Some statistically convergent difference sequence
spaces defined over real 2-normed linear spaces
Hemen Dutta
Abstract. In order to extend the notion of convergence of sequences, statistical convergence of sequences was introduced by Fast [5] and Schoenberg [26]. The main aim of this article is to study the concept of statistical
convergence from difference sequence spaces point of view which are defined over real linear 2-normed spaces.
M.S.C. 2000: 40A05, 46A45, 46E30.
Key words: difference sequence, 2-norm, statistical convergence, paranorm, completeness, solidity, symmetricity.
1
Introduction
Throughout the article w(X), c(X), c0 (X), c̄(X), c̄0 (X), `∞ (X), m(X) and m0 (X)
will represent the spaces of all, convergent, null, statistically convergent, statistically
null, bounded, bounded statistically convergent and bounded statistically null X valued sequences spaces, where (X, k., .k) is a real linear 2-normed space. The zero
sequence is denoted by θ̄ = (θ, θ, θ, . . . ), where θ is the zero element of X.
The notion of difference sequence space was introduced by Kizmaz [18], who studied the difference sequence spaces `∞ (∆), c(∆) and c0 (∆). The notion was further
generalized by Et and Colak [2] by introducing the spaces `∞ (∆n ), c(∆n ) and c0 (∆n ).
Another type of generalization of the difference sequence spaces is due to Tripathy
and Esi [31], who studied the spaces `∞ (∆m ), c(∆m ) and c0 (∆m ).
Tripathy, Esi and Tripathy [32] generalized the above notions and unified these as
follows:
Let m, n be non- negative integers, then for Z a given sequence space we have
Z(∆nm ) = {x = (xk ) ∈ w : (∆nm xk ) ∈ Z} ,
n−1
0
where ∆nm x = (∆nm xk ) = (∆n−1
m xk − ∆m xk+m ) and ∆m xk = xk for all k ∈ N ,
which is equivalent to the following binomial representation:
∆nm xk
Applied Sciences, Vol.12, 2010, pp.
µ ¶
n
X
v n
=
(−1)
xk+mv .
v
v=0
37-47.
c Balkan Society of Geometers, Geometry Balkan Press 2010.
°
38
Hemen Dutta
Taking m = 1, we get the spaces `∞ (∆n ), c(∆n ) and c0 (∆n ) studied by Et and
Colak [2]. Taking n = 1, we get the spaces `∞ (∆m ), c(∆m ) and c0 (∆m ) studied by
Tripathy and Esi [31]. Taking m = n = 1, we get the spaces `∞ (∆), c(∆) and c0 (∆)
introduced and studied by Kizmaz [18].
Recently H. Dutta introduced another type of difference operator ∆n(v,m) , where
m, n are non-negative integers and v = (vk ) is a sequence of non-zero scalars. For
details, one may refer to Dutta [4].
The concept of 2-normed spaces was introduced and studied by Gähler, a German
Mathematician who worked at German Academy of Science, Berlin, in a series of
paper in German language published in Mathematische Nachrichten, see for example
references [3, 9, 10, 11, 12]. This notion which seems to be a two dimensional analogue
of a normed space got the attention of a wider audience after the publication of a
paper by Albert George, White Jr. [34] of USA in 1969 entitled 2-Banach spaces. In
the same year Gähler [12] published another paper on this theme in the same journal.
A.H. Siddiqi delivered a series of lectures on this theme in various conferences in
India and Iran. His joint paper with S. Gähler and S.C. Gupta [17] of 1975 also
provide valuable results related to the theme of this paper. Results up to 1977 were
summarized in the survey paper by A.H. Siddiqi [28]. For other works in this direction
one may refer to [7, 8, 13, 16, 24, 29]. In the recent years, a number of articles devoted
to statistical convergence (see Gürdal and Pehlivan [14]) and its generalization, ideal
convergence(see Gürdal and Şahiner [15]) using 2-norm, have been published.
Let X be a real vector space of dimension d, where 2 ≤ d. A real-valued function
k., .k on X 2 satisfying the following four conditions:
(1)
(2)
(3)
(4)
kx1 , x2 k = 0 if and only if x1 , x2 are linearly dependent,
kx1 , x2 k is invariant under permutation,
kαx1 , x2 k = |α|kx1 , x2 k, for any α ∈ R,
kx + x0 , x2 k ≤ kx, x2 k + kx0 , x2 k
is called a 2-norm on X, and the pair (X, k., .k) is called a 2-normed space.
