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 . A number is called a double limit of a double sequence if for every there exists some such that (see？？). 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？？). 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 ), and , the set of bounded sequences all of whose invariant means are equal, is the set of almost convergent sequences (see ). 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 ) Put assuming that . A straight forward calculation shows (see ) 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 ): Theorem 5. is a Banach space normed by (see ). Subsequently, invariant means have been studied by Ahmad and Mursaleen , Mursaleen et al. [3, 6, 8, 10–14], Raimi , Schaefer , Savas and Rhoades , Vakeel et al. [18–20], and many others [21–23]. For the first time, I-convergence was studied by Kostyrko et al. . Later on, it was studied by ？alát et al. [25, 26], Tripathy and Hazarika , Ebadullah et al. [18–20, 28], and
A. K. Vakeel and S. Tabassum, “On some new double sequence spaces of invariant means defined by Orlicz functions,” Communications de la Faculté des Sciences de l'Université d'Ankara Séries A, vol. 60, no. 2, pp. 11–21, 2011.
M. Mursaleen, S. A. Mohiuddine, and O. H. H. Edely, “On the ideal convergence of double sequences in intuitionistic fuzzy normed spaces,” Computers & Mathematics with Applications, vol. 59, no. 2, pp. 603–611, 2010.
A. K. Vakeel and T. Sabiha, “On ideal convergent difference double sequence spaces in 2-normed spaces defined by Orlicz function,” JMI International Journal of Mathematical Sciences, vol. 1, no. 2, pp. 1–9, 2010.