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 $\{v\in A\mid||v||\leq r\}$ 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 $\mathfrak{T}$ be the set of all those topologies  on $V$ making $V$ a TVS. In other words, if $T\in\mathfrak{T}$, then $(V,T)$ is a topological vector space.

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

(*) for any $T_{1}\in\mathfrak{T}$ 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.

References

• 1 H. H. Schaefer, Topological Vector Spaces, Springer-Verlag, New York (1970).
• 2 R. E. Edwards, Functional Analysis, Theory and Applications, Holt, Reinhart and Winston, New York (1965).
 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