The notion of statistical convergence was studied at the initial stage by Fast [5]
and Schoenberg [26] independently. Later on it was further investigated by S̆alàt [25],
Fridy [6], Buck [1], Sen and Tripathy [33] and many others. Gürdal and Pehlivan [14]
studied statistical convergence in 2-Banach space.
A subset E of N is said to have density δ(E) if
n
1X
χE (k) exists,
n→∞ n
δ(E) = lim
k=1
where χE is the characteristic function of E.
The following inequality will be used throughout the article:
Let p =
ª a positive sequence of real numbers with 0 < pk ≤ sup pk = G,
© (pk ) be
D = max 1, 2G−1 . Then for all ak , bk ∈ C for all k ∈ N , we have
|ak + bk |
and for λ ∈ C,
pk
pk
≤ D {|ak |
p
+ |bk | k }
©
ª
|λ|pk ≤ max 1, |λ|G .
Some statistically convergent difference sequence spaces
39
The notion of paranormed sequence space was studied at the initial stage by
Simons [27] and Nakano [22]. Later on it was further investigated by Maddox [21],
Lascarides [19], Lascarides and Maddox [20], Nanda [23], B.c. Tripathy [30], Tripathy
and Sen [33] and a number of workers in the field of sequence spaces.
2
Definitions and background
A sequence space E is said to be solid (or normal) if (xk ) ∈ E implies (αk xk ) ∈ E
for all sequences of scalars (αk ) with |αk | ≤ 1 for all¡k ∈ N¢ .
A sequence space E is said to be symmetric if xπ(k) ∈ E whenever (xk ) ∈ E,
where π is a permutation on N .
A sequence (xk ) is said to be statistically convergent to L if for every ε > 0,
δ ({k ∈ N : |xk − L| ≥ ε}) = 0.
For L = 0, we say this is statistically null.
Throughout c̄, c¯0 denote the classes of all statistically convergent and statistically
null sequences respectively.
A sequence (xk ) in a 2-normed space (X, k., .k) is said to converge to some L ∈ X
in the 2-norm if
lim kxk − L, u1 k = 0, for every u1 ∈ X.
k→∞
A sequence (xk ) in a 2-normed space (X, k., .k) is said to be Cauchy with respect
to the 2-norm if
lim kxk − xl , u1 k = 0, for every u1 ∈ X.
k, l→∞
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.
We introduce the following definitions in this article. Let m and n be two nonnegative integers and p = (pk ) be a sequence of strictly positive real numbers. Then
³
´
c̄ k., .k, ∆n(m) , p =
n
o
stat
(xk ) ∈ w(X) : (k∆n(m) xk − L, zk)pk −−→ 0, for every z ∈ X and some L ∈ X ,
³
´ n
o
stat
c̄0 k., .k, ∆n(m) , p = (xk ) ∈ w(X) : (k∆n(m) xk , zk)pk −−→ 0, for every z ∈ X ,
We procure the following definition for the sake of completeness:
¾
³
´ ½
n
n
pk
`∞ k., .k, ∆(m) , p = (xk ) ∈ w(X) : sup(k∆(m) xk , zk) < ∞, for every z ∈ X ,
k≥1
The following definition is introduced:
³
´
W k., .k, ∆n(m) , p =
(
(xk ) ∈ w(X) : lim
j→∞
j
X
k=1
)
(k∆n(m) xk
− L, zk)
pk
= 0, for every z ∈ X and some L ∈ X
.
40
Hemen Dutta
We write
and
³
´
³
´
³
´
m k., .k, ∆n(m) , p = c̄ k., .k, ∆n(m) , p ∩ `∞ k., .k, ∆n(m) , p
³
´
³
´
³
´
m0 k., .k, ∆n(m) , p = c̄0 k., .k, ∆n(m) , p ∩ `∞ k., .k, ∆n(m) , p ,
n−1
0
where (∆n(m) xk ) = (∆n−1
(m) xk − ∆(m) xk−m ) and ∆(m) xk = xk for all k ∈ N , which is
equivalent to the following binomial representation:
µ ¶
n
X
n
v n
xk−mv .
∆(m) xk =
(−1)
v
v=0
In the above expansion we take xk = 0 for non-positive values of k.
