Sequence Spaces Defined by Musielak-Orlicz Function over -Normed Spaces

M Mursaleen,Sunil K Sharma,A Kılıçman

doi:10.1155/2013/364743

Abstract

In the present paper we introduce some sequence spaces overn-normed spaces defined by a Musielak-Orlicz function . We also study some topological properties and prove some inclusion relations between these spaces.

Highlights

Introduction and PreliminariesAn Orlicz function M is a function, which is continuous, nondecreasing, and convex with M(0) = 0, M(x) > 0 for x > 0 and M(x) → ∞ as x → ∞.Lindenstrauss and Tzafriri [1] used the idea of Orlicz function to define the following sequence space
In the present paper we introduce some sequence spaces over n-normed spaces defined by a Musielak-Orlicz function M = (Mk)
It may be noted here that the space of strongly summable sequences was discussed by Maddox [16] and recently in [17]

Summary

Introduction and Preliminaries

Lindenstrauss and Tzafriri [1] used the idea of Orlicz function to define the following sequence space. Which is called as an Orlicz sequence space. The space lM is a Banach space with the norm It is shown in [1] that every Orlicz sequence space lM contains a subspace isomorphic to lp (p ≥ 1). A sequence M = (Mk) of Orlicz functions is called a Musielak-Orlicz function (see [2, 3]). Is called the complementary function of a Musielak-Orlicz function M. For a given Musielak-Orlicz function M, the Musielak-Orlicz sequence space tM and its subspace hM are defined as follows: tM = {x ∈ w : IM (cx) < ∞ for some c > 0} , (4).

We consider tM equipped with the Luxemburg norm

This implies that limm

Main Results

Ksup r