Khan et al., Cogent Mathematics (2016), 3: 1122257
http://dx.doi.org/10.1080/23311835.2015.1122257
PURE MATHEMATICS | RESEARCH ARTICLE
On Zweier paranorm I-convergent double sequence
spaces
Vakeel A. Khan1*, Nazneen Khan1 and Yasmeen Khan1
Received: 24 August 2015
Accepted: 31 October 2015
Published: 25 January 2016
*Corresponding author: Vakeel A. Khan
Department of Mathematics, Aligarh
Muslim University, Aligarh 202002, India
E-mail: [email protected]
Abstract: In this article, we introduce the Zweier Paranorm I-convergent double
sequence spaces 2 I (q), 2 I0 (q) and 2 I∞ (q) for q = (qij ), a sequence of positive real
numbers. We study some algebraic and topological properties on these spaces.
Subjects: Engineering Technology; Mathematics Statistics; Science; Technology
Keywords: ideal; filter; I-convergence; I-nullity; paranorm
Reviewing editor:
Lishan Liu, Qufu Normal University,
China
1. Introduction
Additional information is available at
the end of the article
𝜔 = {x = (xk ) : xk ∈ IR or C},
I
I be the sets of all natural, real and complex numbers, respectively. We write
Let IN, IR and C
the space of all real or complex sequences.
Let l∞ , c and c0 denote the Banach spaces of bounded, convergent and null sequences, respectively, normed by ||x||∞ = sup |xk |.
k
The following subspaces of 𝜔 were first introduced and discussed by Maddox (1969).
ABOUT THE AUTHORS
PUBLIC INTEREST STATEMENT
Vakeel A. Khan received the MPhil and PhD
degrees in Mathematics from Aligarh Muslim
University, Aligarh, India. Currently he is a senior
assistant professor at Aligarh Muslim University,
Aligarh, India. A vigorous researcher in the
area of Sequence Spaces , he has published a
number of research papers in reputed national
and international journals, including Numerical
Functional Analysis and Optimization (Taylor’s
and Francis), Information Sciences (Elsevier),
Applied Mathematics Letters Applied Mathematics
(Elsevier), A Journal of Chinese Universities
(Springer- Verlag, China).
Nazneen Khan received the MPhil and PhD
degrees in Mathematics from Aligarh Muslim
University, Aligarh, India. Currently she is an
assistant professor at Taibah University, Kingdom
of Saudi Arabia, Madina. Her research interests are
Functional Analysis, sequence spaces and double
sequences.
Yasmeen Khan received MSc and MPhil from
Aligarh Muslim University, and is currently a PhD
scholar at Aligarh Muslim University. Her research
interests are Functional Analysis, sequence spaces
and double sequences
The term sequence has a great role in analysis.
Sequence spaces play an important role in various
fields of real analysis, complex analysis, functional
analysis and Topology. They are very useful tools
in demonstrating abstract concepts through
constructing examples and counter examples.
Convergence of sequences has always remained a
subject of interest to the researchers. Later on, the
idea of statistical convergence came into existence
which is the generalization of usual convergence.
Statistical convergence has several applications
in different fields of Mathematics like Number
Theory, Trigonometric Series, Summability Theory,
Probability Theory, Measure Theory, Optimization
and Approximation Theory. The notion of Ideal
convergence (I-convergence) is a generalization of
the statistical convergence and equally considered
by the researchers for their research purposes since
its inception.
© 2016 The Author(s). This open access article is distributed under a Creative Commons Attribution
(CC-BY) 4.0 license.
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l(p) : = {x ∈ 𝜔 :
∑
k
|xk |pk < ∞},
l∞ (p) : = {x ∈ 𝜔 : sup |xk |pk < ∞},
k
I
c(p) : = {x ∈ 𝜔 : lim |xk − l|pk = 0, for some l ∈ C},
k
c0 (p) : = {x ∈ 𝜔 : lim |xk |pk = 0, },
k
where p = (pk ) is a sequence of strictly positive real numbers.
After then Lascarides (1971, 1983) defined the following sequence spaces
l∞ {p} = {x ∈ 𝜔 : there exists r < 0 such that sup |xk r|pk tk < ∞},
k
c0 {p} = {x ∈ 𝜔 : there exists r < 0 such that lim |xk r|pk tk = 0, },
k
l{p} = {x ∈ 𝜔 : there exists r < 0 such that
∞
∑
k=1
|xk r|pk tk < ∞},
where tk = pk , for all k ∈ IN.
