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In functional analysis, a discipline within mathematics, a locally convex topological vector space (TVS) is said to be infrabarrelled (also spelled infra barreled) if every bounded barrel is a neighborhood of the origin. [1]
Similarly, quasibarrelled spaces are topological vector spaces (TVS) for which every bornivorous barrelled set in the space is a neighbourhood of the origin. Quasibarrelled spaces are studied because they are a weakening of the defining condition of barrelled spaces, for which a form of the Banach–Steinhaus theorem holds.
A subset of a topological vector space (TVS) is called bornivorous if it absorbs all bounded subsets of ; that is, if for each bounded subset of there exists some scalar such that A barrelled set or a barrel in a TVS is a set which is convex, balanced, absorbing and closed. A quasibarrelled space is a TVS for which every bornivorous barrelled set in the space is a neighbourhood of the origin. [2] [3]
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. [4]
A Hausdorff topological vector space is quasibarrelled if and only if every bounded closed linear operator from into a complete metrizable TVS is continuous. [5] By definition, a linear operator is called closed if its graph is a closed subset of
For a locally convex space with continuous dual the following are equivalent:
If is a metrizable locally convex TVS then the following are equivalent:
Every quasi-complete infrabarrelled space is barrelled. [1]
A locally convex Hausdorff quasibarrelled space that is sequentially complete is barrelled. [6]
A locally convex Hausdorff quasibarrelled space is a Mackey space, quasi-M-barrelled, and countably quasibarrelled. [7]
A locally convex quasibarrelled space that is also a σ-barrelled space is necessarily a barrelled space. [3]
A locally convex space is reflexive if and only if it is semireflexive and quasibarrelled. [3]
Every barrelled space is infrabarrelled. [1] A closed vector subspace of an infrabarrelled space is, however, not necessarily infrabarrelled. [8]
Every product and locally convex direct sum of any family of infrabarrelled spaces is infrabarrelled. [8] Every separated quotient of an infrabarrelled space is infrabarrelled. [8]
Every Hausdorff barrelled space and every Hausdorff bornological space is quasibarrelled. [9] Thus, every metrizable TVS is quasibarrelled.
Note that there exist quasibarrelled spaces that are neither barrelled nor bornological. [3] There exist Mackey spaces that are not quasibarrelled. [3] There exist distinguished spaces, DF-spaces, and -barrelled spaces that are not quasibarrelled. [3]
The strong dual space of a Fréchet space is distinguished if and only if is quasibarrelled. [10]
There exists a DF-space that is not quasibarrelled. [3]
There exists a quasibarrelled DF-space that is not bornological. [3]
There exists a quasibarrelled space that is not a σ-barrelled space. [3]
![]() | This article has multiple issues. Please help
improve it or discuss these issues on the
talk page. (
Learn how and when to remove these template messages)
|
In functional analysis, a discipline within mathematics, a locally convex topological vector space (TVS) is said to be infrabarrelled (also spelled infra barreled) if every bounded barrel is a neighborhood of the origin. [1]
Similarly, quasibarrelled spaces are topological vector spaces (TVS) for which every bornivorous barrelled set in the space is a neighbourhood of the origin. Quasibarrelled spaces are studied because they are a weakening of the defining condition of barrelled spaces, for which a form of the Banach–Steinhaus theorem holds.
A subset of a topological vector space (TVS) is called bornivorous if it absorbs all bounded subsets of ; that is, if for each bounded subset of there exists some scalar such that A barrelled set or a barrel in a TVS is a set which is convex, balanced, absorbing and closed. A quasibarrelled space is a TVS for which every bornivorous barrelled set in the space is a neighbourhood of the origin. [2] [3]
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. [4]
A Hausdorff topological vector space is quasibarrelled if and only if every bounded closed linear operator from into a complete metrizable TVS is continuous. [5] By definition, a linear operator is called closed if its graph is a closed subset of
For a locally convex space with continuous dual the following are equivalent:
If is a metrizable locally convex TVS then the following are equivalent:
Every quasi-complete infrabarrelled space is barrelled. [1]
A locally convex Hausdorff quasibarrelled space that is sequentially complete is barrelled. [6]
A locally convex Hausdorff quasibarrelled space is a Mackey space, quasi-M-barrelled, and countably quasibarrelled. [7]
A locally convex quasibarrelled space that is also a σ-barrelled space is necessarily a barrelled space. [3]
A locally convex space is reflexive if and only if it is semireflexive and quasibarrelled. [3]
Every barrelled space is infrabarrelled. [1] A closed vector subspace of an infrabarrelled space is, however, not necessarily infrabarrelled. [8]
Every product and locally convex direct sum of any family of infrabarrelled spaces is infrabarrelled. [8] Every separated quotient of an infrabarrelled space is infrabarrelled. [8]
Every Hausdorff barrelled space and every Hausdorff bornological space is quasibarrelled. [9] Thus, every metrizable TVS is quasibarrelled.
Note that there exist quasibarrelled spaces that are neither barrelled nor bornological. [3] There exist Mackey spaces that are not quasibarrelled. [3] There exist distinguished spaces, DF-spaces, and -barrelled spaces that are not quasibarrelled. [3]
The strong dual space of a Fréchet space is distinguished if and only if is quasibarrelled. [10]
There exists a DF-space that is not quasibarrelled. [3]
There exists a quasibarrelled DF-space that is not bornological. [3]
There exists a quasibarrelled space that is not a σ-barrelled space. [3]