In
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
, a discipline within mathematics, a
locally convex
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ve ...
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
(TVS) is said to be infrabarrelled (also spelled infra
barreled) if every
bounded absorbing barrel
A barrel or cask is a hollow cylindrical container with a bulging center, longer than it is wide. They are traditionally made of wooden staves and bound by wooden or metal hoops. The word vat is often used for large containers for liquids, ...
is a neighborhood of the origin.
Characterizations
If
is a Hausdorff locally convex space then
the canonical injection from
into its bidual is a topological embedding if and only if
is infrabarrelled.
Properties
Every
quasi-complete
In functional analysis, a topological vector space (TVS) is said to be quasi-complete or boundedly complete if every closed and bounded subset is complete.
This concept is of considerable importance for non- metrizable TVSs.
Properties
* Eve ...
infrabarrelled space is barrelled.
Examples
Every
barrelled space
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 ...
is infrabarrelled.
A closed vector subspace of an infrabarrelled space is, however, not necessarily infrabarrelled.
Every product and locally convex direct sum of any family of infrabarrelled spaces is infrabarrelled.
Every
separated quotient of an infrabarrelled space is infrabarrelled.
See also
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References
Bibliography
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{{Boundedness and bornology
Functional analysis
Topological vector spaces