barrelled

Barrelled is a property used in functional analysis to describe a particular kind of locally convex topological vector space. A space X is called barrelled if every barrel is a neighborhood of zero. This condition helps ensure certain continuity and boundedness results for families of linear operators.

A barrel in a locally convex space is a subset that is closed, convex, balanced, and absorbing. In other words, it is a closed, absolutely convex, absorbing set. The defining requirement for barrelled spaces is that every such barrel must contain a neighborhood of zero, making barrels a control mechanism for the topology of the space.

Many standard spaces are barrelled. Every Banach space and more generally every normed space is barrelled. Fréchet spaces (complete metrizable locally convex spaces) are barrelled as well. Montel spaces, which are barrelled and carry strong forms of compactness, are another important class.

The concept is closely tied to fundamental theorems of functional analysis. In a barrelled space, the Uniform Boundedness Principle (or related forms) holds under appropriate hypotheses for families of continuous linear operators. This makes barrelled spaces a natural setting for many convergence and continuity arguments beyond Banach spaces.

See also: barrel, locally convex space, Fréchet space, Montel space, Uniform Boundedness Principle, closed graph theorem.