Bornivorous set

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]

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]

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]

• Bounded linear operator – Linear transformation between topological vector spacesPages displaying short descriptions of redirect targets
• 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

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