in correspond to each other by polarity. If is metrizable topological vector space then this list may be extended by appending: For any complete metrizable TVS every pointwise bounded in is equicontinuous. If is a locally convex metrizable topological vector space then this list may be extended by appending: (): The weak* topology on is sequentially complete. (): Every weak* bounded subset of is -relatively countably compact. (): Every countable weak* bounded subset of is equicontinuous. (): is not the union of an increase sequence of nowhere dense disks. Examples and sufficient conditions Each of the following topological vector spaces is barreled: TVSs that are Baire space. Consequently, every topological vector space that is of the second category in itself is barrelled. F-spaces, Fréchet spaces, Banach spaces, and Hilbert spaces. However, there exist normed vector spaces that are barrelled. For example, if the -space is topologized as a subspace of then it is not barrelled. Complete pseudometrizable TVSs. Consequently, every finite-dimensional TVS is barrelled. Montel spaces. Strong dual spaces of Montel spaces (since they are necessarily Montel spaces). A locally convex quasi-barrelled space that is also a σ-barrelled space. A sequentially complete quasibarrelled space. A quasi-complete Hausdorff locally convex infrabarrelled space. A TVS is called quasi-complete if every closed and bounded subset is complete. A TVS with a dense barrelled vector subspace. Thus the completion of a barreled space is barrelled. A Hausdorff locally convex TVS with a dense infrabarrelled vector subspace. Thus the completion of an infrabarrelled Hausdorff locally convex space is barrelled. A vector subspace of a barrelled space that has countable codimensional. In particular, a finite codimensional vector subspace of a barrelled space is barreled. A locally convex ultrabarelled TVS. A Hausdorff locally convex TVS such that every weakly bounded subset of its continuous dual space is equicontinuous. A locally convex TVS such that for every Banach space a closed linear map of into is necessarily continuous. A product of a family of barreled spaces. A locally convex direct sum and the inductive limit of a family of barrelled spaces. A quotient of a barrelled space. A Hausdorff sequentially complete quasibarrelled boundedly summing TVS. A locally convex Hausdorff reflexive space is barrelled. Counter examples A barrelled space need not be Montel, complete, metrizable, unordered Baire-like, nor the inductive limit of Banach spaces. Not all normed spaces are barrelled. However, they are all infrabarrelled. A closed subspace of