In functional analysis and related areas of mathematics, a barrelled space (also written barreled space) is a topological vector space (TVS) 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 that 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).
vector. A barrelled set or a barrel in a topological vector space is a set that is convex, balanced, absorbing, and closed. Barrelledspaces are studied...
quasibarrelled space that is sequentially complete is barrelled. A locally convex Hausdorff quasibarrelled space is a Mackey space, quasi-M-barrelled, and countably...
Montel space is a barrelled topological vector space in which every closed and bounded subset is compact. A topological vector space (TVS) has the Heine–Borel...
a topological vector space (TVS) is called a barrel or a barrelled set if it is closed convex balanced and absorbing. Barrelled sets play an important...
Press. Husain, Taqdir; Khaleelulla, S. M. (1978). Barrelledness in Topological and Ordered Vector Spaces. Lecture Notes in Mathematics. Vol. 692. Berlin...
\mathbb {C} [x_{1},\ldots ,x_{n}].} Banach–Mazur game Barrelledspace – Type of topological vector space Blumberg theorem – Any real function on R admits a...
barrelled. Barrelledspace – Type of topological vector space Countably barrelledspace Countably quasi-barrelledspace Infrabarreled space Uniform boundedness...
only if the original set was meagre, and vice versa. Barrelledspace – Type of topological vector space Generic property – Property holding for typical examples...
condition, where the dual space is endowed with the topology of uniform convergence on totally bounded sets. Montel space: a barrelledspace where every closed...
separable, complete, quasi-barrelled, nor σ {\displaystyle \sigma } -quasi-barrelled. Mackey topology Topologies on spaces of linear maps Bourbaki 1987...
X_{b(X^{\prime },X)}^{\prime }} is complete. If X {\displaystyle X} is a barrelledspace, then its topology coincides with the strong topology β ( X , X ′ )...
replaced by metal hoops that were stronger, more durable and took up less space. Barrel has also been used as a standard size of measure, referring to a set...
space that is complete (and so sequentially complete) but neither barrelled nor bornological. Bornological spaces need not be barrelled and barrelled...
of redirect targets Barrelledspace – Type of topological vector space Closed graph – Graph of a map closed in the product spacePages displaying short...
OCLC 589250. Husain, Taqdir; Khaleelulla, S. M. (1978). Barrelledness in Topological and Ordered Vector Spaces. Lecture Notes in Mathematics. Vol. 692. Berlin...
are not Schwartz spaces and there exist Schwartz spaces that are not Montel spaces. Auxiliary normed space Montel space – Barrelledspace where closed and...
locally convex space X {\displaystyle X} onto a barrelledspace Y {\displaystyle Y} (e.g. every complete pseudometrizable space is barrelled) then T {\displaystyle...
ISBN 978-0-387-89488-1. Husain, Taqdir (1978). Barrelledness in topological and ordered vector spaces. Berlin New York: Springer-Verlag. ISBN 3-540-09096-7...
Banach spaces are barrelledspaces, which means that every barrel is neighborhood of the origin (all closed balls centered at the origin are barrels, for...
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