In functional analysis and related areas of mathematics, a barrelled space (also written barreled space) is a topological vector space (TVS) for which every barrelled set in the space is a neighbourhood for the zero vector. A barrelled set or a barrel in a topological vector space is a set that is convex, balanced, absorbing, and closed. Barrelled spaces are studied because a form of the Banach–Steinhaus theorem still holds for them. Barrelled spaces were introduced by . Barrels A convex and balanced subset of a real or complex vector space is called a and it is said to be , , or . A or a in a topological vector space (TVS) is a subset that is a closed absorbing disk; that is, a barrel is a convex, balanced, closed, and absorbing subset. Every barrel must contain the origin. If and if is any subset of then is a convex, balanced, and absorbing set of if and only if this is all true of in for every -dimensional vector subspace thus if then the requirement that a barrel be a closed subset of is the only defining property that does not depend on (or lower)-dimensional vector subspaces of If is any TVS then every closed convex and balanced neighborhood of the origin is necessarily a barrel in (because every neighborhood of the origin is necessarily an absorbing subset). In fact, every locally convex topological vector space has a neighborhood basis at its origin consisting entirely of barrels. However, in general, there exist barrels that are not neighborhoods of the origin; "barrelled spaces" are exactly those TVSs in which every barrel is necessarily a neighborhood of the origin. Every finite dimensional topological vector space is a barrelled space so examples of barrels that are not neighborhoods of the origin can only be found in infinite dimensional spaces. Examples of barrels and non-barrels The closure of any convex, balanced, and absorbing subset is a barrel. This is because the closure of any convex (respectively, any balanced, any absorbing) subset has this same property. A family of examples: Suppose that is equal to (if considered as a complex vector space) or equal to (if considered as a real vector space). Regardless of whether is a real or complex vector space, every barrel in is necessarily a neighborhood of the origin (so is an example of a barrelled space). Let be any function