Bornivorous set

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 (e.g. Bornological spaces).

Definitions

If X is a TVS then a subset S of X is called bornivorous[1] and a bornivore 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).[1]

Infrabornivorous sets and infrabounded maps

A linear map between two TVSs is called infrabounded if it maps Banach disks to bounded disks.[2]

A disk in X 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 (i.e. is "compactivorous").[1]

Properties

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 M is a vector subspace of finite codimension in a locally convex space X and B ⊆ 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 ∩ M.[6]

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.[7]

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.[5]

Counter-examples

Let X be 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

References
    1. Narici & Beckenstein 2011, pp. 441-457.
    2. Narici & Beckenstein 2011, p. 442.
    3. Narici & Beckenstein 2011, p. 443.
    4. Narici & Beckenstein 2011, pp. 172-173.
    5. Wilansky 2013, p. 50.
    6. Narici & Beckenstein 2011, pp. 371-423.
    7. Wilansky 2013, p. 48.

    Bibliography
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