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.

Title	barrel
Canonical name	Barrel
Date of creation	2013-03-22 16:41:26
Last modified on	2013-03-22 16:41:26
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	5
Author	CWoo (3771)
Entry type	Definition
Classification	msc 46A08
Synonym	barreled space
Synonym	infrabarreled space
Synonym	ultrabarreled space
Synonym	barrelled
Synonym	infrabarrelled
Synonym	ultrabarrelled
Defines	barrelled space
Defines	infrabarrelled space
Defines	ultrabarrelled space