−1
A double sequence of complex numbers is defined as a function x : ℕ × ℕ → ℂ. We denote a
double sequence as (xij ) where the two subscripts run through the sequence of natural numbers
independent of each other. A number a ∈ ℂ is called a double limit of a double sequence (xij ) if for
every 𝜖 > 0 there exists some N = N(𝜖) ∈ ℕ such that (Khan & Sabiha, 2011)
|xij − a| < 𝜖,
∀ i, j ≥ N
Therefore we have,
2𝜔
= {x = (xij ) ∈ IR or C},
I
the space of all real or complex double sequences.
Each linear subspace of 𝜔, for example, 𝜆, 𝜇 ⊂ 𝜔 is called a sequence space.
The notion of I-convergence is a generalization of the statistical convergence. At the initial stage
it was studied by Kostyrko, Šalát, and Wilczynski (2000). Later on it was studied by Šalát, Tripathy,
and Ziman (2004), Tripathy and Hazarika (2009) and Demirci (2001).
2. Preliminaries and definitions
Here, we give some preliminaries about the notion of I-convergence and Zweier sequence spaces.
For more details one refer to Das, Kostyrko, Malik, and Wilczyński (2008), Gurdal and Ahmet (2008),
Khan and Khan (2014a, 2014b), Mursaleen and Mohiuddine (2010, 2012), Esi and Sapsizoğlu (2012),
Fadile Karababa and Esi (2012), Khan et al. (2013b).
Definition 2.1 If (X, 𝜌) is a metric space, a set A ⊂ X is said to be nowhere dense if its closure Ā contains no sphere, or equivalently if Ā has no interior points.
Definition 2.2 Let X be a non-empty set. Then a family of sets I⊆ 2X(2X denoting the power set of X)
is said to be an ideal in X if
(i) � ∈I
(ii) I is finitely additive i.e. A, B∈I ⇒ A ∪ B∈I.
(iii) I is hereditary i.e. A ∈I, B ⊆A⇒B∈I.
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An Ideal I⊆ 2X is called non-trivial if I≠ 2X. A non-trivial ideal I⊆ 2X is called admissible if
{{x} : x ∈ X} ⊆I.
A non-trivial ideal I is maximal if there cannot exist any non-trivial ideal J≠I containing I as a subset.
Definition 2.3 A double sequence (xij ) is said to be
(i) I-convergent to a number L if for every 𝜖 > 0,
{(i, j) ∈ ℕ × ℕ : |xij − L| ≥ 𝜖} ∈ I.
In this case we write I − lim xij = L.
(ii) A double sequence (xij ) is said to be I-null if L = 0 . In this case, we write
I − lim xij = 0.
(iii) A double sequence (xij ) is said to be I-cauchy if for every 𝜖 > 0 there exist numbers m = m(𝜖),
n= n(𝜖) such that
{(i, j) ∈ ℕ × ℕ : |xij − xmn | ≥ 𝜖} ∈ I.
(iv) A double sequence (xij ) is said to be I-bounded if there exists M > 0 such that
{(i, j) ∈ ℕ × ℕ : |xij | < M}.
Definition 2.4 A double sequence space E is said to be solid or normal if (xij ) ∈ E implies (𝛼ij xij ) ∈ E for
all sequence of scalars (𝛼ij ) with |𝛼ij | < 1 for all (i,j) ∈ ℕ × ℕ.
Definition 2.5 Let X be a linear space. A function g : X ⟶ R is called a paranorm, if for all x, y, z ∈ X,
(i) g(x) = 0 if x = 𝜃,
(ii) g(−x) = g(x),
(iii) g(x + y) ≤ g(x) + g(y),
(iv) If (𝜆n ) is a sequence of scalars with 𝜆n → 𝜆 (n → ∞) and xn , a ∈ X with xn → a (n → ∞) , in the
sense that g(xn − a) → 0 (n → ∞) , in the sense that g(𝜆n xn − 𝜆a) → 0 (n → ∞).The concept of
paranorm is closely related to linear metric spaces. It is a generalization of that of absolute
value (see Lascarides, 1971; Tripathy & Hazarika, 2009).
A sequence space 𝜆 with linear topology is called a K-space provided each of maps pi ⟶ CI defined
by pi (x) = xi is continuous for all i ∈ IN.
