Bornivorous set#infrabornivorous

In functional analysis, a subset of a real or complex vector space $X$ that has an associated vector bornology $\mathcal{B}$ is called bornivorous and a bornivore if it absorbs every element of $\mathcal{B}.$

If $X$ is a topological vector space (TVS) then a subset $S$ of $X$ is bornivorous if it is bornivorous with respect to the von-Neumann bornology of $X$ .

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

Definitions

If $X$ is a TVS then a subset $S$ of $X$ is called {{visible anchor|bornivorous}}{{sfn|Narici|Beckenstein|2011|pp=441-457}} and a {{visible anchor|bornivore}} if $S$ absorbs every bounded subset of $X.$

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).{{sfn|Narici|Beckenstein|2011|pp=441-457}}

=Infrabornivorous sets and infrabounded maps=

A linear map between two TVSs is called {{visible anchor|infrabounded}} if it maps Banach disks to bounded disks.{{sfn|Narici|Beckenstein|2011|p=442}}

A disk in $X$ is called {{visible anchor|infrabornivorous}} if it absorbs every Banach disk.{{sfn|Narici|Beckenstein|2011|p=443}}

An absorbing disk in a locally convex space is infrabornivorous if and only if its Minkowski functional is infrabounded.{{sfn|Narici|Beckenstein|2011|pp=441-457}}

A disk in a Hausdorff locally convex space is infrabornivorous if and only if it absorbs all compact disks (that is, if it is "{{visible anchor|compactivorous}}").{{sfn|Narici|Beckenstein|2011|pp=441-457}}

Properties

Every bornivorous and infrabornivorous subset of a TVS is absorbing. In a pseudometrizable TVS, every bornivore is a neighborhood of the origin.{{sfn|Narici|Beckenstein|2011|pp=172-173}}

Two TVS topologies on the same vector space have that same bounded subsets if and only if they have the same bornivores.{{sfn|Wilansky|2013|p=50}}

Suppose $M$ is a vector subspace of finite codimension in a locally convex space $X$ and $B \subseteq M.$ If $B$ is a barrel (resp. bornivorous barrel, bornivorous disk) in $M$ then there exists a barrel (resp. bornivorous barrel, bornivorous disk) $C$ in $X$ such that $B = C \cap M.$ {{sfn|Narici|Beckenstein|2011|pp=371-423}}

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.{{sfn|Wilansky|2013|p=48}}

If $X$ is a TVS in which every bounded subset is contained in a finite dimensional vector subspace, then every absorbing set is a bornivore.{{sfn|Wilansky|2013|p=50}}

=Counter-examples=

Let $X$ be $\mathbb{R}^2$ as a vector space over the reals.

If $S$ is the balanced hull of the closed line segment between $(-1, 1)$ and $(1, 1)$ then $S$ is not bornivorous but the convex hull of $S$ is bornivorous.

If $T$ is the closed and "filled" triangle with vertices $(-1, -1), (-1, 1),$ and $(1, 1)$ then $T$ is a convex set that is not bornivorous but its balanced hull is bornivorous.

References

Bibliography

{{Adasch Topological Vector Spaces|edition=2}}
{{Berberian Lectures in Functional Analysis and Operator Theory}}
{{Bourbaki Topological Vector Spaces Part 1 Chapters 1–5}}
{{Conway A Course in Functional Analysis|edition=2}}
{{Edwards Functional Analysis Theory and Applications}}
{{Grothendieck Topological Vector Spaces}}
{{Hogbe-Nlend Bornologies and Functional Analysis}}
{{Jarchow Locally Convex Spaces}}
{{Köthe Topological Vector Spaces I}}
{{Khaleelulla Counterexamples in Topological Vector Spaces}}
{{Kriegl Michor The Convenient Setting of Global Analysis}}
{{Narici Beckenstein Topological Vector Spaces|edition=2}}
{{Schaefer Wolff Topological Vector Spaces|edition=2}}
{{Wilansky Modern Methods in Topological Vector Spaces}}

Category:Topological vector spaces