nLab barreled topological vector space




Barreled spaces are topological vector spaces for which the theorem of Banach–Steinhaus is valid. This theorem says, roughly, that for a set of continuous linear maps L(E,F)L(E, F) from a barreled space EE to a locally convex TVS boundedness in the topology of pointwise convergence implies boundedness in the topology of bounded convergence.


A subset TET \subset E of a TVS E is a barrel if it is

  • absorbing

  • balanced

  • closed

  • convex

A TVS EE is barreled (or barrelled) if every barrel is a neighborhood of zero.

Sometimes locally convex is included in the definition, this is not implied by barreled as defined above, i.e. there are barreled spaces that are not locally convex.

In the definition of quasibarreled or infrabarreled the barrels are replaced by sets that are barrels and which absorb all bounded sets (sets with the latter property are also called bornivorous).



A locally convex TVS which is a Baire space is barreled.


A locally convex TVS is barreled iff its topology is the strong topology.


Since all locally convex TVSes that are Baire spaces are barreled, the examples naturally include Fréchet spaces, Banach spaces and Hilbert spaces.


See the functional analysis bibliography.

The definition of quasibarreled is from

  • S.M. Khaleelulla: Counterexamples in Topological Vector Spaces.

It is called infrabarreled in

  • H.H. Schaefer: Topological vector spaces.

Last revised on May 22, 2013 at 15:13:46. See the history of this page for a list of all contributions to it.