Bornivorous set explained

In functional analysis, a subset of a real or complex vector space

that has an associated vector bornology

l{B}

is called bornivorous and a bornivore if it absorbs every element of

l{B}.

is a topological vector space (TVS) then a subset

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, particularly bornological spaces.

Definitions

is a TVS then a subset

is called and a if

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).

Infrabornivorous sets and infrabounded maps

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

A disk in

is called 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 "").

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

is a vector subspace of finite codimension in a locally convex space

and

B\subseteqM.

is a barrel (resp. bornivorous barrel, bornivorous disk) in

then there exists a barrel (resp. bornivorous barrel, bornivorous disk)

such that

B=C\capM.

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.

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

R²

as a vector space over the reals. If

is the balanced hull of the closed line segment between

(-1,1)

and

(1,1)

then

is not bornivorous but the convex hull of

is bornivorous. If

is the closed and "filled" triangle with vertices

(-1,-1),(-1,1),

and

(1,1)

then

is a convex set that is not bornivorous but its balanced hull is bornivorous