normed space has a natural norm defined on it. However, it is very important in applications because of its compactness properties (see Banach–Alaoglu theorem). Caution: Whenever is a non-normable locally convex space, then the pairing map is never continuous, no matter which vector space topology one chooses on A topological vector space has a non-trivial continuous dual space if and only if it has a proper convex neighborhood of the origin. Properties For any of a TVS the convex (resp. balanced, disked, closed convex, closed balanced, closed disked) hull of is the smallest subset of that has this property and contains The closure (respectively, interior, convex hull, balanced hull, disked hull) of a set is sometimes denoted by (respectively, ). The convex hull of a subset is equal to the set of all of elements in which are finite linear combinations of the form where is an integer, and sum to The intersection of any family of convex sets is convex and the convex hull of a subset is equal to the intersection of all convex sets that contain it. Neighborhoods and open setsProperties of neighborhoods and open setsEvery TVS is connected and locally connected and any connected open subset of a TVS is arcwise connected. If and is an open subset of then is an open set in and if has non-empty interior then is a neighborhood of the origin. The open convex subsets of a TVS (not necessarily Hausdorff or locally convex) are exactly those that are of the form for some and some positive continuous sublinear functional on If is an absorbing disk in a TVS and if is the Minkowski functional of then where importantly, it was assumed that had any topological properties nor that was continuous (which happens if and only if is a neighborhood of the origin). Let and be two vector topologies on Then if and only if whenever a net in converges in then in Let be a neighborhood basis of the origin in let and let Then if and only if there exists a net in (indexed by ) such that in This shows, in particular, that it will often suffice to consider nets indexed by a neighborhood basis of the origin rather than nets on arbitrary directed sets. If is a TVS that is of the second category in itself (that is, a nonmeager space) then any closed convex absorbing