On Zweier paranorm I-convergent double sequence spaces

ABOUT THE AUTHORS 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 (SpringerVerlag, 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 PUBLIC INTEREST STATEMENT 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. Received: 24 August 2015 Accepted: 31 October 2015 Published: 25 January 2016


Introduction
Let IN, IR and I C be the sets of all natural, real and complex numbers, respectively. We write the space of all real or complex sequences.
Let l ∞ , c and c 0 denote the Banach spaces of bounded, convergent and null sequences, respectively, normed by ||x|| ∞ = sup k |x k |.
The following subspaces of were first introduced and discussed by Maddox (1969 where p = (p k ) is a sequence of strictly positive real numbers.
After then Lascarides (1971Lascarides ( , 1983 defined the following sequence spaces A double sequence of complex numbers is defined as a function x : ℕ × ℕ → ℂ. We denote a double sequence as (x ij ) 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 (x ij ) if for every > 0 there exists some N = N( ) ∈ ℕ such that (Khan & Sabiha, 2011) Therefore we have, the space of all real or complex double sequences.
Each linear subspace of , for example, , ⊂ 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), Hazarika (2009) andDemirci (2001).
Definition 2.1 If (X, ) 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⊆ 2 X (2 X 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.
A non-trivial ideal I is maximal if there cannot exist any non-trivial ideal J≠I containing I as a subset.
In this case we write I − lim x ij = L.
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, is a sequence of scalars with n → (n → ∞) and x n , a ∈ X with x n → a (n → ∞) , in the sense that g(x n − a) → 0 (n → ∞) , in the sense that g( n x n − a) → 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 with linear topology is called a K-space provided each of maps A K-space is called an FK-space provided is a complete linear metric space.
An FK-space whose topology is normable is called a BK-space.
Let and be two sequence spaces and A = (a nk ) is an infinite matrix of real or complex numbers a nk , where n, k ∈ IN. Then we say that A defines a matrix mapping from to , and we denote it by writing A : ⟶ . (1) By ( : ), we denote the class of matrices A such that A : ⟶ .
Thus, A ∈ ( : ) if and only if series on the right side of (1) converges for each n ∈ IN and every x ∈ .
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 = (y i ) which is frequently used as the Z p transform of the sequence where x −1 = 0, 1 < p < ∞ and Z p denotes the matrix Z p = (z ik ) defined by Following Basar and Altay (2003), Şengönül (2007), introduced the Zweier sequence spaces  and  0 as follows 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 2.3 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 ⊂ 2 N and M⊆N. If M ∉I then M∩N ∉ I (Şengönül, 2007).
We also denote by In this article, we introduce the following sequence spaces.
For any > 0, we have We also denote by and where q = (q ij ) 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 .
Proof We shall prove the result for the space 2  I (q).
The proof for the other spaces will follow similarly.
Let (x ij ), (y ij ) ∈ 2  I (q) and let , be scalars. Then for a given > 0. we have where and Let be such that A c 1 , A c 2 ∈ I. Then . Therefore 2  I (q) is a linear space. Proof of 2 Z I 0 (q) follows since it is a special case of 2 Z I (q).
Remark The sequence spaces 2 m I Z (q), 2 m I Z 0 (q) , are linear spaces since each is an intersection of two of the linear spaces in Theorem 3.1.
(3) Since q ij M ≤ 1 and M > 1, using Minkowski's inequality, we have . That is to say that the scalar multiplication is continuous. Hence 2 m I z (q) is a paranormed space. Theorem 3.3 2 m I  (q) is a closed subspace of 2 l ∞ (q).
Proof Let (x �(mn) ij ) be a Cauchy sequence in 2 m I  (q) such that x �(mn) → x � . We show that x � ∈ 2 m I  (q). Since (x �(mn) ij ) ∈ 2 m I  (q), then there exists (a mn ) such that We need to show that (1) (a mn ) converges to a. ( Since (x �(mn) ij ) is a Cauchy sequence in 2 m I  (q) then for a given > 0, there exists (i 0 , For a given > 0, we have Then B c ∈ I. We choose (i 0 , j 0 ) ∈ B c , then for each (m, n), (p, q) ≥ (i 0 , j 0 ), we have Then (a mn ) is a Cauchy sequence of scalars in I C, so there exists a scalar a ∈ I C such that a mn → a, as (m, n) → ∞.
For the next part let 0 < < 1 be given. Then we show that which implies that P c ∈ I. The number (p 0 , q 0 ) can be so chosen that together with (1), we have Therefore for each (i, j) ∈ U c , we have Then the result follows.
Theorem 3.4 The spaces 2 m I  (q) and 2 m I  0 (q) are nowhere dense subsets of 2 l ∞ (q).
Proof Since the inclusions 2 m I  (q) ⊂ 2 l ∞ (q) and 2 m I  0 (q) ⊂ 2 l ∞ (q) are strict so in view of Theorem 3.3 we have the following result.
Theorem 3.5 The spaces 2 m I  (q) and 2 m I Proof We shall prove the result for the space 2 m I  (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 Clearly P 0 is uncountable.

Consider the class of open balls
Let C 1 be an open cover of 2 m I  (q) containing B 1 .
Since B 1 is uncountable, so C 1 cannot be reduced to a countable subcover for 2 m I  (q).
Proof Suppose that H < ∞ and h > 0, then the inequalities min{1, s h } ≤ s q ij ≤ max{1, s H } hold for any s > 0 and for all (i, j) ∈ IN × IN. Therefore, the equivalence of (a) and (b) is obvious. (r). Then there exists > 0 such that q ij > r ij , for all sufficiently large (i, j) ∈ K. Since (x ij ) ∈ 2 m I  0 (r) for a given > 0, we have Then for all sufficiently large (i, j) ∈ G 0 , Therefore (x � ij ) ∈ 2 m I  0 (q).
The converse part of the result follows obviously.
The other inclusion follows by symmetry of the two inequalities.

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.