A Generalization of Orlicz Sequence Spaces by Cesàro Mean of Order One

In this paper, we introduce the Orlicz sequence spaces generated by Cesàro mean of order one associated with a fixed multiplier sequence of non-zero scalars. Furthermore, we emphasize several algebraic and topological properties relevant to these spaces. Finally, we determine the Köthe-Toeplitz dual of the spaces `M (C, Λ) and hM (C, Λ). 1. Preliminaries, Background and Notation By ω, we denote the space of all complex valued sequences. Any vector subspace of ω which contains φ, the set of all finitely non–zero sequences is called a sequence space. We write `∞, c and c0 for the classical sequence spaces of all bounded, convergent and null sequences which are Banach spaces with the sup-norm ‖x‖∞ = supk∈N |xk|, where N = {0, 1, 2, . . . }, the set of natural numbers. A sequence space X with a linear topology is called a K−space provided each of the maps pi : X → C defined by pi(x) = xi is continuous for all i ∈ N. A K-space X is called an FK-space provided X is a complete linear metric space. An FK-space whose topology is normable is called a BK-space. A function M : [0,∞) → [0,∞) which is convex with M(u) ≥ 0 for u ≥ 0, and M(u)→∞ as u→∞, is called as an Orlicz function. An Orlicz function M can always be represented in the following integral form