nLab barreled topological vector space

Idea

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 $L(E, F)$ from a barreled space $E$ to a locally convex TVS boundedness in the topology of pointwise convergence implies boundedness in the topology of bounded convergence.

Definition

A subset $T \subset E$ of a TVS E is a barrel if it is

• absorbing

• balanced

• closed

• convex

A TVS $E$ 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).

Properties

Proposition

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

Proposition

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

Examples

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

References

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.