The main aim behind considering the generalized difference operator ∆n(m) is that
we can derive several other spaces from the above constructed spaces for particular
values of m and n. In particular for n = 0, the above spaces reduce to the spaces
c̄ (k., .k, p), c̄0 (k., .k, p), `∞ (k., .k, p), W (k., .k, p), m (k., .k, p) and m0 (k., .k, p) respectively.
Again if we replace the base space X, which is a real³ linear ´
2-normed
space
³
´
n
n
by C, complete normed linear space, we get the spaces c̄ ∆(m) , p , c̄0 ∆(m) , p ,
³
´
³
´
³
´
³
´
`∞ ∆n(m) , p , W ∆n(m) , p , m ∆n(m) , p and m0 ∆n(m) , p respectively.
Further if we take X = C, pk = l, a constant for all k ∈ N and n = 0, we get the
spaces c̄, c̄0 , `∞ , W , m and m0 respectively.
First we procure some known results; those will help in establishing the results of
this article.
Lemma 2.1. ([33]) For two sequences pk and (tk ) we have m0 (p) ⊇ m0 (t) if and
only if lim inf ptkk > 0, where K⊆ N such that δ(K) = 1.
k∈K
Lemma 2.2. ([33]) Let h = inf pk and G = sup pk , then the following are equivalent:
(i) G < ∞ and h> 0,
(ii) m(p) = m.
We now cite the following two known 2-normed spaces.
Example 2.1. Consider the spaces Z where Z=`∞ , c and c0 of real sequences. Let
us define:
kx, yk = sup sup |xi yj − xj yi |, where x = (x1 , x2 , . . . ) and y = (y1 , y2 , . . . ) ∈ Z.
i∈N j∈N
Then k., .kE is a 2-norm on Z.
Example 2.2. Let us take X = R2 and consider the function on X defined as:
¯¶
µ¯
¯x11 x12 ¯
¯
¯ , where xi = (xi1 , xi2 ) ∈ R2 f or each i = 1, 2.
kx1 , x2 kE = abs ¯
x21 x22 ¯
Then k., .kE is a 2-norm on X known as Euclidean 2-norm.
Remark 2.1. Every closed linear subspace of an arbitrary linear normed space E,
different from E, is a nowhere dense set in E.
Some statistically convergent difference sequence spaces
3
41
Main results
In this section we mainly investigate
several
topological
³
´ linear
³
´ and³algebraic properties
´
relevant to the spaces c̄0 k., .k, ∆n(m) , p , c̄ k., .k, ∆n(m) , p , m k., .k, ∆n(m) , p and
³
´
m0 k., .k, ∆n(m) , p respectively.
Theorem 3.1. Let p = (pk ) be³a bounded sequence
´
³real numbers.´
´ ³ of strictly positive
n
n
Then the classes of sequences c̄0 k., .k, ∆(m) , p , c̄ k., .k, ∆(m) , p , m0 k., .k, ∆n(m) , p
³
´
and m k., .k, ∆n(m) , p are linear spaces.
³
´
Proof. We prove the theorem only for the space c̄ k., .k, ∆n(m) , p and for the other
spaces it will follow on
arguments.
³ applying similar
´
Let (xk ), (yk ) ∈ c̄ k., .k, ∆n(m) , p . Then there exist L, J ∈ X such that for every
z∈X
³
´pk
stat
−−→ 0
k∆n(m) xk − L, zk
and
³
´pk
k∆n(m) yk − J, zk
stat
−−→ 0.
Let α, β be scalars. Then we have for every z ∈ X
³
´pk ³
´p k
k∆n(m) (αxk + βyk ) − (αL + βJ), zk
= k∆n(m) α(xk − L) + ∆n(m) β(yk − J), zk
³
´pk
≤ |α|k∆n(m) xk − L, zk + |β|k∆n(m) yk − J, zk
³
´pk
³
´pk
≤ D|α|G k∆n(m) xk − L, zk
+ D|β|G k∆n(m) yk − J, zk
, where G = sup pk .
stat
³
−−→ 0, as k → ∞.
´
Hence c̄ k., .k, ∆n(m) , p is a linear space.
