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Concept
Barrelled space
Summary
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 .
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.
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).
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