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 Authors : Narayan Prasad Pahari
 Paper ID : IJERTV3IS031181
 Volume & Issue : Volume 03, Issue 03 (March 2014)
 Published (First Online): 08042014
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
On Certain Topological Properties of Double Paranormed Null Sequence Space
Narayan Prasad Pahari
Central Department of Mathematics Tribhuvan University, Kirtipur, Kathmandu, Nepal
Abstract The aim of this paper is to introduce and study a
new class c ((S ,  .  ),
(amn) in various sequence spaces c0, c, , p depending upon the mode of m and n tending to infinity lead to several
0 mn
mn , u ) of double sequences with
u
their terms in a normed space S as a generalization of the familiar sequence space c0 . We investigate the condition in terms of and so that a class is contained in or equal to another class of same kind and thereby derive the conditions of their equality. We further explore some of the preliminary results that characterize the linear topological structures of
spaces, see Maddox [12].
A paranormed space (S, G) is a linear space S with zero element together with a function G : S R+ (called a paranorm on S) which satisfies the following axioms:
the space c0
((Smn
,  . mn
),
, u ) when topologized it with
PN1: G () = 0; PN2: G () = G () for all S;
suitable natural paranorm .
Keywords: Sequence space, Double sequence, Paranormed space, GKspace.
I.INTRODUCTION
We begin with recalling some notations and basic definitions that are used in this paper.
s
Let S be a normed space over C , the field of complex numbers. Let (S) denotes the linear space of all sequences = (sk ) with sk S , k 1 with usual coordinate wise operations .We shall denote (C) by . Any subspace S of is then called a sequence space. A normed space valued sequence space or a generalized sequence space is a linear space of sequences with their terms in a normed space. Several workers like Kamthan and Gupta [6], Khan [7], Kolk [8], KÃ¶the [9], Maddox [11], Malkowski and Rakocevic [13], Pahari [16,17,18], Ruckle

etc. have introduced and studied some properties of vector and scalar valued single sequence spaces, when sequences are taken from a Banach space.
The theory of single sequence spaces has also been extended to the spaces of double sequences and studied by several workers. Boos Leiger [3], Gupta and Kamthan [4], Milovidov and Povolotzki [14], Morics [15], Rao [20] and many others have made their significant contributions and enriched the theories in this direction. In the recent years, Savas [22], Subramanian et al [23] and many others have
PN3: G ( + ) G () + G ( ) for all , S; and
PN4: Scalar multiplication is continuous.
Note that the continuity of scalar multiplication is equivalent to

if G (n) 0 and n as n , then
G (n n) 0 as n , and

if n 0 as n and be any element in S, then

G (n ) 0, see Wilansky [24].
A paranorm is called total if G () = 0 implies =
The concept of paranorm is closely related to linear metric space; see Wilansky [24] and its studies on sequence spaces were initiated by Maddox [10] and many others. Basariv and Altundag [1], Bhardwaj and Bala [2], Khan [7], Parasar and Choudhary [19], and many others further studied various types of paranormed sequence spaces .
Concerning Kproperty of scalar sequence spaces, see Kamthan and Gupta [6], GAKspace have been defined for vector valued sequence spaces and also they are defined for Banach space valued function space, see Gupta and Patterson [5]. We now introduce the following definition for double sequence spaces:
Let V(Smn) be a class of sequences { s = (smn), smn Smn ; m, n 1}.The topological sequence space (V(Smn), ) equipped with the linear topology is said to be a GKspace if the
introduced and studied various types of double sequence spaces using orlicz function.
The notion of convergence of a single sequence (an) leads to various notions of convergence for a double sequence (amn) by using many senses. The double sequence
map Pij : V(Smn) Sij, defined by P
for each i, j 1.
ij (s) = sij
is continuous
1
k 1/ umn z
m = m(k) n = n(k) k 1 and

