ls4 ws16"> (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.
Copyright © 2011 SciRes. AM
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.
Copyright © 2011 SciRes. AM
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.
Copyright © 2011 SciRes. AM
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.
Copyright © 2011 SciRes. AM
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