Abstract

In the theory of classes of sequence, a wonderful application of Hahn-Banach extension theorem gave rise to the concept of Banach limit, i.e., the limit functional defined on c can be extended to the whole space l ∞ and this extended functional is called as the Banach limit. After that, in 1948 Lorentz used this concept of a week limit to introduce a new type of convergence, named as the almost convergence. Later on, Raimi generalized the concept of almost convergence known as σ − convergence and the sequence space BV σ was introduced and studied by Mursaleen. The main aim of this chapter is to study some new double sequence spaces of invariant means defined by ideal, modulus function and Orlicz function. Furthermore, we also study several properties relevant to topological structures and inclusion relations between these spaces.

Highlights

  • The concept of convergence of a sequence of real numbers has been extended to statistical convergence independently by Fast [1] and Schoenberg [2]

  • There has been an effort to introduce several generalizations and variants of statistical convergence in different spaces. One such very important generalization of this notion was introduced by Kostyrko et al [3] by using an ideal I of subsets of the set of natural numbers, which they called I-convergence

  • An ideal I ⊂ 2X is said to be nontrivial if I 61⁄4 2X, this non trivial ideal is said to be admissible if I ⊇ ffxg : x ∈ Xg and is said to be maximal if there cannot exist any nontrivial ideal J 61⁄4 I containing I as a subset

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Summary

Introduction

The concept of convergence of a sequence of real numbers has been extended to statistical convergence independently by Fast [1] and Schoenberg [2]. A double sequence x 1⁄4 ÀxijÁ ∈2ω is said to be I-convergent [5–8] to a number L if for every ε>0, we have. Mþ where σmðkÞ denote the mth-iterate of σðkÞ at k In this case σ is the translation mapping, that is, σðkÞ 1⁄4 k þ 1, σÀ mean is called a Banach limit [11] and Vσ, the set of bounded sequences of all whose invariant means are equal, is the set of almost convergent sequences. We define and study the concepts of I-convergence for double sequences defined by Orlicz function and present some basic results on the above definitions [8, 20]. Theorem 2.1 Let M1, M2 be two Orlicz functions with Δ2 condition, (a) χðM2Þ ⊆ χðM1M2Þ (b) χðM1Þ ∩ χðM2Þ ⊆ χðM1 þ M2Þ for χ 1⁄42 BVIσ, 2À0BVIσÁ: Proof. (a) Let x 1⁄4 ÀxijÁ ∈ 2À0BVIσðM2ÞÁ be an arbitrary element, so there exists ρ > 0 such that

À lim M2
À0 BV Iσ ðM1 þ M2ÞÁ
Paranorm bounded variation sequence spaces
Bounded variation sequence spaces defined by modulus function
Conclusion
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