In functional analysis, a subset of a real or complex vector space
is a topological vector space (TVS) then a subset
Bornivorous sets play an important role in the definitions of many classes of topological vector spaces, particularly bornological spaces.
absorbs every bounded subset of
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] A linear map between two TVSs is called infrabounded if it maps Banach disks to bounded disks.
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] 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.
is a vector subspace of finite codimension in a locally convex space
then there exists a barrel (resp.
[6] Every neighborhood of the origin in a TVS is bornivorous.
is a TVS in which every bounded subset is contained in a finite dimensional vector subspace, then every absorbing set is a bornivore.
is the balanced hull of the closed line segment between
is not bornivorous but the convex hull of
is the closed and "filled" triangle with vertices