Bornivorous set
In functional analysis, a subset of a real or complex vector space [math]\displaystyle{ X }[/math] that has an associated vector bornology [math]\displaystyle{ \mathcal{B} }[/math] is called bornivorous and a bornivore if it absorbs every element of [math]\displaystyle{ \mathcal{B}. }[/math] If [math]\displaystyle{ X }[/math] is a topological vector space (TVS) then a subset [math]\displaystyle{ S }[/math] of [math]\displaystyle{ X }[/math] is bornivorous if it is bornivorous with respect to the von-Neumann bornology of [math]\displaystyle{ X }[/math].
Bornivorous sets play an important role in the definitions of many classes of topological vector spaces, particularly bornological spaces.
Definitions
If [math]\displaystyle{ X }[/math] is a TVS then a subset [math]\displaystyle{ S }[/math] of [math]\displaystyle{ X }[/math] is called bornivorous[1] and a bornivore if [math]\displaystyle{ S }[/math] absorbs every bounded subset of [math]\displaystyle{ X. }[/math]
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 [math]\displaystyle{ X }[/math] 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 (that is, if it 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 [math]\displaystyle{ M }[/math] is a vector subspace of finite codimension in a locally convex space [math]\displaystyle{ X }[/math] and [math]\displaystyle{ B \subseteq M. }[/math] If [math]\displaystyle{ B }[/math] is a barrel (resp. bornivorous barrel, bornivorous disk) in [math]\displaystyle{ M }[/math] then there exists a barrel (resp. bornivorous barrel, bornivorous disk) [math]\displaystyle{ C }[/math] in [math]\displaystyle{ X }[/math] such that [math]\displaystyle{ B = C \cap M. }[/math][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 [math]\displaystyle{ X }[/math] 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 [math]\displaystyle{ X }[/math] be [math]\displaystyle{ \mathbb{R}^2 }[/math] as a vector space over the reals. If [math]\displaystyle{ S }[/math] is the balanced hull of the closed line segment between [math]\displaystyle{ (-1, 1) }[/math] and [math]\displaystyle{ (1, 1) }[/math] then [math]\displaystyle{ S }[/math] is not bornivorous but the convex hull of [math]\displaystyle{ S }[/math] is bornivorous. If [math]\displaystyle{ T }[/math] is the closed and "filled" triangle with vertices [math]\displaystyle{ (-1, -1), (-1, 1), }[/math] and [math]\displaystyle{ (1, 1) }[/math] then [math]\displaystyle{ T }[/math] is a convex set that is not bornivorous but its balanced hull is bornivorous.
See also
- Bounded set (topological vector space) – Generalization of boundedness
- Bornological space – Space where bounded operators are continuous
- Bornology – Mathematical generalization of boundedness
- Ultrabornological space
- Vector bornology
References
- ↑ 1.0 1.1 1.2 1.3 Narici & Beckenstein 2011, pp. 441-457.
- ↑ Narici & Beckenstein 2011, p. 442.
- ↑ Narici & Beckenstein 2011, p. 443.
- ↑ Narici & Beckenstein 2011, pp. 172-173.
- ↑ 5.0 5.1 Wilansky 2013, p. 50.
- ↑ Narici & Beckenstein 2011, pp. 371-423.
- ↑ Wilansky 2013, p. 48.
Bibliography
- 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.
- Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.
- Bourbaki, Nicolas (1987). Topological Vector Spaces: Chapters 1–5. Éléments de mathématique. 2. Berlin New York: Springer-Verlag. ISBN 978-3-540-42338-6. OCLC 17499190. http://www.numdam.org/item?id=AIF_1950__2__5_0.
- Conway, John B. (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.
- Edwards, Robert E. (Jan 1, 1995). Functional Analysis: Theory and Applications. New York: Dover Publications. ISBN 978-0-486-68143-6. OCLC 30593138.
- Grothendieck, Alexander (January 1, 1973). Topological Vector Spaces. New York: Gordon and Breach Science Publishers. ISBN 978-0-677-30020-7. OCLC 886098. https://archive.org/details/topologicalvecto0000grot.
- Template:Hogbe-Nlend Bornologies and Functional Analysis
- Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.
- Köthe, Gottfried (1969). Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. 159. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. OCLC 840293704.
- 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.
- Template:Kriegl Michor The Convenient Setting of Global Analysis
- 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.
- Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.
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