Barrelled space

In functional analysis and related areas of mathematics barrelled spaces are topological vector spaces where every barrelled set in the space is a neighbourhood for the zero vector. They are studied because the Banach-Steinhaus theorem still holds for them.

Examples

• Fréchet spaces are barelled and in particular Banach spaces. But generally a normed vector space is not barrelled.
• Montel spaces are barrelled
• locally convex spaces which are Baire spaces are barrelled.

Properties

• A locally convex space <math>X<math> with continuous dual <math>X'<math> is barralled if and only if it carries the strong topology <math>\beta(X, X')<math>.
• A locally convex space <math>X<math> with continuous dual <math>X'<math> is barralled if and only if it carries the Mackey topology <math>\tau(X, X')<math>.