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.
A barrel or a barrelled set 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.
is a convex, balanced, and absorbing set of
then the requirement that a barrel be a closed subset of
is the only defining property that does not depend solely on
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.
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.
(if considered as a complex vector space) or equal to
(if considered as a real vector space).
is a real or complex vector space, every barrel in
denote the closed line segment from the origin to the point
(a real vector space) but it is an absorbing subset of
(a complex vector space) if and only if it is a neighborhood of the origin.
if and only it is an open or closed ball centered at the origin (of radius
are exactly those closed balls centered at the origin with radius in
the space of continuous linear maps from
is locally convex space then this list may be extended by appending: If
is a Hausdorff locally convex space then this list may be extended by appending: If
is metrizable topological vector space then this list may be extended by appending: If
is a locally convex metrizable topological vector space then this list may be extended by appending: Each of the following topological vector spaces is barreled: The importance of barrelled spaces is due mainly to the following results.
The following are equivalent: The Banach-Steinhaus theorem is a corollary of the above result.
consists of the complex numbers then the following generalization also holds.
is a barrelled TVS over the complex numbers and
, then the following are equivalent: Recall that a linear map
Closed Graph Theorem[22] — Every closed linear operator from a Hausdorff barrelled TVS into a complete metrizable TVS is continuous.