¤
³
´
³
´
Theorem 3.2. The spaces m0 k., .k, ∆n(m) , p and m k., .k, ∆n(m) , p are paranormed
spaces, paranormed by
³
´ pHk
n
g(x) = sup
k∆(m) xk , zk
, where H = max{1, sup pk }.
k∈N, z∈X
k
Proof. Clearly g(x) = g(−x); x = θ implies g(θ) = 0. Now
³
´ pHk
g(x + y) = sup
k∆n(m) (xk + yk ), zk
.
k∈N, z∈X
³
≤
sup
k∈N, z∈X
´ pHk
k∆n(m) xk , zk
This implies that
g(x + y) ≤ g(x) + g(y).
³
+
sup
k∈N, z∈X
´ pHk
k∆n(m) yk , zk
.
42
Hemen Dutta
The continuity of the scalar multiplication follows from the following equality:
³
´ pHk
g(λx) = sup
k∆n(m) (λxk ), zk
.
k∈N, z∈X
=
sup
k∈N, z∈X
³
´ pHk
|λ|k∆n(m) xk , zk
≤ max (1, |λ|)
sup
k∈N, z∈X
.
´ pHk
³
|λ|k∆n(m) xk , zk
.
= max (1,
³ |λ|) g(x).
´
³
´
Hence the spaces m0 k., .k, ∆n(m) , p and m k., .k, ∆n(m) , p are paranormed by g. ¤
Remark 3.1. For any two sequences p = (pk ) and t = (tk ) of ³positive real num´
bers and for any two 2-norms k., .k1 and k., .k2 on X we have Z k., .k1 , ∆n(m) , p ∩
´
³
Z k., .k2 , ∆n(m) , t 6= φ, where Z = c̄, m, c̄0 , m0 .
Proof. The proof follows from the fact that the zero element belongs to each of the
classes of sequences involved in the intersection.
¤
³
´
Theorem 3.3. The spaces Z k., .k, ∆n(m) , p are not solid in general, where Z =
c̄, m, c̄0 , m0 .
Proof. To show that the spaces are not solid in general, consider the following examples.
Example 3.1. Let m = 3, n = 1 and consider the 2-normed space as defined in
Example 2.1. Let pk = 5 for all k ∈ N . Consider the sequence (xk ), where
xk = (xik´)
³
is defined by (xik ) = (k, k, k, . . . ) for each fixed k ∈ N . Then (xk ) ∈ Z k., .k, ∆1(3) , p
³
´
for Z = c̄, m. Let αk = (−1)k , then (αk xk ) ∈
/ Z k., .k, ∆1(3) , p for Z = c̄, m. Thus
³
´
Z k., .k, ∆n(m) , p for Z = c̄, m are not solid in general.
¤
Example 3.2. Let m = 3, n = 1 and consider the 2-normed space as defined in
Example 2.1. Let pk = 1 for all k odd and pk = 2 for all k even. Consider the
i
sequence (xk ), where
) is defined by (xik ) = (3, 3, 3, . . . ) for each fixed k ∈ N .
³ xk = (xk´
Then (xk ) ∈ Z k., .k, ∆1(3) , p for Z = c̄0 , m0 . Let αk = (−1)k , then (αk xk ) ∈
/
³
´
³
´
Z k., .k, ∆1(3) , p for Z = c̄0 , m0 . Thus Z k., .k, ∆n(m) , p for Z = c̄0 , m0 are not
solid in general.
¤
³
´
Theorem 3.4. The spaces Z k., .k, ∆n(m) , p are not symmetric in general, where
Z = c̄, m, c̄0 , m0 .
Proof. To show that the spaces are not symmetric in general, consider the following
example.
Example 3.3. Let m = 2, n = 2 and consider the 2-normed space as defined in
Example 2.2. Let pk = 2 for all k odd and pk = 1 for all k even. Consider the
sequence (xk ) defined by xk = (k, k) for each fixed k ∈ N . Then ∆2(2) xk = xk −
Some statistically convergent difference sequence spaces
43
³
´
2xk−2 + xk−4 , k ∈ N . Hence (xk ) ∈ Z k., .k, ∆2(2) , p for Z = c̄, m, c̄0 , m0 . Let (yk )
be a rearrangement of (xk ), which is defined as follows:
(yk ) = {x1 , x2 , x4 , x3 , x9 , x5 , x16 , x6 , x25 , x7 , x36 , x8 , x49 , x10 , . . . }.
