A New Aspect for Some Sequence Spaces Derived Using the Domain of the Matrix $\widehat{\widehat{B}}$

Year 2022, Volume: 5 Issue: 1, 51 - 62, 01.03.2022

Murat Candan

https://doi.org/10.33401/fujma.1003752

Abstract

This study serves for analysing algebraic and topological characteristics of the sequence spaces $X(\widehat{\widehat{B}}(r,s))$ constituted by using non-zero real number $r$ and $s$, where $X$ denotes arbitrary of the classical sequence spaces $\ell_{\infty}, c, c_{0} $ and $\ell_{p}$ $(1<p<\infty)$ of bounded, convergent, null and absolutely $p$-summable sequences, respectively and $X(\widehat{\widehat{B}})$ also is the domain of the matrix $\widehat{\widehat{B}}(r,s)$ in the sequence space $X$. Briefly, the $\beta$- and $\gamma$-duals of the space $X(\widehat{\widehat{B}})$ are computed, and Schauder bases for the spaces $c(\widehat{\widehat{B}})$, $c_{0}(\widehat{\widehat{B}})$ and $\ell_{p}(\widehat{\widehat{B}})$ are determined, and some algebraic and topological properties of the spaces $c_{0}(\widehat{\widehat{B}})$, $\ell_{1}(\widehat{\widehat{B}})$ and $\ell_{p}(\widehat{\widehat{B}})$ are studied. Additionally, it is observed that all these spaces have some remarkable features, including the classes $(X_{1}(\widehat{\widehat{B}})$: $X_{2})$ and $(X_{1}(\widehat{\widehat{B}})
: X_{2}(\widehat{\widehat{B}}))$ of infinite matrices which are characterized, in which $X_{1}\in\{ \ell_{\infty},c,c_{0},\ell_{p},\ell_{1}\}$ and $X_{2}\in\{\ell_{\infty},c,c_{0},\ell_{1}\}$.

Keywords

Matrix domain of a sequence space, Schauder basis, $\beta-$ and $% \gamma-$duals and matrix transformations

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