On Generalized Paranormed Statistically Convergent Sequence Spaces Defined by Orlicz Function
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Research Article On Generalized Paranormed Statistically Convergent Sequence Spaces Defined by Orlicz Function Metin Bas¸arir and Selma Altundag˘ Department of Mathematics, Faculty of Science and Arts, Sakarya University, 54187 Sakarya, Turkey Correspondence should be addressed to Metin Bas¸arir, [email protected] Received 8 May 2009; Revised 3 August 2009; Accepted 26 August 2009 Recommended by Andrei Volodin We define generalized paranormed sequence spaces cσ, M, p, q, s, c0 σ, M, p, q, s, mσ, M, p, q, s, and m0 σ, M, p, q, s defined over a seminormed sequence space X, q. We establish some inclusion relations between these spaces under some conditions. Copyright q 2009 M. Bas¸arir and S. Altundag. ˘ This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction wX, cX, c0 X, cX, c0 X, l∞ X, mX, m0 X will represent the spaces of all, convergent, null, statistically convergent, statistically null, bounded, bounded statistically convergent, and bounded statistically null X-valued sequence spaces throughout the paper, where X, q is a seminormed space, seminormed by q. For X C, the space of complex numbers, these spaces represent the w, c, c0 , c, c0 , l∞ , m, m0 which are the spaces of all, convergent, null, statistically convergent, statistically null, bounded, bounded statistically convergent, and bounded statistically null sequences, respectively. The zero sequence is denoted by θ θ, θ, θ, . . ., where θ is the zero element of X. The idea of statistical convergence was introduced by Fast 1 and studied by various authors see 2–4. The notion depends on the density of subsets of the set N of natural numbers. A subset E of N is said to have density δE if n 1 χE k exists, n→∞n k1
δE lim
where χE is the characteristic function of E.
1.1
2
Journal of Inequalities and Applications
A sequence x xk is said to be statistically convergent to the number L i.e., xk ∈ c if for every ε > 0 δ{k ∈ N : |xk − L| ≥ ε} 0.
1.2
stat
In this case, we write xk → L or stat − lim x L. Let σ be a mapping of the set of positive integers into itself. A continuous linear functional φ on l∞ , the space of real bounded sequences, is said to be an invariant mean or σ-mean if and only if 1 φx ≥ 0 when the sequence x xn has xn ≥ 0 for all n ∈ N, 2 φe 1, where e 1, 1, . . ., 3 φxσn φx for all x ∈ l∞ . n for all positive integers n and k, The mappings σ are one to one and such that σ k n / where σ k n denotes the kth iterate of the mapping σ at n. Thus φ extends the limit functional on c, the space of convergent sequences, in the sense that φx lim x for all x ∈ c. In that case σ is translation mapping n → n 1, a σ-mean is often called a Banach limit, and Vσ , the set of bounded sequences all of whose invariant means are equal, is the set of almost convergent se
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