Barrelled space

In functional analysis and related areas of mathematics, a barrelled space (also written space) is a topological vector spaces (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.
Barrels.
Let be a topological vector space (TVS).
A convex and balanced subset of a real or complex vector space is called a disk and it is said to be disked, absolutely convex, or convex balanced.
A barrel or a barrelled set in a TVS is a subset that is a closed absorbing disk.
The only topological requirement on a barrel is that it be a closed subset of the TVS; all other requirements (i.e. being a disk and being absorbing) are purely algebraic properties.
Characterizations of barreled spaces.
If formula_2 is a topological vector space (TVS) with continuous dual formula_3 then the following are equivalent:
If formula_2 is Hausdorff then we may add to this list:
Properties of barreled spaces.
Banach–Steinhaus generalization.
The importance of barrelled spaces is due mainly to the following results.
The Banach-Steinhaus theorem is a corollary of the above result. When the vector space consists of the complex numbers then the following generalization also holds.
Recall that a linear map formula_5 is called closed if its graph is a closed subset of formula_6
History.
Barrelled spaces were introduced by .