On some locally convex FK spaces

Abstract

We provide necessary and/or sufficient conditions on vector spaces $V$ of real sequences to be a Fréchet 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 $\mathcal{I}$.