Bornivorous set

Short description: A set that can absorb any bounded subset

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

Infrabornivorous sets and infrabounded maps

A linear map between two TVSs is called infrabounded if it maps Banach disks to bounded disks.

A disk in $X$ is called infrabornivorous if it absorbs every Banach disk.

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

Properties

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

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

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 $B=C\cap M.$

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.

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.

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.

See also

• Bounded set (topological vector space) – Generalization of boundedness
• Bornological space – Space where bounded operators are continuous
• Bornology – Mathematical generalization of boundedness
• Ultrabornological space
• Vector bornology

Bibliography

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