³
´
Then (yk ) ∈
/ Z k., .k, ∆2(2) , p for Z = c̄, m, c̄0 , m0 . Hence for Z = c̄, m, c̄0 , m0 , the
´
³
¤
spaces Z k., .k, ∆n(m) , p are not symmetric in general.
Remark 3.2. For two sequences (pk ) and (tk ) we have
³
´
³
´
m0 k., .k, ∆n(m) , p ⊇ m0 k., .k, ∆n(m) , t
pk
k∈K tk
if and only if lim inf
> 0, where K ⊆ N such that δ(K) = 1.
³ ∆n x
´
k
Proof. If we take (yk ) = k (m)
, zk for all k ∈ N , then the result follows from the
ρ
Lemma 2.1.
¤
Remark 3.3. For two sequences (pk ) and (tk ) we have
³
´
³
´
m0 k., .k, ∆n(m) , p = m0 k., .k, ∆n(m) , t
pk
k∈K tk
if and only if lim inf
tk
k∈K pk
> 0 and lim inf
> 0, where K ⊆ N such that δ(K) = 1.
Proof. This result is a consequence of the above result.
¤
Remark 3.4. Let h = inf pk and G = sup pk , then the following are equivalent:
(i) G =³ sup pk and h´> 0, ³
´
(ii) m k., .k, ∆n(m) , p = m k., .k, ∆n(m)
³ ∆n x
´
k
Proof. Taking (yk ) = k (m)
,
zk
for all k ∈ N and using the Lemma 2.2, we get
ρ
the result.
¤
Theorem 3.5. Let p = (p
bounded
real
³ k ) be a sequence
´ of non-negative
³
´
³ numbers such
´
n
n
that inf pk > 0. Then m k., .k, ∆(m) , p = W k., .k, ∆(m) , p ∩ `∞ k., .k, ∆n(m) , p .
³
´
³
´
Proof. Let (xk ) ∈ W k., .k, ∆n(m) , p ∩ `∞ k., .k, ∆n(m) , p . Then for a given ε > 0,
we have
j ³
X
´pk
k∆n(m) xk − L, zk
o
n
≥ card k ≤ j : (k∆n(m) xk − L, zk)pk ≥ ε ε.
k=1
³
´
From the above inequality, it follows that (xk ) ∈ m k., .k, ∆n(m) , p .
³
´
Conversely let (xk ) ∈ m k., .k, ∆n(m) , p and ρ > 0 be such that
stat
(k∆n(m) xk − L, zk)pk −−→ 0, for every z ∈ X and some L ∈ X.
44
Hemen Dutta
³
´ pHk
k∆n(m) xk
For a given ε > 0, let B = sup
− L, zk
< ∞, where H = max{1, sup pk }.
k
n
o
³
´
Let Lj = k ≤ j : (k∆n(m) xk − L, zk)pk ≥ 2ε . Since (xk ) ∈ m k., .k, ∆n(m) , p , so
Card{Lj }
j
→ 0 as j → ∞. Let n0 > 0 be such that
Then for all j > n0 , we have
1
j
j ¡
¢pk
P
k∆n
(m) xk − L, zk
P ¡
¢pk
k∆n
+
(m) xk − L, zk
=
1
j
≤
j−Card{Lj } ε
.2
j
k=1
³
k∈L
/ j
Card{Lj }
j
´
Hence (xk ) ∈ W k., .k, ∆n(m) , p ∩ `∞
³
+
Card{Lj }
.B H
j
1
j
≤
ε
2B H
<
P ¡
k∈Lj
ε
2
+
for all j > n0 .
k∆n
(m) xk − L, zk
ε
2
¢pk
= ε.
´
k., .k, ∆n(m) , p .
¤
The following result is a consequence of the above theorem.
Corollary 3.1. Let (pk ) and (tk ) be two bounded sequences of real numbers such that
inf pk > 0 and inf tk > 0. Then
³
´
³
´
³
´
³
´
W k., .k, ∆n(m) , p ∩ `∞ k., .k, ∆n(m) , p = W k., .k, ∆n(m) , t ∩ `∞ k., .k, ∆n(m) , t .
³
´
Theorem 3.6. Let (X, k., .k) be a 2-Banach space, then the spaces m k., .k, ∆n(m) , p
³
´
and m0 k., .k, ∆n(m) , p are complete.