THE CLASS c0 ((Smn ,  . mn), , u ) OF DOUBLE
SEQUENCES
smn
= mn mn
otherwise.
Let u = (umn) and v = (vmn) be any double sequences of
Then we see that s c0 ((Smn ,  . mn), , u ) . Since
u 1 v
m(k)n(k) k mn
strictly positive real numbers and = (mn) and = (mn) be
double sequences of nonzero complex numbers. Let (S , 
 m(k) n(k) s m(k) n(k)  m(k)n(k) = , k 1 and mn smn mn =
mn
. mn) ,m, n 1 be normed space over the field C of complex
, otherwise. But for each k 1,
numbers with zero element .We now introduce and study
the following class of Banach space valued double sequences:

m(k) n(k)
s m(k) n(k)
um(k)n(k) vm(k) n(k) um(k) n(k)
 = k /
m(k)n(k)
> k 1/k > e1/2
c ((S
,  . 
),
shows that s c0 ((Smn ,  . mn), , u ) , a
0 mn
mn , u ) = {s = (smn): smn Smn, m, n 1, and
contradiction. This cmpletes the proof.
 mn smn umn 0 as m + n }.
mn Theorem 3.2: For any = (mn) , c0 ((Smn ,  . mn), , v
Further, by u = (umn)
, we mean sup umn
< We
) c0 ((Smn ,  . mn),
, u ) if and only if
denote A() = max(1, ) and the zero element of this class
by = ( ) for all m,n .
lim sup
m + n
vmn < .
u
mn
mn

SOME CONTAINMENT RELATIONS
Proof:
s
Let the condition hold, and = (smn) c0 ((Smn ,  .
In this section we investigate the conditions in terms of u
 ),
and , so that a class c ((S ,  .  ),
mn , v ) . Then there exists a constant L > 0 such
0 mn
mn , u ) is contained in
that vmn < L umn for all sufficiently large values of m,n.

or equal to another class of same kind and thereby derive the conditions of their equality.
Further  mn
smn
umn mn
0 as m + n together
Theorem 3.1: For any = (mn), c0 ((Smn ,  . mn), , )
u
with
 s umn ( s
vmn)1/Lfor all sufficiently large
lim inf v
mn mn mn
mn mn mn
c0 ((Smn ,  . mn), , v ) if and only if
mn >
m + n
0.
umn
values of m, n implies that s c0 ((Smn ,  . mn), , u
) and hence
Proof: For the sufficiency of the condition, suppose that
c ((S
,  . 
),
0 mn
mn , v ) c0 ((Smn ,  . mn), , u .
the condition hold, and s = (smn) c0 ((Smn ,  . mn), , u ) . Then there exists a constant K > 0 such that vmn > K umn for
Conversely let the inclusion hold but
lim sup
m + n
all
sufficiently large values of m,n. Further  mn smn umn < 1
vmn
umn
= Then there exists subsequences (m(k)) of
mn
for all sufficiently large values of m,n and so
 s umn  s umn )K for all sufficiently large
(m) and (n(k)) of (n) respectively for each k 1;
v m(k) n(k) < k m(k) n(k).
mn mn mn
mn mn mn
Thus for zmn
Smn
with z
mnmn
= 1 the sequence
values of m,n which implies that s c0 ((Smn ,  . mn), , v
). Hence
s = (smn) defined as
c ((S
,  . 
),
1
smn = mn k
1/vmn z m = m(k) n = n(k) k 1 and
mn
0 mn
mn , u ) c0 ((Smn ,  . mn), , v ).
otherwise.
For the necessity of the condition suppose the inclusion
holds, but lim inf
vmn
= 0. Then there exist
is in c0 ((Smn ,  . mn), , u ) but for each k 1,
m + n
umn
 m(k) n(k) s m(k) n(k) um(k)n(k)
= kvm(k) n(k)/um(k) n(k)
subsequences (m(k)) of (m) and (n(k)) of (n) respectively such that
k vm(k) n(k) < u m(k) n(k), k 1.
Now taking zmn Smn with zmnmn = 1, we define a
m(k)n(k)
> k 1/k
> e1/2.
sequence s = (smn) by
This shows that s c0 ((Smn ,. mn), , u ), a contradiction. This completes the proof.
Theorem 3.3: For any = (mn),c0 ((Smn ,  . mn), c0 ((Smn ,  . mn),
) = c0 ((Smn ,  . mn),
, u which shows that s
c0 ((Smn ,  . mn),
, u ) but s
0 <
lim inf
m + n
vmn
umn
, v ) if and only if
lim sup
m + n
vmn < .
u
mn
, u ), a contradiction. This
completes the proof.
u u
Theorem 3.5: For any = ( umn), c0 ((Smn ,  . mn), , )
Theorem 3.4: For any u
= (umn), c0 ((Smn ,  . mn), , u )
c0 ((Smn ,  . mn), , u )
c0 ((Smn ,  . mn),
lim sup
mn umn
, u )
if and only if
m + n
.
lim inf
mn umn mn
Proof:
if and only if
m + n mn
> 0.
Proof:
For the sufficiency of the condition, suppose that
Let the condition hold, and s = (smn) c0 ((Smn ,  . mn),
lim inf
umn
, u ) . Then there exists 0 < L < such that
mn
> 0 and s = (smn)
 umn < L
umn, for all sufficiently large values of
mn
m + n mn
mn mn
c ((S ,  .  ),
m,n and so  mn smn umn
 mn smn umn
mn
0 mn
mn , u ) . Then there exists a constant
umn
K > 0 such that K < mn
for sufficiently large
implies that s c0 ((Smn ,  . mn), , u ) . This shows
mn
values of m, n. Thus
mn
mn
K  mn smn umn <  mn smn umn large
for all sufficiently
that c0 ((Smn ,  . mn), , u ) c0 ((Smn ,  . mn), , u )
.
Conversely let
values of m, n and so  s
umn 0 implies that
c0 ((Smn ,  . mn), , u ) c0 ((Smn ,  . mn), , u )
mn mn mn
but
lim sup
mn
umn
= .Then there exist
mn
mn smn umn 0 and hence s c0 ((Smn ,  . mn), , u ).
This proves that
m + n mn
subsequences (m(k)) of (m) and n(k)) of (n) respectively such that for each k 1
c0 ((Smn ,  . mn), , u ) c0 ((Smn ,  . mn), , u ) .
 m(k) n(k)
um(k)n(k) > k 
m(k) n(k)
um(k)n(k).
For the necessity, suppose that
Now taking z
S , such that  z
 = 1, we
lim inf
mn
umn
=0. Then there exist
mn mn
mn mn
m + n mn
define the sequence s = (smn) by
m(k) n(k) 
subsequences (m(k)) of (m) and (n(k)) of (n) respectively such that for each k 1, k  m(k) n(k)
1 1/ umn
k
smn = mn
otherwise.
zmn m = m(k) n = n(k) k 1 and

