A note on the inheritance of properties of locally convex spaces by subspaces of countable codimension

Abstract:

A locally convex space E is said to be $\omega$-barrelled if every countable $\mathrm {weak}^*$ bounded subset of its topological dual $E’$ is equicontinuous; to have property (C) if every $\mathrm {weak}^*$ bounded subset of $E’$ is relatively $\mathrm {weak}^*$ compact; to have property (S) if $E’$ is $\mathrm {weak}^*$ sequentially complete. If a locally convex space possesses any of the above properties, then so do all of its linear subspaces of countable codimension. Examples are furnished to show that the mentioned properties are distinct from each other.