a barreled space is not necessarily countably quasi-barreled (and thus not necessarily barrelled). There exists a dense vector subspace of the Fréchet barrelled space that is not barrelled. There exist complete locally convex TVSs that are not barrelled. The finest locally convex topology on an infinite-dimensional vector space is a Hausdorff barrelled space that is a meagre subset of itself (and thus not a Baire space). Properties of barreled spaces Banach–Steinhaus generalization The importance of barrelled spaces is due mainly to the following results. The Banach-Steinhaus theorem is a corollary of the above result. When the vector space consists of the complex numbers then the following generalization also holds. Recall that a linear map is called closed if its graph is a closed subset of Other properties Every Hausdorff barrelled space is quasi-barrelled. A linear map from a barrelled space into a locally convex space is almost continuous. A linear map from a locally convex space to a barrelled space is almost open. A separately continuous bilinear map from a product of barrelled spaces into a locally convex space is hypocontinuous. A linear map with a closed graph from a barreled TVS into a -complete TVS is necessarily continuous. See also References Bibliography Topological vector spaces