Asian Bulletin of Mathe-
matics, Vol. 33, No. 2, 2009, pp. 257-264.
[10] A. R. Freedman and I. J. Sember, “Densities and Summa-
bility,” Pacific Journal of Mathematics, Vol. 95, 1981, pp.
293-305.
[11] J. A. Fridy, “On Statistical Convergence,” Analysis, Vol.
5, No. 4, 1985, pp. 301-313.
[12] I. J. Maddox, “Sequence Spaces Defined by a Modulus,”
Mathematical Proceedings of the Cambridge Philosophi-
cal Society, Vol. 100, No. 1, 1986, pp. 161-166.
doi:10.1017/S0305004100065968
[13] J. Lindenstrauss and L. Tzafiri, “On Orlicz Sequence
Spaces,” Israel Journal of Mathematics, Vol. 10, No. 3,
1971, pp. 379-390. doi:10.1007/BF02771656
[14] V. A. Khan and Q. M. D. Lohani, “Statistically Pre-Cau-
chy Sequence and Orlicz Functions,” Southeast Asian
Bulletin of Mathematics, Vol. 31, No. 6, 2007, pp. 1107-
1112.
[15] V. A. Khan, “On a New Sequence Space Defined by
Orlicz Functions,” Communication, Faculty of Science,
University of Ankara, Series Al, Vol. 57, No. 2, 2008, pp.
25-33.
[16] V. A. Khan, “On a New Sequence Space Related to the

V. A. KHAN ET AL. 402
Orlicz Sequence Space,” Journal of Mathematics and Its
Applications, Vol. 30, 2008, pp. 61-69.
[17] V. A. Khan, “On a New Sequence Spaces Defined by
Musielak Orlicz Functions,” Studia Mathematica, Vol. 55
No. 2, 2010, pp. 143-149.
[18] V. A. Khan, “Quasi almost Convergence in a Normed
Space for Double Sequences,” Thai Journal of Mathema-
tics, Vol. 8, No. 1, 2010, pp. 227-231.
[19] V. A. Khan and S. Tabassum, “Statistically Pre-Cauchy
Double Sequences and Orlicz Functions,” Accepted by
Southeast Asian Bulletin of Mathema tics.
[20] V. A. Khan and S. Tabassum, “Some Vector Valued Mul-
tiplier Difference Double Sequence Spaces in 2-Normed
Spaces Defined by Orlicz Function,” Submitted to Jour-
nal of Mathematics and Applications.
[21] F. Moricz and B. E. Rhoades, “Almost Convergence of
Double Sequences and Strong Regularity of Summability
Matrices,” Mathematical Proceedings of the Cambridge
Philosophical Society, Vol. 104, No. 2, 1988, pp. 283-294.
doi:10.1017/S0305004100065464