Countable-Codimensional Subspaces of c0-Barrelled Spaces

A Hausdorff locally convex space is said to be c₀-barrelled (respectively ω-barrelled) if each sequence in the dual space that converges weakly to 0 (respectively that is weakly bounded), is equicontinuous. It is proved that if a c₀-barrelled space E has dual E′ weakly sequentially complete, then every subspace of countable codimension of E is c₀-barrelled. We prove that the hypothesis on E′ cannot be dropped and we supply an example of a complete c₀-barrelled space with dual weakly sequentially complete that is not ω-barrelled.