%0 Journal Article
%T On a New I-Convergent Double-Sequence Space
%A Vakeel A. Khan
%A Nazneen Khan
%J International Journal of Analysis
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/126163
%X The sequence space was introduced and studied by Mursaleen (1983). In this article we introduce the sequence space 2 and study some of its properties and inclusion relations. 1. Introduction and Preliminaries Let , , and be the sets of all natural, real, and complex numbers, respectively. We write showing the space of all real or complex sequences. Definition 1. A double sequence of complex numbers is defined as a function . We denote a double sequence as where the two subscripts run through the sequence of natural numbers independent of each other [1]. A number is called a double limit of a double sequence if for every there exists some such that (see£¿£¿[2]). Let and denote the Banach spaces of bounded and convergent sequences, respectively, with norm . Let denote the space of sequences of bounded variation; that is, where is a Banach space normed by (see£¿£¿[3]). Definition 2. Let be a mapping of the set of the positive integers into itself having no finite orbits. A continuous linear functional on is said to be an invariant mean or -mean if and only if(i) when the sequence has for all ;(ii) , where ;(iii) for all . In case is the translation mapping , a -mean is often called a Banach limit (see [4]), and , the set of bounded sequences all of whose invariant means are equal, is the set of almost convergent sequences (see [5]). If , then . Then it can be shown that where , . Consider where denote the th iterate of at . The special case of (5) in which was given by Lorentz [5, Theorem 1], and that the general result can be proved in a similar way. It is familiar that a Banach limit extends the limit functional on . Theorem 3. A -mean extends the limit functional on in the sense that for all if and only if has no finite orbits; that is to say, if and only if, for all , , (see [3]) Put assuming that . A straight forward calculation shows (see [6]) that For any sequence , , and scalar , we have Definition 4. A sequence is of -bounded variation if and only if (i) converges uniformly in ;(ii) , which must exist, should take the same value for all . We denote by , the space of all sequences of -bounded variation (see [7]): Theorem 5. is a Banach space normed by (see [8]). Subsequently, invariant means have been studied by Ahmad and Mursaleen [9], Mursaleen et al. [3, 6, 8, 10¨C14], Raimi [15], Schaefer [16], Savas and Rhoades [17], Vakeel et al. [18¨C20], and many others [21¨C23]. For the first time, I-convergence was studied by Kostyrko et al. [24]. Later on, it was studied by £¿al¨¢t et al. [25, 26], Tripathy and Hazarika [27], Ebadullah et al. [18¨C20, 28], and
%U http://www.hindawi.com/journals/ijanal/2013/126163/