A K-space 𝜆 is called an FK-space provided 𝜆 is a complete linear metric space.
An FK-space whose topology is normable is called a BK-space.
Let 𝜆 and 𝜇 be two sequence spaces and A = (ank ) is an infinite matrix of real or complex numbers
ank, where n, k ∈ IN. Then we say that A defines a matrix mapping from 𝜆 to 𝜇, and we denote it by
writing A : 𝜆 ⟶ 𝜇.
If for every sequence x = (xk ) ∈ 𝜆 the sequence Ax = {(Ax)n }, the A transform of x is in 𝜇, where
(Ax)n =
∑
k
ank xk ,
(n ∈ IN)
(1)
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By (𝜆 : 𝜇), we denote the class of matrices A such that A : 𝜆 ⟶ 𝜇.
Thus, A ∈ (𝜆:𝜇) if and only if series on the right side of (1) converges for each n ∈ IN and every x ∈ 𝜆.
The approach of constructing new sequence spaces by means of the matrix domain of a particular
limitation method have been recently employed by Altay, Başar, and Mursaleen (2006), Başar and
Altay (2003), Malkowsky(1997), Ng and Lee (1978) and Wang (1978).
Şengönül (2007) defined the sequence y = (yi ) which is frequently used as the Z p transform of the
sequence x = (xi ) i.e.
yi = pxi + (1 − p)xi−1
where x−1 = 0, 1 < p < ∞ and Z p denotes the matrix Z p = (zik ) defined by
zik =
{
p, (i = k), 1 − p,
(i − 1 = k); (i, k ∈ IN), 0, otherwise.
Following Basar and Altay (2003), Şengönül (2007), introduced the Zweier sequence spaces  and
0 as follows
 = {x = (xk ) ∈ 𝜔 : Z p x ∈ c}
0 = {x = (xk ) ∈ 𝜔 : Z p x ∈ c0 }
Here, we quote below some of the results due to Şengönül (2007) which we will need in order to
establish the results of this article.
Theorem 2.1 The sets  and 0 are the linear spaces with the co-ordinate wise addition and scalar
multiplication which are the BK-spaces with the norm
||x|| = ||x|| = ||Z p x||c .
0
Theorem 2.2 The sequence spaces  and 0 are linearly isomorphic to the spaces c and c0, respectively, i.e.  ≅ c and 0 ≅ c0.
Theorem 2.3 The inclusions 0 ⊂  strictly hold for p ≠ 1.
The following Lemma and the inequality has been used for establishing some results of this
article.
Lemma 2.4 If I ⊂ 2N and M⊆N. If M ∉I then M∩N ∉ I (Şengönül, 2007).
Let p = (pk ) be the bounded sequence of positive reals numbers. For any complex
𝜆, whenever H = sup pk < ∞, we have |𝜆|pk ≤ max(1, |𝜆|H ). Also, whenever H = sup pk we have
k
k
|ak + bk |pk ≤ C(|ak |pk + |bk |pk ) where C = max(1;2H−1 ). (Maddox, 1969) cf. (Khan Ebadullah, Ayhan Esi,
Khan, & Shafiq, 2013a; Khan & Khan, 2014b; Khan & Sabiha, 2011; Malkowsky, 1997; Ng & Lee, 1978).
Recently Khan Ebadullah, Ayhan Esi, Khan, and Shafiq (2013a) introduced various Zweier sequence
spaces the following sequence spaces.
I = {x = (xk ) ∈ 𝜔 : {k ∈ IN : I − lim Z p x = L, for some L} ∈ I},
I0 = {x = (xk ) ∈ 𝜔 : {k ∈ IN : I − lim Z p x = 0} ∈ I},
I∞ = {x = (xk ) ∈ 𝜔 : {k ∈ IN : sup |Z p x| < ∞} ∈ I}.
k
We also denote by
mI = I∞ ∩ I and mI = I∞ ∩ I0 .
0
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In this article, we introduce the following sequence spaces.
For any 𝜖 > 0, we have
I
2  (q)
I
2 0 (q)
I
= {x = (xij ) ∈2 𝜔 : {(i, j) ∈ IN × IN : |Z p x − L|qij ≥ 𝜖} ∈ I, for some L ∈ C};
2 ∞ (q)
= {x = (xij ) ∈ 2 𝜔 : {(i, j) ∈ IN × IN : |Z p x|qij ≥ 𝜖} ∈ I};
= {x = (xij ) ∈2 𝜔 : sup |Z p x|qij < ∞}.
i,j
We also denote by
I
2 m (q)
=2 ∞ (q) ∩2 I (q)
and
I
2 m0 (q)
=2 ∞ (q) ∩2 I0 (q)
where q = (qij ) is a double sequence of positive real numbers.
