A metrizable locally convex topological vector space (TVS) X (for example, a Fréchet space) is a normable space if and only if its strong dual space X^{\prime}_b is a Fréchet–Urysohn space,Gabriyelyan, S.S. "On topological spaces and topological groups with certain local countable networks (2014) or equivalently, if and only if X^{\prime}_b is a normable space.
