A Study of Bounded Variation Sequence Spaces

Vakeel Ahmad Khan, Hira Fatima and Mobeen Ahmad

### 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<sup>∞</sup> 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<sup>σ</sup> 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.

Keywords: invariant mean, bounded variation, ideal, filter, I-convergence, Orlicz function, modulus function, paranorm
