Английская Википедия:Bornivorous set

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Шаблон:Short description In functional analysis, a subset of a real or complex vector space <math>X</math> that has an associated vector bornology <math>\mathcal{B}</math> is called bornivorous and a bornivore if it absorbs every element of <math>\mathcal{B}.</math> If <math>X</math> is a topological vector space (TVS) then a subset <math>S</math> of <math>X</math> is bornivorous if it is bornivorous with respect to the von-Neumann bornology of <math>X</math>.

Bornivorous sets play an important role in the definitions of many classes of topological vector spaces, particularly bornological spaces.

Definitions

If <math>X</math> is a TVS then a subset <math>S</math> of <math>X</math> is called Шаблон:Visible anchorШаблон:Sfn and a Шаблон:Visible anchor if <math>S</math> absorbs every bounded subset of <math>X.</math>

An absorbing disk in a locally convex space is bornivorous if and only if its Minkowski functional is locally bounded (i.e. maps bounded sets to bounded sets).Шаблон:Sfn

Infrabornivorous sets and infrabounded maps

A linear map between two TVSs is called Шаблон:Visible anchor if it maps Banach disks to bounded disks.Шаблон:Sfn

A disk in <math>X</math> is called Шаблон:Visible anchor if it absorbs every Banach disk.Шаблон:Sfn

An absorbing disk in a locally convex space is infrabornivorous if and only if its Minkowski functional is infrabounded.Шаблон:Sfn A disk in a Hausdorff locally convex space is infrabornivorous if and only if it absorbs all compact disks (that is, if it is "Шаблон:Visible anchor").Шаблон:Sfn

Properties

Every bornivorous and infrabornivorous subset of a TVS is absorbing. In a pseudometrizable TVS, every bornivore is a neighborhood of the origin.Шаблон:Sfn

Two TVS topologies on the same vector space have that same bounded subsets if and only if they have the same bornivores.Шаблон:Sfn

Suppose <math>M</math> is a vector subspace of finite codimension in a locally convex space <math>X</math> and <math>B \subseteq M.</math> If <math>B</math> is a barrel (resp. bornivorous barrel, bornivorous disk) in <math>M</math> then there exists a barrel (resp. bornivorous barrel, bornivorous disk) <math>C</math> in <math>X</math> such that <math>B = C \cap M.</math>Шаблон:Sfn

Examples and sufficient conditions

Every neighborhood of the origin in a TVS is bornivorous. The convex hull, closed convex hull, and balanced hull of a bornivorous set is again bornivorous. The preimage of a bornivore under a bounded linear map is a bornivore.Шаблон:Sfn

If <math>X</math> is a TVS in which every bounded subset is contained in a finite dimensional vector subspace, then every absorbing set is a bornivore.Шаблон:Sfn

Counter-examples

Let <math>X</math> be <math>\mathbb{R}^2</math> as a vector space over the reals. If <math>S</math> is the balanced hull of the closed line segment between <math>(-1, 1)</math> and <math>(1, 1)</math> then <math>S</math> is not bornivorous but the convex hull of <math>S</math> is bornivorous. If <math>T</math> is the closed and "filled" triangle with vertices <math>(-1, -1), (-1, 1),</math> and <math>(1, 1)</math> then <math>T</math> is a convex set that is not bornivorous but its balanced hull is bornivorous.

See also

References

Шаблон:Reflist Шаблон:Reflist

Bibliography

Шаблон:Functional analysis Шаблон:Boundedness and bornology Шаблон:Topological vector spaces

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