um(k)n(k)
<  um(k)n(k).
Then,  m(k) n(k) s m(k) n(k) um(k)n(k) =
1, k 1 and
Now for zmn Smn with zmnmn = 1, define the
sequence s = (smn) by
 mn
smn
m(k)n(k) k
mn
umn = 0, otherwise
1
=
k 1/ u
mn zmn m = m(k) n = n(k) k 1 and
smn
mn
m(k)n(k)
otherwise.
shows that s c0 ((Smn ,  . mn), , u ). But on the other hand
Now,  m(k) n(k) s m(k) n(k) um(k)n(k) =
1, k 1
 m(k) n(k) s m(k) n(k) um(k)n(k)
> 1, for all k = 1
m(k)n(k) k
mn
and  mn smn umn = 0, otherwise. But for each k 1,
 m(k) n(k) s m(k) n(k) um(k) n(k)
implies that s c0 ((Smn ,.mn), , u ) ,a contradiction. This completes the proof.
On combining Theorems 3.4 and 3.5 we get :
= 
1
m(k)n(k)
k 1/ um(k)n(k) u
z um(k) n(k)
Theorem 3.6: For any u = (umn), c0 ((Smn ,  . mn), , u )
m(k) n(k)
m(k)n(k)
m(k) n(k)
m(k) n(k)
m(k)n(k)
= c0 ((Smn
,  . mn), , ) if and only if
u
m(k) n(k)um(k) n(k) 1
lim inf
umn
lim sup
umn
0 <
=
m(k) n(k)
. k > 1
mn
m + n mn
mn
m + n mn
< .

