On some locally convex FK spaces (Q2105020)

Recall that a locally convex FK space is a Fréchet vector subspace \(V\subseteq \mathbb R^{\omega}\) for which the inclusion map is continuous (\(\omega\) here denotes the set of nonnegative integers). Recall also that an ideal \(\mathcal{I}\) on \(\omega\) is said to be tall if every infinite subset of \(\omega\) contains an infinite subset which is in \(\mathcal{I}\). The main results of this paper are the following: \(1^{\circ}\) If \(\mathcal{I}\) is a tall ideal on \(\omega\) and \(V\) is a proper vector subspace of \(\mathbb R^{\omega}\) which contains \(c_{00}(\mathcal{I})\), then \(V\) does not admit a locally convex FK topology. \(2^{\circ}\) If \(\mathcal{I}\) is a nontall ideal on \(\omega\) and \(V\) is a dense vector subspace of \(\mathbb R^{\omega}\) contained in \(\ell_{\infty}(\mathcal{I})\), then \(V\) admits a locally convex FK topology. \(3^{\circ}\) If \(\mathcal{I}\) is an ideal on \(\omega\) and \(V\) is a vector subspace such that \(c_{00}(\mathcal{I})\subseteq V\subseteq \ell_{\infty}(\mathcal{I})\), then \(V\) admits a locally convex FK topology if and only if \(\mathcal{I}\) is not tall. Several interesting consequences are also deduced.