On some locally convex FK spaces

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}$)$forsomeideal$$\mathcal{I}$$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 $\mathcal{I}$.