Barrelled space

In functional analysis and related areas of mathematics, barrelled spaces are Hausdorff topological vector spaces for which 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 a form of the Banach–Steinhaus theorem still holds for them.

Barrelled spaces were introduced by Bourbaki (1950).

For a Hausdorff locally convex space ${\displaystyle X}$ with continuous dual ${\displaystyle X'}$ the following are equivalent:

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