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Some paranormed sequence spaces of non-absolute type derived by weighted mean. (English) Zbl 1105.46005
Summary: The sequence spaces $\ell_\infty(p)$, $c(p)$ and $c_0(p)$ were introduced and studied by {\it I. J. Maddox}, Proc. Camb. Philos. Soc. 64, 335--340 (1968; Zbl 0157.43503)]. In the present paper, the sequence spaces $\lambda(u,v;p)$ of non-absolute type which are derived by the generalized weighted mean are defined and it is proved that the spaces $\lambda(u,v;p)$ and $\lambda(p)$ are linearly isomorphic, where $\lambda$ denotes one of the sequence spaces $\ell_\infty$, $c$ or $c_0$. Besides this, the $\beta$- and $\gamma$-duals of the spaces $\lambda(u,v;p)$ are computed and basis of the spaces $c_0(u,v;p)$ and $c(u,v;p)$ is constructed. Additionally, it is established that the sequence space $c_0(u,v)$ has the AD property and the $f$-dual of the space $c_0(u,v;p)$ is given. Finally, the matrix mappings from the sequence spaces $\lambda(u,v;p)$ to the sequence space $\mu$ and from the sequence space $\mu$ to the sequence spaces $\lambda(u,v;p)$ are characterized.

MSC:
 46A45 Sequence spaces
Full Text:
References:
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