AIMS Mathematics, 2020, 5(4): 2858-2868. doi: 10.3934/math.2020183

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Distinguished subspaces in topological sequence spaces theory

Department of Mathematics, Kahramanmaraş Sütçü İmam University, Kahramanmaraş-46100, Turkey

In this paper, we study $R_{\lambda}$-semiconservative $FK$-spaces for Riesz-method defined by the Riesz matrix $(R)$ and give some characterizations. We show that if $\ell_{A}$ is $\ell$-replaceable, then $A$ can not be $R_{\lambda}$-semiconservative and also if $X_{A}$ is $R_{\lambda}$-conull $FK$-space then it must be $R_{\lambda}$-semiconservative space. In addition, we determine a new $r(\lambda)$ and $rb(\lambda)$ type duality of a sequence space $X$ containing $\varphi$. The paper aims to develop some new subspaces which each one has its own value on topological sequence spaces theory. These subspaces are called as $R_{\lambda}S; R_{\lambda}W; R_{\lambda}F^{+};$ and $R_{\lambda}B^{+}$ for a locally convex $FK$-space X containing $\varphi$. The subspaces mentioned in the work requires some serious studies and they arose independently from the literature which was done at the recent stage of the development of summability through functional analysis.
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