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 and if A and B are subsets of X, then we say that A absorbs B if there exists a real number r > 0 such that B⊆sA for all scalars s such that |s| ≥ r.

If X is a TVS then a subset S of X is bornivorous if S absorbs every (von-Neumann) bounded subset of X. A disk in a TVS is called infrabornivorous if it absorbs every Banach disk. In a Hausdorff locally convex TVS, a disk is infrabornivorous if and only if it absorbs all compact disks.

Properties

Every bornivorous and infrabornivorous subset of a TVS is absorbing. In a pseudometrizable TVS, every bornivore is a neighborhood of the origin.[1]

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.

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 2011, pp. 172-173.
  • Hogbe-Nlend, Henri (1977). Bornologies and functional analysis. Amsterdam: North-Holland Publishing Co. pp. xii+144. ISBN 0-7204-0712-5. MR 0500064.
  • Khaleelulla, S. M. (1982). Written at Berlin Heidelberg. Counterexamples in topological vector spaces. GTM. 936. Berlin New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.CS1 maint: ref=harv (link)
  • Narici, Lawrence (2011). Topological vector spaces. Boca Raton, FL: CRC Press. ISBN 1-58488-866-0. OCLC 144216834.
  • H.H. Schaefer (1970). Topological Vector Spaces. GTM. 3. Springer-Verlag. pp. 61–63. ISBN 0-387-05380-8.
  • Khaleelulla, S.M. (1982). Counterexamples in Topological Vector Spaces. GTM. 936. Berlin Heidelberg: Springer-Verlag. pp. 29–33, 49, 104. ISBN 9783540115656.
  • Kriegl, Andreas; Michor, Peter W. (1997). The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs. American Mathematical Society. ISBN 9780821807804.
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