Barrelled space
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In functional analysis and related areas of mathematics barrelled spaces are Hausdorff 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.
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[edit] History
Barrelled spaces were introduced by Bourbaki in an article in Ann. Inst. Fourier , 2 (1950), pp. 5-16.
[edit] 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.
[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] References
- Robertson, A.P.; W.J. Robertson (1964). Topological vector spaces. Cambridge Tracts in Mathematics. 53. Cambridge University Press. pp. 65–75.
- Schaefer, Helmuth H. (1971). Topological vector spaces. GTM. 3. New York: Springer-Verlag. p. 60. ISBN 0-387-98726-6.
