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Some new sequence spaces which include the spaces $\ell_p$ and $\ell_\infty$. (English) Zbl 1096.46005
Summary: We introduce the sequence space $a_p^r$ of non-absolute type and prove that the spaces $a_p^r$ and $\ell_p$ are linearly isomorphic for $0<p\le\infty$. We also show that $a_p^r$, which includes the space $\ell_p$, is a $p$-normed space and a BK space in the cases of $0<p<1$ and $1\le p\le\infty$, respectively. Furthermore, we give some inclusion relations and determine the $\alpha$-, $\beta$- and $\gamma$-duals of the space $a_p^r$ and construct its basis. We devote the last section of the paper to the characterization of the matrix mappings from the space $a_p^r$ to some of the known sequence spaces and to some new sequence spaces.

##### MSC:
 46A45 Sequence spaces 46B45 Banach sequence spaces 46A35 Summability and bases in topological linear spaces