Journal of Inequalities and Applications
Volume 2010 (2010), Article ID 457892, 7 pages
doi:10.1155/2010/457892
Research Article

Generalized Vector-Valued Sequence Spaces Defined by Modulus Functions

Mahmut Işik

Department of Statistics, Firat University, 23119 Elazığ, Turkey

Received 17 June 2010; Accepted 16 December 2010

Academic Editor: Alberto Cabada

Copyright © 2010 Mahmut Işik. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We introduce the vector-valued sequence spaces 𝑤 ∞ ( Δ 𝑚 , 𝐹 , 𝑄 , 𝑝 , 𝑢 ) , 𝑤 1 ( Δ 𝑚 , 𝐹 , 𝑄 , 𝑝 , 𝑢 ) , and 𝑤 0 ( Δ 𝑚 , 𝐹 , 𝑄 , 𝑝 , 𝑢 ) , 𝑆 𝑞 𝑢 and 𝑆 𝑞 0 𝑢 , using a sequence of modulus functions and the multiplier sequence 𝑢 = ( 𝑢 𝑘 ) of nonzero complex numbers. We give some relations related to these sequence spaces. It is also shown that if a sequence is strongly Δ 𝑚 𝑢 𝑞 -Cesàro summable with respect to the modulus function 𝑓 then it is Δ 𝑚 𝑢 𝑞 -statistically convergent.