Bornivorous set


In functional analysis, a subset of a real or complex vector space that has an associated vector bornology is called bornivorous and a bornivore if it absorbs every element of.
If is a topological vector space then a subset of is bornivorous if it is bornivorous with respect to the von-Neumann bornology of.
Bornivorous sets play an important role in the definitions of many classes of topological vector spaces.

Definitions

An absorbing disk in a locally convex space is bornivorous if and only if its Minkowski functional is locally bounded.
;Infrabornivorous and infrabounded
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.

Properties

Examples and sufficient conditions

Counter-examples

Let be as a vector space over the reals.
If is the balanced hull of the closed line segment between and then is not bornivorous but the convex hull of is bornivorous.
If is the closed and "filled" triangle with vertices,, and then is a convex set that is not bornivorous but its balanced hull is bornivorous.