barrel

Let $V$ be a topological vector space (TVS). A barrel $B$ is a subset of $V$ that is closed, convex, balanced (http://planetmath.org/BalancedSet), and absorbing. For example, in a Banach space $A$, any ball $\{v\in A\mid ||v||\le r\}$ for some $r>0$ is a barrel.

A topological vector space is said to be a barrelled space if it is locally convex (http://planetmath.org/LocallyConvexTopologicalVectorSpace), and every barrel is a neighborhood of $0$. Every Banach space is a barrelled space.

A weaker form of a barrelled space is that of an infrabarrelled space. A TVS is said to be infrabarrelled if it is locally convex, and every barrel that absorbs every bounded set is a neighborhood of $0$.

Let $V$ be a vector space and $𝔗$ be the set of all those topologies on $V$ making $V$ a TVS. In other words, if $T\in 𝔗$, then $(V,T)$ is a topological vector space.

Let $V$ and $T\in 𝔗$ be defined as above. Then $(V,T)$ being barrelled has an equivalent characterization below:

(*) for any $T_1\in 𝔗$ such that there is a neighborhood base (http://planetmath.org/LocalBase) of $0$ consisting of $T$-closed sets, then $T_1$ is coarser than $T$.

A variation of a barrelled space is that of an ultrabarrelled space. A topological vector space is said to be ultrabarrelled if it satisfies (*) above. A locally convex ultrabarrelled space is barrelled.