On a class of double difference sequences, their statistical convergence in 2-normed spaces and their duals

(3s.) v. 32 1 (2014): 123–136. ISSN-00378712 in press doi:10.5269/bspm.v32i1.19174 Bol. Soc. Paran. Mat. c SPM –ISSN-2175-1188 on line SPM: www.spm.uem.br/bspm On A class of Double Difference Sequences, their Statistical convergence in 2-normed spaces and their duals P. Baliarsingh abstract: In this article, we determine a new class of difference double se0 quence spaces ℓ∞ 2 (∆ν ), c2 (∆ν ) and c2 (∆ν ) by defining a double difference ∆ν = (xmn ν mn − xm,n+1 ν m,n+1 ) − (xm+1,n ν m+1,n − xm+1,n+1 ν m+1,n+1 ), where ν = (ν mn ) is a fixed double sequence of non-zero real numbers satisfying some conditions and m, n ∈ N, the set of natural numbers. Moreover, we have studied their topological properties and certain inclusion relations. We have also discussed the concept of the statistical convergence of these classes in 2-normed space and found their pα−, pβ−, pγ−duals. Key Words: Difference double sequence space; 2-normed space; natural density; statistical convergence; pα−, pβ−, and pγ− duals. Contents 1 Introduction 123 2 Preliminaries and definitions 124 3 Main results 126 4 The dual spaces 131 1. Introduction Let ω2 be the set of all double sequence spaces of real numbers and ℓ∞ , c and c0 denote the set of linear spaces that are bounded, convergent and null sequences respectively. A double sequence x = (xmn ) is said to be bounded if and only if k x k∞ = sup |xmn | < ∞. m,n Let ℓ∞ 2 denote the space of all bounded double sequence spaces and it is known ∞ that ℓ2 is Banach space (see [18]). A double sequence x = (xmn ) is called convergent (with the limit L) if and only if for every ε > 0, there exists a positive integer 2000 Mathematics Subject Classification: 40A05, 40C05, 40D05 123 Typeset by BSP style. M c Soc. Paran. de Mat. 124 P. Baliarsingh n0 = n0 (ε) such that |xmn − L| < ε ; for all m, n ≥ n0 . The limit L is called double limit or Pringsheim limit of the double sequence x = (xmn ). By c2 and c02 , we denote the space of all convergent and null double sequences. A double sequence x = (xmn ) is called Cauchy sequence if and only if for every ε > 0, there exists n0 = n0 (ε) such that |xmn − xpq | < ε for all m, n, p, q ≥ n0 . In [1] it is known that a double sequence is Cauchy if and only if it is convergent. P Throughout this paper we write supm,n , limm , limm,n and m,n instead of P∞,∞ supm,n≥1 , limm→∞ , limm,n→∞,∞ and m,n=1,1 respectively. The notion of difference sequence space was first introduced by Kızmaz [15] by defining the sequence space X(∆) = {x = (xk ) : ∆x ∈ X}, (1.1) for X = ℓ∞ , c and c0 , where ∆x = (xk − xk+1 ). Let ν = (ν mn ) be any fixed double sequence of non zero real numbers satisfying 1 lim inf |ν mn | m+n = r, m,n (0 < r ≤ ∞) and ν 1m = ν n1 = 0 for all m, n ∈ N \ {1}. Now, we define a class of double difference sequence spaces as follows: n o 2 ℓ∞ (∆ ) = x = (x ) ∈ ω : sup |∆ x | < ∞ , ν mn ν mn 2 m,n n o c02 (∆ν ) = x = (xmn ) ∈ ω2 : lim |∆ν xmn | = 0 , m,n n o 2 c2 (∆ν ) = x = (xmn ) ∈ ω : lim |∆ν xmn − L| = 0, for some L ∈ C . m,n where ∆ν xmn = xmn ν mn − xm,n+1 ν m,n+1 − xm+1,n ν m+1,n + xm+1,n+1 ν m+1,n+1 . 