�
Throughout the article, for the sake of convenience now we will denote by Z p x = x for all x ∈2 𝜔.
3. Main results
Theorem 3.1 The sequence spaces 2 I0 (q), 2 I (q), 2 I∞ (q) are linear spaces.
Proof We shall prove the result for the space 2 I (q).
The proof for the other spaces will follow similarly.
Let (xij ), (yij ) ∈2 I (q) and let 𝛼, 𝛽 be scalars. Then for a given 𝜖 > 0. we have
𝜖
, for some L1 ∈ C}
I ∈I
2M1
�
𝜖
{(i, j) ∈ IN × IN : |yij − L2 |qij ≥
, for some L2 ∈ C}
I ∈I
2M2
�
{(i, j) ∈ IN × IN : |xij − L1 |qij ≥
where
M1 = D.max{1, sup |𝛼|qij }
i,j
M2 = D.max{1, sup |𝛽|qij }
i,j
and
D = max{1, 2H−1 } where H = sup qij ≥ 0.
i,j
Let
𝜖
, for some L1 ∈ C}
I
2M1
�
𝜖
A2 = {(i, j) ∈ IN × IN : |yij − L2 |qij <
, for some L2 ∈ C}
I
2M2
�
A1 = {(i, j) ∈ IN × IN : |xij − L1 |qij <
be such that Ac1 , Ac2 ∈ I. Then
�
�
A3 = {(i, j) ∈ IN × IN : |(𝛼xij + 𝛽yij ) − (𝛼L1 + 𝛽L2 )|qij ) < 𝜖}
�
𝜖
|𝛼|qij .D}
⊇ {(i, j) ∈ IN × IN : |𝛼|qij |xij − L1 |qij <
2M1
�
𝜖
∩ {(i, j) ∈ IN × IN : |𝛽|qij |yij − L2 |qij <
|𝛽|qij .D}
2M2
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�
�
Thus Ac3 ⊆ Ac1 ∪ Ac2 ∈ I. Hence (𝛼xij + 𝛽yij ) ∈ 2 I (q). Therefore 2 I (q) is a linear space. Proof of 2 Z0I (q)
follows since it is a special case of 2 Z I (q).
Remark The sequence spaces 2 mIZ (q), 2 mIZ (q) , are linear spaces since each is an intersection of
0
two of the linear spaces in Theorem 3.1.
Theorem 3.2 Let (qij ) ∈ 2 l∞. Then 2 mI (q) and 2 mI (q) are paranormed spaces, paranormed by
0
q
ij
g(x� ) = sup |xij� | M where M = max{1, sup qij }.
i,j
i,j
Proof Let x� = (xij� ), y � = (yij� ) ∈ 2 mI (q).
(1) Clearly, g(x� ) = 0 if and only if x� = 0.
(2) g(x� ) = g(−x� ) is obvious.
(3) Since
qij
M
≤ 1 and M > 1, using Minkowski’s inequality, we have
q
ij
q
ij
q
ij
q
ij
q
ij
g(x� + y � ) = g(xij� + yij� ) = sup ∣ xij� + yij� ∣ M ≤ sup(∣ xij� ∣ M + ∣ yij� ∣ M ) ≤ sup ∣ xij� ∣ M + sup ∣ yij� ∣ M
ij
ij
ij
ij
= g(xij� ) + g(yij� ) = g(x� ) + g(y � ).
Therefore, g(x� + y � ) ≤ g(x� ) + g(y � ), for all x� , y � ∈ 2 mIz (q).
(4) Let (𝜆ij ) be a double sequence of scalars with (𝜆ij ) → 𝜆, (i, j → ∞) and x� = (xij� ), x0� = (xi� j ) ∈ 2 mIz (q)
0 0
with g(xij� ) → g(x0 ), (i, j → ∞).
Note that g(𝜆x� ) ≤ max{1, ∣ 𝜆 ∣}g(x� ). Then since the inequality g(xij� ) ≤ g(xij� − x0� ) + g(x0� ) holds by
bounded.
