On \(B({\mathcal I})\)-spaces (Q1092363)

There exist Hausdorff locally convex spaces \(F, G\) such that (i) any continuous linear map from \(F\) onto a Fréchet space need not be open, (ii) any continuous linear map from \(G\) onto any Fréchet space is open, but a continuous linear map from \(G\) onto a Hausdorff barrelled space need not be open. The above are among inclusion relations obtained in this note. The following is first proved. Let \((E,u)\) be a Hausdorff locally convex space, and \(w\) the associated weak topology. If \(F\) is a Banach space, then a continuous linear map from \((E,w)\) onto \(F\) is open. When \((E,u)\) is normed, then F may instead be a Hausdorff locally convex space such that any continuous linear map onto F is open.