LINEAR TOPOLOGICAL STRUCTURES
Then for each m = m(k) n = n(k) k 1, we have
 mn smn umn
=  m(k) n(k) s m(k) n(k) um(k)n(k)
OF c0 ((Smn ,  . mn), , u )
In this section, we shall investigate some results that
mn
1
= k, k 1
m(k)n(k)
characterize the linear topological structure of
and  mn smn umn = 0, otherwise,
c ((S
,  .  ), mn
0 mn
mn , u ) when topologized it with suitable
natural paranorm . As far as the linear space structure of
u
c0 ((Smn ,  . mn), , ) over the field C of complex numbers is concerned, we throughout take the
shows that s c0 ((Smn ,  . mn), , u ) .But on the otherhand, for each m = m(k) n = n(k) k 1and for scalar = 2 we have
umn
um(k) n(k)
coordinatewise operations i.e., for s
scalar ,
= (smn), t
= (tmn) and
 mn ( smn) mn =  m(k) n(k) 2 s m(k) n(k) m(k)n(k)
2k
s + t = (smn + tmn) and s = (smn)
u
and we see below that is necessary and sufficient
> k 1
s u
showing that c0 ((Smn ,  . mn), , ), a
condition for linearity of c0 ((Smn ,  . mn), , ).
u
contradiction. This completes the proof. Hence c0
((Smn ,  . mn),
if and only if u =
Theorem 4.1: c0 ((Smn ,  . mn), , u ) forms a linear space over the set of complex number C
= (u
, u ) is a linear space if and only if u
mn) .
(umn) .
Proof:
Consider u
,we define
= (umn) and s c0 ((Smn ,  . mn), , u )
For the sufficiency of the condition, assume that
umn
G, u(s) = supmn mn smn mn (4.1)
u = (umn) and s = (smn), t = (tmn) c0 ((Smn ,  .
mn),
mn mn
umn
Theorem 4.2 : Let u = (umn) and Smn be a normed
, u ), m, n 1.So that 
s mn 0
mn
and mn tmn umn 0 as m + n . Then we have
mn mn mn
 mn (smn + tmn) umn  mn smn umn +  mn tmn umn
0 , as m + n .
space for each m,n 1. Then (c0 ((Smn ,  . m), , u ) , G,u)
forms
a total paranormed space.
Proof:
It can be easily verified that G,u defined by (4.1) satisfy following properties of paranormed space.
Hence s + t c0 ((Smn ,  . mn), , u ).
PN1: G,u(s) 0 and G,u(s) = 0 if and only if s =
Also it is clear that for any scalar , c0 ((Smn ,  . mn), ,
u ) , since
s
PN2: G,u(s + t ) G
,u
(s) + G
,u( t )
mn
 mn smn umn
= umn  mn smn umn
mn
mn
A() mn smn umn
0 as m + n .
PN3: G,u(s) A() G,u(s) where C.
Throughout the proof of the theorem, G,u will be denoted by G. Here we prove the continuity of scalar
multiplication i.e., PN4. Further for continuity of scalar multiplication, it is sufficient to show that
Conversely if u
= (umn) then there exist subsequences
(k)

if s
in G and k as k
(m(k)) of (m) and (n(k)) of (n) respectively such that
u m(k) n(k)
> k for each k 1.

if k 0 as k
and s c0 ((Smn ,  . mn),
, u )
Now taking zmn
sequence s = (smn) by
Smn
with z
mnmn
= 1, we a define
then k in G as k .
s
Now (a) is easily proved if we suppose that k L
1 k 1/ umn z
m = m(k) n = n(k) k 1 and
for all k 1 and consider,
s = mn mn
(k) u u
mn
mn otherwise.
G(k s ) supm,nk mn sup m,n mn smn  mn
(k)
A(L) G (s ).
s u
Now let c0 ((Smn ,  . mn), , ) , k 1 for all k
N and > 0. Then there exists K such that

Maddox, I.J., Infinite matrices of operators; Lecture Notes in Mathematics 786, Springer Verlag Berlin, Heidelberg, New York, ( 1980).

