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# On a New I-Convergent Double-Sequence Space

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Abstract:

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

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