Let be a topological vector space (TVS). A barrel is a subset of that is closed, convex, balanced (http://planetmath.org/BalancedSet), 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 (http://planetmath.org/LocallyConvexTopologicalVectorSpace), and every barrel is a neighborhood of . 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 .
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.
- 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).
|Date of creation||2013-03-22 16:41:26|
|Last modified on||2013-03-22 16:41:26|
|Last modified by||CWoo (3771)|