Let be a topological vector space (TVS). A barrel is a subset of that is closed, convex, balanced, and absorbing. For example, in a Banach space , any ball for some is a barrel.

A topological vector space is said to be a barrelled space if it is locally convex, 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 be a vector space and be the set of all those topologies on making a TVS. In other words, if , then is a topological vector space.

Let and be defined as above. Then being barrelled has an equivalent characterization below:

(*) for any such that there is a neighborhood base of 0 consisting of -closed sets, then is coarser than .

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.

Bibliography

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).