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. A barrelled set or a barrel in a topological vector space is a set which is convex, balanced, absorbing and closed. Barrelled spaces are studied because the Banach-Steinhaus theorem still holds for them.

History

Bourbaki invented terms such as "barrel" and "barrelled" space (from wine barrels), as well as "bornographic" space...

Examples

• In a semi normed vector space the unit ball is a barrel.
• Every locally convex topological vector space has a neighbourhood basis consisting of barrelled sets.
• Fréchet spaces, and in particular Banach spaces, are barrelled, but generally a normed vector space is not barrelled.
• Montel spaces are barrelled
• locally convex spaces which are Baire spaces are barrelled.
• a separated, complete Mackey space is barrelled.

Properties

• A locally convex space $X$ with continuous dual $X'$ is barrelled if and only if it carries the strong topology $\beta (X,X')$.