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Concept
Bornivorous set
Summary
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 (TVS) 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, particularly bornological spaces.
If is a TVS then a subset of 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).
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 "").
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 If is a barrel (resp. bornivorous barrel, bornivorous disk) in then there exists a barrel (resp. bornivorous barrel, bornivorous disk) in such that
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 is a TVS in which every bounded subset is contained in a finite dimensional vector subspace, then every absorbing set is a bornivore.
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.
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