Barreled space - Imported Knowlpedia Content

Definition

Let $E$ be a Hausdorff locally convex space. A barrel in $E$ is a closed, convex, balanced, and absorbing subset of $E$ . The space $E$ is barreled if every barrel is a neighborhood of $0$ . Thus a set satisfying the algebraic size and symmetry conditions, together with topological closedness, cannot be anomalously thin near the origin. This property is unchanged by the alternative British spelling barrelled.

Equivalent characterizations

A Hausdorff locally convex space $E$ is barreled exactly when every lower-semicontinuous seminorm on $E$ is continuous. It is also equivalent to the following uniform-boundedness formulation: every pointwise bounded subset of the topological dual $E'$ is equicontinuous. These equivalences explain why barrels, rather than arbitrary absorbing sets, occur in the definition Schaefer–Wolff, Chapter III, §4.

Uniform boundedness and examples

If $E$ is barreled and $F$ is locally convex, every pointwise bounded family of continuous linear maps from $E$ to $F$ is equicontinuous. This is the locally convex Banach–Steinhaus principle. Every Banach space and every Fréchet space is barreled by the Baire category theorem. Locally convex inductive limits of barreled spaces, including LF-spaces, are also barreled.

Conventions and nearby notions

Some authors build Hausdorffness into “locally convex space”; it is stated explicitly here. Requiring only barrels that absorb every bounded subset gives the weaker notion of a quasibarreled or infrabarreled space. Barreledness does not itself imply metrizability, completeness, or bornologicality, although familiar function spaces often possess several of these properties simultaneously.

References

Helmut H. Schaefer and Manfred P. Wolff, Topological Vector Spaces, 2nd ed., Springer, 1999. Springer DOI record. Relevant: Chapter III, §4 on barreled spaces and uniform boundedness.
Nicolas Bourbaki, Topological Vector Spaces: Chapters 1–5, Springer, 2003. Springer DOI record. Relevant: Chapter III on spaces of continuous linear maps and equicontinuity.