Scholium

In the dual of a barrelled space, the class of equicontinuous sets, the class of strongly bounded sets, the class of weakly bounded sets and the class of relatively compact sets for the weak topology are all identical. If E is Hausdorff and barrelled, and if E'_b is its strong dual, the polars of the neighbourhoods of 0 in one of the spaces form a base of the canonical bornology of the other, and the polars of bounded subsets of one of the spaces form a base for the filter of neighbourhoods of 0 of the other space.