View source for Bornivorous set

{{Short description|A set that can absorb any bounded subset}}
In [[Functional analysis|functional analysis]], a subset of a real or complex vector space <math>X</math> that has an associated [[Vector bornology|vector bornology]] <math>\mathcal{B}</math> is called '''bornivorous''' and a '''bornivore''' if it [[Absorbing set|absorbs]] every element of <math>\mathcal{B}.</math>  
If <math>X</math> is a [[Topological vector space|topological vector space]] (TVS) then a subset <math>S</math> of <math>X</math> is '''bornivorous''' if it is bornivorous with respect to the [[Bornological space|von-Neumann bornology of <math>X</math>]].  

Bornivorous sets play an important role in the definitions of many classes of topological vector spaces, particularly [[Bornological space|bornological space]]s.  

==Definitions==

If <math>X</math> is a TVS then a subset <math>S</math> of <math>X</math> is called '''{{visible anchor|bornivorous}}'''{{sfn|Narici|Beckenstein|2011|pp=441-457}} and a '''{{visible anchor|bornivore}}''' if <math>S</math> [[Absorbing set|absorbs]] every [[Bounded set (topological vector space)|bounded subset]] of <math>X.</math>  

An [[Absorbing set|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).{{sfn|Narici|Beckenstein|2011|pp=441-457}}  

===Infrabornivorous sets and infrabounded maps===

A linear map between two TVSs is called '''{{visible anchor|infrabounded}}''' if it maps [[Auxiliary normed space|Banach disks]] to bounded disks.{{sfn|Narici|Beckenstein|2011|p=442}}  

A disk in <math>X</math> is called '''{{visible anchor|infrabornivorous}}''' if it [[Absorbing set|absorbs]] every Banach disk.{{sfn|Narici|Beckenstein|2011|p=443}}  

An [[Absorbing set|absorbing]] disk in a locally convex space is infrabornivorous if and only if its [[Minkowski functional]] is infrabounded.{{sfn|Narici|Beckenstein|2011|pp=441-457}}  
A disk in a Hausdorff locally convex space is infrabornivorous if and only if it absorbs all compact disks (that is, if it is "{{visible anchor|compactivorous}}").{{sfn|Narici|Beckenstein|2011|pp=441-457}}

==Properties==

Every bornivorous and infrabornivorous subset of a TVS is [[Absorbing set|absorbing]].  In a pseudometrizable TVS, every bornivore is a neighborhood of the origin.{{sfn|Narici|Beckenstein|2011|pp=172-173}}  

Two TVS topologies on the same vector space have that same bounded subsets if and only if they have the same bornivores.{{sfn|Wilansky|2013|p=50}}  

Suppose <math>M</math> is a vector subspace of finite codimension in a locally convex space <math>X</math> and <math>B \subseteq M.</math> If <math>B</math> is a barrel (resp. bornivorous barrel, bornivorous disk) in <math>M</math> then there exists a barrel (resp. bornivorous barrel, bornivorous disk) <math>C</math> in <math>X</math> such that <math>B = C \cap M.</math>{{sfn|Narici|Beckenstein|2011|pp=371-423}}  

==Examples and sufficient conditions==

Every neighborhood of the origin in a TVS is bornivorous.  
The convex hull, closed convex hull, and [[Balanced set|balanced hull]] of a bornivorous set is again bornivorous.  
The preimage of a bornivore under a bounded linear map is a bornivore.{{sfn|Wilansky|2013|p=48}}

If <math>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.{{sfn|Wilansky|2013|p=50}}

===Counter-examples===

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

==See also==

* {{annotated link|Bounded set (topological vector space)}}
* {{annotated link|Bornological space}}
* {{annotated link|Bornology}}
* {{annotated link|Ultrabornological space}}
* {{annotated link|Vector bornology}}

==References==

{{reflist|group=note}}
{{reflist}}

==Bibliography==

* {{Adasch Topological Vector Spaces|edition=2}} <!--{{sfn|Adasch|Ernst|Keim|1978|p=}}-->
* {{Berberian Lectures in Functional Analysis and Operator Theory}} <!--{{sfn|Berberian|1974|p=}}-->
* {{Bourbaki Topological Vector Spaces Part 1 Chapters 1–5}} <!--{{sfn|Bourbaki|1987|p=}}-->
* {{Conway A Course in Functional Analysis|edition=2}} <!--{{sfn|Conway|1990|p=}}-->
* {{Edwards Functional Analysis Theory and Applications}} <!--{{sfn|Edwards|1995|p=}}-->
* {{Grothendieck Topological Vector Spaces}} <!--{{sfn|Grothendieck|1973|p=}}-->
* {{Hogbe-Nlend Bornologies and Functional Analysis}} <!--{{sfn|Hogbe-Nlend|1977|p=}}-->
* {{Jarchow Locally Convex Spaces}} <!--{{sfn|Jarchow|1981|p=}}-->
* {{Köthe Topological Vector Spaces I}} <!--{{sfn|Köthe|1969|p=}}-->
* {{Khaleelulla Counterexamples in Topological Vector Spaces}} <!--{{sfn|Khaleelulla|1982|p=}}--> 
* {{Kriegl Michor The Convenient Setting of Global Analysis}} <!--{{sfn|Kriegl|Michor|1997|p=}}-->
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!--{{sfn|Narici|Beckenstein|2011|p=}}-->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!--{{sfn|Schaefer|Wolff|1999|p=}}-->
* {{Wilansky Modern Methods in Topological Vector Spaces}} <!--{{sfn|Wilansky|2013|p=}}-->

{{Functional analysis}}

[[Category:Topological vector spaces]]

{{Sourceattribution|Bornivorous set}}