Therefore,
{g(xij� )} q is
subadditivity
of
g,
the
sequence
q
ij
ij
�
�
�
�
�
�
∣ g(𝜆ij xij ) − g(𝜆x0 ) =∣ g(𝜆ij xij ) − g(𝜆xij ) + g(𝜆xij ) − g(𝜆x0 ) ∣ ≤∣ 𝜆ij − 𝜆 ∣ M ∣ g(xij� ) ∣ + ∣ 𝜆 ∣ M ∣ g(xij� ) − g(x0� ) ∣→ 0
as (i, j → ∞). That is to say that the scalar multiplication is continuous. Hence 2 mIz (q) is a paranormed
space.
Theorem 3.3 I
2 m (q)
is a closed subspace of 2 l∞ (q).
Proof Let (xij�(mn) ) be a Cauchy sequence in 2 mI (q) such that x�(mn) → x�. We show that x� ∈ 2 mI (q).
Since (xij�(mn) ) ∈ 2 mI (q), then there exists (amn ) such that
{(i, j) ∈ IN × IN : |x�(mn) − amn | ≥ 𝜖} ∈ I
We need to show that
(amn ) converges to a.
(1) (2) If U = {(i, j) ∈ IN × IN : |xij − a| < 𝜖} then Uc ∈ I.
�
Since (xij�(mn) ) is a Cauchy sequence in 2 mI (q) then for a given 𝜖 > 0, there exists (i0 , j0 ) ∈ IN × IN
such that
�(pq)
sup |xij�(mn) − xij
i,j
|<
𝜖
, for all (m,n),(p,q) ≥ (i0 , j0 )
3
For a given 𝜖 > 0, we have
�(pq)
Bmn,pq = {(i, j) ∈ IN × IN : |xij�(mn) − xij
𝜖
}
3
𝜖
}
3
𝜖
= {(i, j) ∈ IN × IN : |xij�(mn) − amn | < }
3
�(pq)
Bpq = {(i, j) ∈ IN × IN : |xij
Bmn
|<
− apq | <
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Then Bcmn,pq , Bcpq , Bcmn ∈ I.
Let Bc = Bcmn,pq ∩ Bcpq ∩ Bcmn, where
B = {(i, j) ∈ IN × IN : |apq − amn | < 𝜖}.
Then Bc ∈ I.
We choose (i0 , j0 ) ∈ Bc, then for each (m, n), (p, q) ≥ (i0 , j0 ), we have
𝜖
}
3
𝜖
�(pq)
∩ {(i, j) ∈ IN × IN : |xij�(mn) − xij | < }
3
𝜖
∩ {(i, j) ∈ IN × IN : |xij�(mn) − amn | < }
3
�(pq)
{(i, j) ∈ IN × IN : |apq − amn | < 𝜖} ⊇ {(i, j) ∈ IN × IN : |xij
− apq | <
Then (amn ) is a Cauchy sequence of scalars in CI , so there exists a scalar a ∈ CI such that amn → a, as
(m, n) → ∞.
For the next part let 0 < 𝛿 < 1 be given. Then we show that if U = {(i, j) ∈ IN × IN : |xij� − a|qij < 𝛿}, then
Uc ∈ I.
Since x�(mn) → x�, then there exists (p0 , q0 ) ∈ IN × IN such that
�(p0 ,q0 )
P = {(i, j) ∈ IN × IN : |xij
− x� | < (
𝛿 M
) }
3D
which implies that P c ∈ I. The number (p0 , q0 ) can be so chosen that together with (1), we have
Q = {(i, j) ∈ IN × IN : |ap
0 q0
− a|qij < (
𝛿 M
) }
3D
such that Qc ∈ I
Since {(i, j) ∈ IN × IN : |x�(p0 q0 ) − ap q |qij ≥ 𝛿} ∈ I. Then we have a subset S of IN × IN such that IN × IN,
0 0
where
S = {(i, j) ∈ IN × IN : |x�(p0 q0 ) − ap
0 q0
|qij < (
𝛿 M
) }.
3D
Let Uc = P c ∩ Qc ∩ Sc, where U = {(i, j) ∈ IN × IN : |xij� − a|qij < 𝛿}.
Therefore for each (i, j) ∈ Uc, we have
𝛿 M
) }
3D
𝛿
∩ {(i, j) ∈ IN × IN : |x�(p0 q0 ) − ap q |qij < ( )M }
0 0
3D
𝛿
∩ {(i, j) ∈ IN × IN : |ap q − a|qij < ( )M }.
