AbstractA Hausdorff locally convex space is said to bec0‐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 ac0‐barrelled spaceEhas dualE′ weakly sequentially complete, then every subspace of countable codimension ofEisc0‐barrelled. We prove that the hypothesis onE′ cannot be dropped and we supply an example of a completec0‐barrelled space with dual weakly sequentially complete that is n