On some locally convex FK spaces

TL;DR

This paper characterizes conditions under which certain vector spaces of real sequences are locally convex FK spaces, focusing on the inclusion relations with $c_{00}( ext{I})$ and $ ext{ell}_ ext{infinity}( ext{I})$, and the properties of the ideal $ ext{I}$.

Contribution

It provides necessary and sufficient conditions for vector spaces of sequences to be locally convex FK spaces, especially in relation to ideals on $ ext{omega}$ and inclusion properties.

Findings

Characterization of when $V$ is a locally convex FK space based on ideal properties.

Equivalence of FK space condition with the existence of an infinite set $S$ with specific ideal properties.

Conditions linking sequence space inclusions with the structure of the ideal $ ext{I}$.

Abstract

We provide necessary and/or sufficient conditions on vector spaces $V$ of real sequences to be a Fr\'{e}chet space such that each coordinate map is continuous, that is, to be a locally convex FK space. In particular, we show that if $c_{00}(\mathcal{I})\subseteq V\subseteq \ell_\infty(\mathcal{I})$ for some ideal $\mathcal{I}$ on $\omega$, then $V$ is a locally convex FK space if and only if there exists an infinite set $S\subseteq \omega$ for which every infinite subset does not belong to $\mathcal{I}$.