barrel in is a neighborhood of the origin. This definition is similar to a characterization of Baire TVSs proved by Saxon [1974], who proved that a TVS with a topology that is not the indiscrete topology is a Baire space if and only if every absorbing balanced subset is a neighborhood of point of (not necessarily the origin). For any Hausdorff TVS every pointwise bounded subset of is equicontinuous. For any F-space every pointwise bounded subset of is equicontinuous. An F-space is a complete metrizable TVS. Every closed linear operator from into a complete metrizable TVS is continuous. A linear map is called closed if its graph is a closed subset of Every Hausdorff TVS topology on that has a neighborhood basis of the origin consisting of -closed set is course than If is locally convex space then this list may be extended by appending: There exists a TVS not carrying the indiscrete topology (so in particular, ) such that every pointwise bounded subset of is equicontinuous. For any locally convex TVS every pointwise bounded subset of is equicontinuous. It follows from the above two characterizations that in the class of locally convex TVS, barrelled spaces are exactly those for which the uniform boundedness principal holds. Every -bounded subset of the continuous dual space is equicontinuous (this provides a partial converse to the Banach-Steinhaus theorem). carries the strong dual topology Every lower semicontinuous seminorm on is continuous. Every linear map into a locally convex space is almost continuous. A linear map is called if for every neighborhood of the origin in the closure of is a neighborhood of the origin in Every surjective linear map from a locally convex space is almost open. This means that for every neighborhood of 0 in the closure of is a neighborhood of 0 in If is a locally convex topology on such that has a neighborhood basis at the origin consisting of -closed sets, then is weaker than If is a Hausdorff locally convex space then this list may be extended by appending: Closed graph theorem: Every closed linear operator into a Banach space is continuous. The linear operator is called if its graph is a closed subset of For every subset of the continuous dual space of the following properties are equivalent: is equicontinuous; relatively weakly compact; strongly bounded; weakly bounded. The 0-neighborhood bases in and the fundamental families of bounded sets