Bornivorous set

Template:Short description In functional analysis, a subset of a real or complex vector space $X$ that has an associated vector bornology $ℬ$ is called bornivorous and a bornivore if it absorbs every element of $ℬ.$ If $X$ is a topological vector space (TVS) then a subset $S$ of $X$ is bornivorous if it is bornivorous with respect to the von-Neumann bornology of $X$.

Bornivorous sets play an important role in the definitions of many classes of topological vector spaces, particularly bornological spaces.

Definitions

If $X$ is a TVS then a subset $S$ of $X$ is called Template:Visible anchorTemplate:Sfn and a Template:Visible anchor if $S$ absorbs every bounded subset of $X.$

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).Template:Sfn

Infrabornivorous sets and infrabounded maps

A linear map between two TVSs is called Template:Visible anchor if it maps Banach disks to bounded disks.Template:Sfn

A disk in $X$ is called Template:Visible anchor if it absorbs every Banach disk.Template:Sfn

An absorbing disk in a locally convex space is infrabornivorous if and only if its Minkowski functional is infrabounded.Template:Sfn A disk in a Hausdorff locally convex space is infrabornivorous if and only if it absorbs all compact disks (that is, if it is "Template:Visible anchor").Template:Sfn

Properties

Every bornivorous and infrabornivorous subset of a TVS is absorbing. In a pseudometrizable TVS, every bornivore is a neighborhood of the origin.Template:Sfn

Two TVS topologies on the same vector space have that same bounded subsets if and only if they have the same bornivores.Template:Sfn

Suppose $M$ is a vector subspace of finite codimension in a locally convex space $X$ and $B\subseteq M.$ If $B$ is a barrel (resp. bornivorous barrel, bornivorous disk) in $M$ then there exists a barrel (resp. bornivorous barrel, bornivorous disk) $C$ in $X$ such that $B=C\cap M.$Template:Sfn

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.Template:Sfn

If $X$ is a TVS in which every bounded subset is contained in a finite dimensional vector subspace, then every absorbing set is a bornivore.Template:Sfn

Counter-examples

Let $X$ be ${ℝ}^2$ as a vector space over the reals. If $S$ is the balanced hull of the closed line segment between $(-1,1)$ and $(1,1)$ then $S$ is not bornivorous but the convex hull of $S$ is bornivorous. If $T$ is the closed and "filled" triangle with vertices $(-1,-1),(-1,1),$ and $(1,1)$ then $T$ is a convex set that is not bornivorous but its balanced hull is bornivorous.

See also

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Bibliography

• Template:Adasch Topological Vector Spaces
• Template:Berberian Lectures in Functional Analysis and Operator Theory
• Template:Bourbaki Topological Vector Spaces Part 1 Chapters 1–5
• Template:Conway A Course in Functional Analysis
• Template:Edwards Functional Analysis Theory and Applications
• Template:Grothendieck Topological Vector Spaces
• Template:Hogbe-Nlend Bornologies and Functional Analysis
• Template:Jarchow Locally Convex Spaces
• Template:Köthe Topological Vector Spaces I
• Template:Khaleelulla Counterexamples in Topological Vector Spaces
• Template:Kriegl Michor The Convenient Setting of Global Analysis
• Template:Narici Beckenstein Topological Vector Spaces
• Template:Schaefer Wolff Topological Vector Spaces
• Template:Wilansky Modern Methods in Topological Vector Spaces