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.
- 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 β(X,X').