Bornivorous set

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

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

Definitions

If ${\displaystyle X}$ is a TVS then a subset ${\displaystyle S}$ of ${\displaystyle X}$ is called bornivorous ^[1] and a bornivore if ${\displaystyle S}$ absorbs every bounded subset of ${\displaystyle X.}$

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). ^[1]

Infrabornivorous sets and infrabounded maps

A linear map between two TVSs is called infrabounded if it maps Banach disks to bounded disks. ^[2]

A disk in ${\displaystyle X}$ is called infrabornivorous if it absorbs every Banach disk. ^[3]

An absorbing disk in a locally convex space is infrabornivorous if and only if its Minkowski functional is infrabounded. ^[1] 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"). ^[1]

Properties

Every bornivorous and infrabornivorous subset of a TVS is absorbing. In a pseudometrizable TVS, every bornivore is a neighborhood of the origin. ^[4]

Two TVS topologies on the same vector space have that same bounded subsets if and only if they have the same bornivores. ^[5]

Suppose ${\displaystyle M}$ is a vector subspace of finite codimension in a locally convex space ${\displaystyle X}$ and ${\displaystyle B\subseteq M.}$ If ${\displaystyle B}$ is a barrel (resp. bornivorous barrel, bornivorous disk) in ${\displaystyle M}$ then there exists a barrel (resp. bornivorous barrel, bornivorous disk) ${\displaystyle C}$ in ${\displaystyle X}$ such that ${\displaystyle B=C\cap M.}$ ^[6]

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. ^[7]

If ${\displaystyle X}$ is a TVS in which every bounded subset is contained in a finite dimensional vector subspace, then every absorbing set is a bornivore. ^[5]

Counter-examples

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

See also

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

References

1. Narici & Beckenstein 2011, pp. 441–457.
2. Narici & Beckenstein 2011, p. 442.
3. Narici & Beckenstein 2011, p. 443.
4. Narici & Beckenstein 2011, pp. 172–173.
5. Wilansky 2013, p. 50.
6. Narici & Beckenstein 2011, pp. 371–423.
7. Wilansky 2013, p. 48.

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