0 0
3D
{(i, j) ∈ IN × IN : |xij� − a|qij < 𝛿} ⊇ {(i, j) ∈ IN × IN : |x�(p0 q0 ) − x|qij < (
Then the result follows.
Theorem 3.4 The spaces 2 mI (q) and 2 mI (q) are nowhere dense subsets of 2 l∞ (q).
0
Proof Since the inclusions 2 mI (q) ⊂2 l∞ (q) and 2 mI (q) ⊂2 l∞ (q) are strict so in view of Theorem 3.3
0
we have the following result.
Theorem 3.5 The spaces 2 mI (q) and 2 mI (q) are not separable.
0
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Proof We shall prove the result for the space 2 mI (q). The proof for the other spaces will follow
similarly.
Let M be an infinite subset of IN × IN of such that M ∈ I. Let
qij =
{
1, if (i, j) ∈ M,
2, otherwise.
�
�
Let P0 = {x� = (xij ) : xij = 0 or 1, for (i, j) ∈ M and xij� = 0, otherwise}.
Clearly P0 is uncountable.
Consider the class of open balls B1 = {B(x� , 12 ) : x� ∈ P0 }.
Let C1 be an open cover of 2 mI (q) containing B1.
Since B1 is uncountable, so C1 cannot be reduced to a countable subcover for 2 mI (q).
Thus 2 mI (q) is not separable.
Theorem 3.6 Let h = inf qij and H = sup qij . Then the following results are equivalent.
i,j
i,j
(a) H < ∞ and h > 0.
(b) 2 I0 (q) = 2 I0 .
Proof Suppose that H < ∞ and h > 0, then the inequalities min{1, sh } ≤ sqij ≤ max{1, sH } hold for
any s > 0 and for all (i, j) ∈ IN × IN. Therefore, the equivalence of (a) and (b) is obvious.
Theorem 3.7 Let (qij ) and (rij ) be two sequences of positive real numbers. Then 2 mIZ (r) = 2 mIZ (q) if
0
0
q
r
and only if lim inf r ij > 0, and lim inf qij > 0, where K ⊆ IN × IN such that Kc ∈ I.
(i,j)∈K
Proof Let lim inf
(i,j)∈K
(i,j)∈K
ij
qij
rij
> 0 and
(xij� )
ij
∈ 2 mI (r). Then there exists 𝛽 > 0 such that qij > 𝛽rij , for all suf0
ficiently large (i, j) ∈ K. Since (xij ) ∈ 2 mI (r) for a given 𝜖 > 0, we have
0
B0 = {(i, j) ∈ IN × IN : |xij |rij ≥ 𝜖} ∈ I
Let G0 = Kc ∪ B0 . Then G0 ∈ I.
Then for all sufficiently large (i, j) ∈ G0 ,
{(i, j) ∈ IN × IN : |xij |qij ≥ 𝜖} ⊆ {(i, j) ∈ IN × IN : |xij |𝛽rij ≥ 𝜖} ∈ I.
Therefore (xij� ) ∈ 2 mI (q).
0
The converse part of the result follows obviously.
The other inclusion follows by symmetry of the two inequalities.
4. Conclusion
The notion of Ideal convergence (I-convergence) is a generalization of the statical convergence and
equally considered by the researchers for their research purposes since its inception. Along with this
the very new concept of double sequences has also found its place in the field of analysis. It is also
being further discovered by mathematicians all over the world. In this article, we introduce paranorm ideal convergent double sequence spaces using Zweier transform. We study some topological
and algebraic properties. Further we prove some inclusion relations related to these new spaces.
Page 8 of 9
Khan et al., Cogent Mathematics (2016), 3: 1122257
http://dx.doi.org/10.1080/23311835.2015.1122257
Acknowledgements
The authors would like to record their gratitude to the
reviewer for his careful reading and making some useful
corrections which improved the presentation of the paper.
Funding
The authors received no direct funding for this research.
Author details
Vakeel A. Khan1
E-mail: [email protected]
Nazneen Khan1
E-mail: [email protected]
Yasmeen Khan1
E-mail: [email protected]
1
Department of Mathematics, Aligarh Muslim University,
Aligarh, 202002, India.
Citation information
Cite this article as: On Zweier paranorm I-convergent
double sequence spaces, Vakeel A. Khan, Nazneen Khan &
Yasmeen Khan, Cogent Mathematics (2016), 3: 1122257.
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