In functional analysis, a discipline within mathematics, a locally convex topological vector space (TVS) is said to be infrabarrelled (also spelled infrabarreled) if every bounded barrel is a neighborhood of the origin. 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.
Definition A subset B {\displaystyle B} of a topological vector space (TVS) X {\displaystyle X} is called bornivorous if it absorbs all bounded subsets of X {\displaystyle X} ; that is, if for each bounded subset S {\displaystyle S} of X , {\displaystyle X,} there exists some scalar r {\displaystyle r} such that S ⊆ r B . {\displaystyle S\subseteq rB.} 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.
Characterizations If X {\displaystyle X} is a Hausdorff locally convex space then the canonical injection from X {\displaystyle X} into its bidual is a topological embedding if and only if X {\displaystyle X} is infrabarrelled. A Hausdorff topological vector space X {\displaystyle X} is quasibarrelled if and only if every bounded closed linear operator from X {\displaystyle X} into a complete metrizable TVS is continuous. By definition, a linear F : X → Y {\displaystyle F:X\to Y} operator is called closed if its graph is a closed subset of X × Y . {\displaystyle X\times Y.} For a locally convex space X {\displaystyle X} with continuous dual X ′ {\displaystyle X^{\prime }} the following are equivalent:
X {\displaystyle X} is quasibarrelled. Every bounded lower semi-continuous semi-norm on X {\displaystyle X} is continuous. Every β ( X ′ , X ) {\displaystyle \beta (X',X)} -bounded subset of the continuous dual space X ′ {\displaystyle X^{\prime }} is equicontinuous. If X {\displaystyle X} is a metrizable locally convex TVS then the following are equivalent:
The strong dual of X {\displaystyle X} is quasibarrelled. The strong dual of X {\displaystyle X} is barrelled. The strong dual of X {\displaystyle X} is bornological.
Properties Every quasi-complete infrabarrelled space is barrelled. A locally convex Hausdorff quasibarrelled space that is sequentially complete is barrelled. A locally convex Hausdorff quasibarrelled space is a Mackey space, quasi-M-barrelled, and countably quasibarrelled. A locally convex quasibarrelled space that is also a σ-barrelled space is necessarily a barrelled space. A locally convex space is reflexive if and only if it is semireflexive and quasibarrelled.
Examples Every barrelled space 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. Every Hausdorff barrelled space and every Hausdorff bornological space is quasibarrelled. Thus, every metrizable TVS is quasibarrelled. Note that there exist quasibarrelled spaces that are neither barrelled nor bornological. There exist Mackey spaces that are not quasibarrelled. There exist distinguished spaces, DF-spaces, and σ {\displaystyle \sigma } -barrelled spaces that are not quasibarrelled. The strong dual space X b ′ {\displaystyle X_{b}^{\prime }} of a Fréchet space X {\displaystyle X} is distinguished if and only if X {\displaystyle X} is quasibarrelled.
Counter-examples There exists a DF-space that is not quasibarrelled. There exists a quasibarrelled DF-space that is not bornological. There exists a quasibarrelled space that is not a σ-barrelled space.
See also Barrelled space – Type of topological vector space Reflexive space – Locally convex topological vector space Semi-reflexive space
References
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