### Domain of Generalized Difference Matrix B(r, s) on Some Maddox’s Spaces

#### Abstract

In the present paper, the sequence spaces $\widehat{\ell}_\infty(p),~\widehat{c}_0(p)$ and $\widehat{c}(p)$ of non-absolute type have been introduced and proved that the spaces $\widehat{\ell}_\infty(p),~\widehat{c}_0(p)$ and $\widehat{c}(p)$ are linearly isomorphic to the spaces $\ell_\infty(p),~c_0(p)$ and $c(p)$, respectively. The $\beta$- and

$\gamma$-duals of the spaces $\widehat{\ell}_\infty(p),~\widehat{c}_0(p)$ and $\widehat{c}(p)$ have been computed and their basis have been constructed. Finally, some matrix mappings from $\widehat{\ell}_\infty(p),~\widehat{c}_0(p)$ and $\widehat{c}(p)$ to the some sequence spaces of Maddox have been characterized and relationship between the modular $\sigma_p$ and the Luxemburg norm on the sequence space $\widehat{\ell}_\infty(p)$ has been discussed.

$\gamma$-duals of the spaces $\widehat{\ell}_\infty(p),~\widehat{c}_0(p)$ and $\widehat{c}(p)$ have been computed and their basis have been constructed. Finally, some matrix mappings from $\widehat{\ell}_\infty(p),~\widehat{c}_0(p)$ and $\widehat{c}(p)$ to the some sequence spaces of Maddox have been characterized and relationship between the modular $\sigma_p$ and the Luxemburg norm on the sequence space $\widehat{\ell}_\infty(p)$ has been discussed.

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