Barrelled space explained

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

\dim X \geq 2 and if S is any subset of X, then S is a convex, balanced, and absorbing set of X if and only if this is all true of S \cap Y in Y for every 2 -dimensional vector subspace Y; thus if \dim X > 2 then the requirement that a barrel be a closed subset of X is the only defining property that does not depend on 2 (or lower)-dimensional vector subspaces of X.

If

X is any TVS then every closed convex and balanced neighborhood of the origin is necessarily a barrel in X (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

X is equal to \Complex (if considered as a complex vector space) or equal to \R ^{2} (if considered as a real vector space). Regardless of whether X is a real or complex vector space, every barrel in X is necessarily a neighborhood of the origin (so X is an example of a barrelled space). Let R : [ 0 , 2 \pi) \to ( 0 , inf ty] be any function and for every angle \theta \in [ 0 , 2 \pi), let S _{\theta} denote the closed line segment from the origin to the point R(\theta) e ^{i} \in \Complex. Let $S\; :=\; \backslash bigcup\_$