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
Definition: 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.
Definition:[1] If X is a TVS then a subset S of X is bornivorous 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 and infrabounded
Definition:[1] A linear map between two TVSs is called infrabounded if it maps Banach disks to bounded disks.
Definition:[1] A disk in X is called infrabornivorous 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.[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] 7}}

Properties
  • Every bornivorous and infrabornivorous subset of a TVS is absorbing.
  • In a pseudometrizable TVS, every bornivore is a neighborhood of the origin.[2]
  • 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.[3]
  • Two TVS topologies on the same vector space have that same bounded subsets if and only if they have the same bornivores.[4]

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.[5]
  • 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.[4]

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. 441-457.
  2. Narici 2011, pp. 172-173.
  3. Narici 2011, pp. 371-423.
  4. Wilansky 2013, p. 50.
  5. Wilansky 2013, p. 48.
  • Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978). Topological Vector Spaces: The Theory Without Convexity Conditions. Lecture Notes in Mathematics. {3834. Berlin New York: Springer-Verlag. ISBN 978-3-540-08662-8. OCLC 297140003.CS1 maint: ref=harv (link)
  • Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.CS1 maint: ref=harv (link)
  • Bourbaki, Nicolas (1987) [1981]. Topological Vector Spaces: Chapters 1–5 [Sur certains espaces vectoriels topologiques]. Annales de l'Institut Fourier. Elements of mathematics (in French). 2. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. ISBN 978-3-540-42338-6. OCLC 17499190.CS1 maint: ref=harv (link)
  • Conway, John (1990). A course in functional analysis. Graduate Texts in Mathematics. 96 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908.CS1 maint: ref=harv (link)
  • Edwards, Robert E. (Jan 1, 1995). Functional Analysis: Theory and Applications. New York: Dover Publications. ISBN 978-0-486-68143-6. OCLC 30593138.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
  • Grothendieck, Alexander (January 1, 1973). Topological Vector Spaces. Translated by Chaljub, Orlando. New York: Gordon and Breach Science Publishers. ISBN 978-0-677-30020-7. OCLC 886098.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
  • Hogbe-Nlend, Henri (1977). Bornologies and functional analysis. Amsterdam: North-Holland Publishing Co. pp. xii+144. ISBN 0-7204-0712-5. MR 0500064.
  • Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.CS1 maint: ref=harv (link)
  • Köthe, Gottfried (1969). Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704.CS1 maint: ref=harv (link)
  • Khaleelulla, S. M. (July 1, 1982). Written at Berlin Heidelberg. Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. 936. Berlin New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
  • Kriegl, Andreas; Michor, Peter W. (1997). The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs. American Mathematical Society. ISBN 9780821807804.
  • Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
  • Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.CS1 maint: ref=harv (link)
  • Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.CS1 maint: ref=harv (link)
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