Almost Sequence Spaces Derived by the Domain of the Matrix A r

Abstract

By using $A^r$, we introduce the sequence spaces $a_f^r$, $a_{f_0}^r$, and $a_{\mathrm{fs}}^r$ of normed space and $\mathrm{BK}$-space and prove that $a_f^r,a_{f_0}^r$, and $a_{\mathrm{fs}}^r$ are linearly isomorphic to the sequence spaces $f$, $f_0$, and $\mathrm{fs}$ , respectively. Further, we give some inclusion relations concerning the spaces $a_f^r$, $a_{f_0}^r$, and the nonexistence of Schauder basis of the spaces $\mathrm{fs}$ and $a_{\mathrm{fs}}^r$ is shown. Finally, we determine the $\beta$- and $\gamma$-duals of the spaces $a_f^r$ and $a_{\mathrm{fs}}^r$. Furthermore, the characterization of certain matrix classes on new almost convergent sequence and series spaces has exhaustively been examined.