Some Topological and Geometrical Properties of a New Difference Sequence Space

Abstract

We introduce the new difference sequence space $a_p^r\left(\Delta \right)$ . Further, it is proved that the space $a_p^r\left(\Delta \right)$ is the BK-space including the space $b{v}_p$, which is the space of sequences of pbounded variation. We also show that the spaces $a_p^r\left(\Delta \right)$, and ${\ell }_p$ are linearly isomorphic for $1\le p<\infty$. Furthermore, the basis and the $\alpha \text{-}$, $\beta \text{-}$ and $\gamma \text{-}$ duals of the space $a_p^r\left(\Delta \right)$ are determined. We devote the final section of the paper to examine some geometric properties of the space $a_p^r\left(\Delta \right)$.