Malkowski, E. and Rakocevic, V. , An introduction into the theory of
sequence spaces and measures of noncompactness, (2004).
mn
 mn smn umn
and hence
< , for all m + n K

Milovidov, S.P. and Povolotzki, A.I. , Dual spaces for conditional KÃ¶the spaces of double real
k
mn
smn
umn < , for all m + n K and k N.
sequences; Izvestiya, VUZ. Mathematica, 35(2),
1991), 9091.
mn
mn
Now choose N1 so that kumn  mn smn umn < , for all
k N1, and 2 m + n K.

Morics, F. , Extensions of the spaces c and c0 from single to double sequences; Acta Math.
Hungar. 57 (12), (1991),129136.
s)
Thus G (k
for all k max(N, N1) which proves

Pahari,N.P., On Banach space valued sequence space l (X,M, l , p, L)
(b). Hence G forms a total paranorm on
c0 ((Smn), mn
defined by Orlicz function , Nepal Jour. of Science and Tech. , 12, ( 2011), 252259.

Pahari , N.P., On Certain Topological Structures of Paranormed
, u,  .  ).
Theorem 4.3 : Let u = (umn) and Smn be a normed space for each m, n 1. Then
Orlicz Space (S ((X ,.),, , u ), F ) of Vector Valued Sequences , International Jour. of Mathematical Archive. , 4(11) ( 2013) 231 241.

Pahari , N.P., On certain topological structures of
(c0 ((Smn ,  . mn),
Proof:
, u ) , G
,u
) is a GKspace.
normed space valued generalized Orlicz function space. International Jour. of Scientific and Research (IJSER) 2(1) ( 2014) 6166.
For each m,n 1, the continuity of linear map

Parashar, S.D. and Choudhary, B., Sequence spaces
defined by Orlicz functions, Indian
Pij : c0 ((Smn ,  . mn), , ) Sij where P
u
ij (s) = sij
J. Pure Appl. Maths., 25(4), (1994), 419428.

Rao, K Chandrasekhra , Space of matrix operators;
1 1/uij
follows from  Pij (s) ij [G(s)]
and so
Bull. Cal. Math. Soc. 80, (1988), 9195.
c0 ((Smn
,  . mn), , ) is a GKspace.
u
REFERENCES

Ruckle. W.H., Sequence spaces ; Pitman Advanced Publishing Programme, (1981).

SavasÂ¸E., Some New Double Sequence Spaces Defined

Basariv, M. and Altundag, S., On generalized paranormed statistically convergent sequence spaces defined by Orlicz function, Handawi. Pub. Cor., J. of Inequality and Applications (2009).

Bhardwaj,V.N.and Bala, I., Banach space valued sequence space M
(X, p), Int. J. of Pure and Appl. Maths., 41(5), (2007), 617626.

Boos, J., Leiger, T. and Zeller, K. ,Consistency theory for SM methods; Acta Math. Hunger.76 (1,2),
(1997), 109142.

Gupta, M., and Kamthan, P.K., Infinite matrices and tensorial transformations; Acta Mathematic a Vietnamica Tom 5 (1), (1980),3341.

Gupta, M. and Patterson, J., The generalized p spaces; Tamkang. Jour. Maths.; 13(2), (1982), 161179.

Kamthan, P.K. and Gupta, M. ,Sequence and series ,
lecture notes; 65 Marcel Dekker Inc., (1980).

Khan, V.A., On a new sequence space defined by Orlicz functions; Common Fac. Sci. Univ. Ankseries;
57( 2), (2008), 2533.

Kolk,E., Topologies in generalized Orlicz sequence spaces; Filomat, 25(4), (2011), 191211.

KÃ¶the, G., Topological vector spaces ; Springer Verlag, Berlin Heidelberg, New York, ( 1970).

Maddox, I.J., Some properties of paranormed sequence spaces, London. J. Math. Soc. , 2(1), (1969),316322.

Maddox, I.J. , Elements of Functional Analysis; Cambridge University, Press, (1970).
by Orlicz Function in nNormed Space;
Hindawi Publishing Corporation, 2011, 592840.

Subramanian, N., Nallaswamy, R. and Saivaraju, Characterization of Entire Sequences via Double Orlicz Space, Hindawi Pub. Corp.Int. Jou.of Math. and Math. Sc.,(2007) ,( 59681).

Wilansky, A. , Modern methods in topological vector spaces; McGrawHill Book Co. Inc. New York, (1978).