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 (I) ⊆ V ⊆ ℓ ∞ (I) for some ideal I on ω, then V is a locally convex FK space if and only if there exists an infinite set S ⊆ ω for which every infinite subset does not belong to I.