³
´
Proof. We prove the result for the space m0 k., .k, ∆n(m) , p and for the other space
i
it will
³ follow on ´applying similar arguments. Let (x ) be a Cauchy sequence in
m0 k., .k, ∆n(m) , p . Then for a given ε(0 < ε < 1), there exists a positive integer n0
such that g(xi − xj ) < ε, for all i, j ≥ n0 . This implies that
³
sup
k∈N, z∈X
k∆n(m) xik − ∆n(m) xjk , zk
´ pHk
< ε,
for all i, j ≥ n0 . It follows that for every z ∈ X,
³
´
k∆n(m) (xik − xjk ), zk < ε, for each k ≥ 1 and i, j ≥ n0 .
Hence (∆n(m) xik ) is a Cauchy sequence in the 2-Banach space X for all k ∈ N . Thus
(∆n(m) xik ) is convergent in X for all k ∈ N . For simplicity, let lim ∆n(m) xik = yk for
each k ∈ N . Let k = 1, then we have
(3.1)
lim
i→∞
∆n(m) xi1
i→∞
µ ¶
n
X
v n
= lim
(−1)
xi1−mv = lim xi1 = y1 .
i→∞
i→∞
v
v=0
Similarly, we have,
(3.2)
lim ∆n(m) xik = lim xik = yk , for k = 1, . . . , nm.
i→∞
i→∞
Some statistically convergent difference sequence spaces
45
Thus from (3.1) and (3.2), we have lim xi1+nm exists. Let lim xi1+nm = x1+nm .
i→∞
i→∞
Proceeding in this way inductively, we have lim xik = xk , say exists for each k ∈ N .
i→∞
Now we have for all i, j ≥ n0 ,
³
sup
k∈N, z∈X
´ pHk
k∆n(m) (xik − xjk ), zk
(
³
⇒ lim
j→∞
sup
k∈N, z∈X
³
⇒
sup
k∈N, z∈X
k∆n(m) (xik
k∆n(m) (xik
<ε
´ pHk
−
xjk ), zk
− xk ), zk
´ pHk
)
< ε, for all i ≥ n0
< ε, for all i ≥ n0 .
³
´
³
´
It follows that (xi − x) ∈ m0 k., .k, ∆n(m) , p . Since (xi ) ∈ m0 k., .k, ∆n(m) , p and
³
´
³
´
m0 k., .k, ∆n(m) , p is a linear space, so we have x = xi −(xi −x) ∈ m0 k., .k, ∆n(m) , p .
This completes the proof.
¤
³
´
³
´
As consequence, it follows that m0 k., .k, ∆n(m) , p and m k., .k, ∆n(m) , p are closed
³
´
subspaces of `∞ k., .k, ∆n(m) , p . Since the inclusion relations
³
´
³
´
m0 k., .k, ∆n(m) , p ⊂ `∞ k., .k, ∆n(m) , p ,
³
´
³
´
m k., .k, ∆n(m) , p ⊂ `∞ k., .k, ∆n(m) , p
are strict, we have the following result.
³
´
³
´
Corollary 3.2. The spaces m0 k., .k, ∆n(m) , p and m k., .k, ∆n(m) , p are nowhere
³
´
dense subsets of `∞ k., .k, ∆n(m) , p .
4
Conclusions
In this paper we introduce the difference sequence spaces c̄0 (k., .k, ∆n(m) , p),
c̄(k., .k, ∆n(m) , p), m(k., .k, ∆n(m) , p), m0 (k., .k, ∆n(m) , p), `∞ (k., .k, ∆n(m) , p) and W (k., .k,
∆n(m) , p) with base space a real linear 2-normed space. We study the spaces c̄0 (k., .k,
∆n(m) , p), c̄(k., .k, ∆n(m) , p), m(k., .k, ∆n(m) , p) and m0 (k., .k, ∆n(m) , p) with the help of the
spaces `∞ (k., .k, ∆n(m) , p) and W (k., .k, ∆n(m) , p) for different properties including linearity, existence of paranorm and investigate the spaces for solidity and symmetricity.
Further we prove that the spaces m(k., .k, ∆n(m) , p) and m0 (k., .k, ∆n(m) , p) are complete
paranormed spaces when the base space is a 2-Banach space.
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Authors’ addresses:
Hemen Dutta
Department of Mathematics, Gauhati University,
Kokrajhar Campus, Assam, India.
E-mail: hemen dutta08@rediffmail.com