Some Properties of the Sequence Space

Kuldip Raj,Sunil K Sharma

doi:10.1155/2013/904838

Abstract

We introduce the sequence space defined by a Musielak-Orlicz function . We also study some topological properties and prove some inclusion relations involving this space.

Highlights

Introduction and PreliminariesThe concept of 2-normed spaces was initially developed by Gahler [1] in the mid-1960s, while one can see that of n-normed spaces in Misiak [2]
A real valued function ‖⋅, . . . , ⋅‖ on Xn satisfying the following four conditions: (1) ‖x1, x2, . . . , xn‖ = 0 if and only if x1, x2, . . . , xn are linearly dependent in X; (2) ‖x1, x2, . . . , xn‖ is invariant under permutation; (3) ‖αx1, x2, . . . , xn‖ = |α|‖x1, x2, . . . , xn‖ for any α ∈ K; and
We may take X = Rn being equipped with the n-norm ‖x1, x2, . . . , xn‖E = the volume of the n-dimensional parallelopiped spanned by the vectors x1, x2, . . . , xn which may be given explicitly by the formula

Summary

Introduction

Introduction and PreliminariesThe concept of 2-normed spaces was initially developed by Gahler [1] in the mid-1960s, while one can see that of n-normed spaces in Misiak [2]. If every Cauchy sequence in X converges to some L ∈ X, X is said to be complete with respect to the n-norm. Lindenstrauss and Tzafriri [6] used the idea of Orlicz function to define the following sequence space.

Results

Conclusion