Barrelled space

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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.

Contents 1 History 2 Examples 3 Properties 4 Reference

[edit] History

[Bourbaki] invented terms such as "barrel" and "barrelled" space (from wine barrels), as well as "bornographic" space...^[1]

[edit] Examples

• 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.

[edit] Properties

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

[edit] Reference

1. ^ Liliane Beaulieu, Bourbaki's Art of Memory (in Commemorating Scientific Disciplines: Memorializing Objectivity), Osiris, 2nd Series, Vol. 14, Commemorative, Practices in Science: Historical Perspectives on the Politics of Collective Memory. (1999), pp. 219-251.