2. Preliminaries and definitions In this part, we give the definition of 2-normed space and investigate some rela0 tions of ℓ∞ 2 (∆ν ), c2 (∆ν ) and c2 (∆ν ) in 2-normed space by introducing the concept of statistical convergence. The idea of 2-normed spaces was first introduced by Gahler [7,8,9]. Later on concept of double sequence spaces and 2-normed spaces have been studied and extended by many authors such as [11,12,13,14,16,17,25] etc. Let X be a real vector space of dimension d, where 2 ≤ d ≤ ∞. A mapping on X, defined by k., .k : X × X → R which satisfies the following four conditions: (i) kx1 , x2 k = 0 if and only if x1 and x2 are linearly dependent; On A class of Double Difference Sequences 125 (ii) kx1 , x2 k = kx2 , x1 k; (iii) kαx1 , x2 k = |α|kx1 , x2 k for any α ∈ R and (iv) kx1 + x′1 , x2 k ≤ kx1 , x2 k + kx′1 , x2 k. The pair (X, k., .k) is called 2-normed space. Standard examples of 2-normed space are R2 equipped with the following 2-norm • kx, yk := |x1 y2 − x2 y1 |, where x = (x1 , x2 ), y = (y1 , y2 ), • kx, yk :=the area of the triangle having vertices 0, x and y (see [11]). In this case, we have the following observations: 1. kx, yk ≥ 0; 2. kx, y + αxk = kx, yk for all x, y ∈ X and α ∈ R; 3. kx, y + zk = kx, yk + kx, zk if x, y, z are linearly independent with dimension d=2. The notion of statistical convergence was introduced by Fast [5] and studied by various authors [2,4,6,19,20,21,22,23,24]. We recall some concepts connecting with statistical convergence. Let K be a subset of N, set of natural numbers and Kn be a set i.e. Kn = {k ∈ K : k < n}, then the natural density of K is given by δ(K) = limn→∞ |Knn | , provided the limit exists, where |Kn | denotes the number of elements in Kn . Finite subsets have natural density zero. Definition 2.1. A sequence x = (xk ) is said to be statistically convergent to a number L, if for every ǫ > 0 lim n 1 |{k < n : |xk − L| ≥ ǫ}| = 0. n Equivalently, the natural density of the given set i.e. δ({k < n : |xk − L| ≥ ǫ}) = 0. In this case we write st − limk xk = L or xk → L(S) and S = {x ∈ ω : st − lim xk = L, for some L} k Definition 2.2. A double sequence x = (xmn ) in 2-normed space (X, k., .k), is said to be double statistically convergent to a number L, if for every ǫ > 0 and z ∈ X lim 1 m,n mn |{p ≤ m, q ≤ n : kxpq − L, zk ≥ ǫ}| = 0. 126 P. Baliarsingh Equivalently, the natural density of the given set i.e. δ({(p, q) ∈ N × N : kxpq − L, zk ≥ ǫ}) = 0. In this case we write st2 − limmn kxmn , zk = kL, zk or xmn → LS(X, k., .k) . Definition 2.3. A double sequence x = (xmn ) in 2-normed space (ℓ∞ 2 (∆ν ), k., .k), is said to be double ∆ν -statistically convergent to a number L, if for every ǫ > 0 and z ∈ ℓ∞ 2 (∆ν ) lim 1 m,n mn |{p ≤ m, q ≤ n : k∆ν xpq − L, zk ≥ ǫ}| = 0. Equivalently, the natural density of the given set i.e. δ({(p, q) ∈ N × N : k∆ν xpq − L, zk ≥ ǫ}) = 0. In this case we write st2 −limmn k∆ν xmn , zk = kL, zk or xmn → LS(ℓ∞ 2 (∆ν ), k., .k). Definition 2.4. A double sequence x = (xmn ) in 2-normed space (ℓ∞ 2 (∆ν ), k., .k), is said to be double ∆ν -statistically Cauchy if for every ǫ > 0, there exist M (ǫ), N (ǫ) and z ∈ ℓ∞ 2 (∆ν ) such that δ(|{(p, q) ∈ N × N : k∆ν (xpq − xMN ), zk ≥ ǫ}|) = 0. 3. Main results In this section, we discuss some inclusion relations and basic topological prop0 erties of the spaces ℓ∞ 2 (∆ν ), c2 (∆ν ) and c2 (∆ν ). Moreover, we determine some interesting results of the space ℓ∞ 2 (∆ν ) by introducing the concept of double statistical convergence in 2-normed space. Now, we state the following two theorems without proof. ∞ Theorem 3.1. c02 (∆ν ) ⊂ c2 (∆ν ) ⊂ ℓ∞ 2 (∆ν ) ⊂ ℓ2 and the inclusion is strict. 0 Theorem 3.2. The spaces ℓ∞ 2 (∆ν ), c2 (∆ν ) and c2 (∆ν ) are normed linear spaces, normed by kxk∆ν = sup |∆ν xmn |. (3.1) m,n 0 Theorem 3.3. The spaces ℓ∞ 2 (∆ν (...truncated)