Let V be a topological vector spaceMathworldPlanetmath (TVS). A barrel B is a subset of V that is closed, convex, balanced (, and absorbingPlanetmathPlanetmath. For example, in a Banach spaceMathworldPlanetmath A, any ball {vA||v||r} for some r>0 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 V be a vector space and 𝔗 be the set of all those topologiesMathworldPlanetmath on V making V a TVS. In other words, if T𝔗, then (V,T) is a topological vector space.

Let V and T𝔗 be defined as above. Then (V,T) being barrelled has an equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath characterization below:

(*) for any T1𝔗 such that there is a neighborhood base ( of 0 consisting of T-closed setsPlanetmathPlanetmath, then T1 is coarserPlanetmathPlanetmath 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.


  • 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