View source for Ultrabornological space

In [[functional analysis]], a [[topological vector space]] (TVS) <math>X</math> is called '''ultrabornological''' if every [[bounded linear operator]] from <math>X</math> into another TVS is necessarily [[Continuous map|continuous]]. A general version of the [[closed graph theorem]] holds for ultrabornological spaces. 
Ultrabornological spaces were introduced by [[Alexander Grothendieck]] (Grothendieck [1955, p. 17] "espace du type (β)").{{sfn|Narici|Beckenstein|2011|p=441}} 

== Definitions ==

Let <math>X</math> be a [[topological vector space]] (TVS). 

=== Preliminaries ===

A '''[[Absolutely convex set|disk]]''' is a convex and [[Balanced set|balanced]] set.  
A disk in a TVS <math>X</math> is called '''[[Bornivorous set|bornivorous]]'''{{sfn|Narici|Beckenstein|2011|pp=441-457}} if it [[Absorbing set|absorbs]] every bounded subset of <math>X.</math>  

A linear map between two TVSs is called '''infrabounded'''{{sfn|Narici|Beckenstein|2011|pp=441-457}} if it maps [[Banach disk]]s to bounded disks.   

A disk <math>D</math> in a TVS <math>X</math> is called '''infrabornivorous''' if it satisfies any of the following equivalent conditions:
<ol>
<li><math>D</math> [[Absorbing set|absorbs]] every [[Banach disk]]s in <math>X.</math></li>
</ol>

while if <math>X</math> locally convex then we may add to this list:
<ol start=2>
<li>the [[Minkowski functional|gauge]] of <math>D</math> is an infrabounded map;{{sfn|Narici|Beckenstein|2011|pp=441-457}}</li>
</ol>

while if <math>X</math> locally convex and Hausdorff then we may add to this list:
<ol start=3>
<li><math>D</math> absorbs all compact disks;{{sfn|Narici|Beckenstein|2011|pp=441-457}} that is, <math>D</math> is "compactivorious".</li>
</ol>

=== Ultrabornological space ===

A TVS <math>X</math> is '''ultrabornological''' if it satisfies any of the following equivalent conditions:
<ol>
<li>every infrabornivorous disk in <math>X</math> is a neighborhood of the origin;{{sfn|Narici|Beckenstein|2011|pp=441-457}}</li>
</ol>

while if <math>X</math> is a locally convex space then we may add to this list:
<ol start=2>
<li>every bounded linear operator from <math>X</math> into a complete [[metrizable TVS]] is necessarily continuous;</li>
<li>every infrabornivorous disk is a neighborhood of 0;</li>
<li><math>X</math> be the inductive limit of the spaces <math>X_D</math> as {{mvar|D}} varies over all compact disks in <math>X</math>;</li>
<li>a seminorm on <math>X</math> that is bounded on each Banach disk is necessarily continuous;</li>
<li>for every locally convex space <math>Y</math> and every linear map <math>u : X \to Y,</math> if <math>u</math> is bounded on each Banach disk then <math>u</math> is continuous;</li>
<li>for every Banach space <math>Y</math> and every linear map <math>u : X \to Y,</math> if <math>u</math> is bounded on each Banach disk then <math>u</math> is continuous.</li>
</ol>

while if <math>X</math> is a Hausdorff locally convex space then we may add to this list:
<ol start=8>
<li><math>X</math> is an inductive limit of Banach spaces;{{sfn|Narici|Beckenstein|2011|pp=441-457}}</li>
</ol>

== Properties ==

Every [[locally convex]] ultrabornological space is [[Barrelled space|barrelled]],{{sfn|Narici|Beckenstein|2011|pp=441-457}} [[quasi-ultrabarrelled space]], and a [[bornological space]] but there exist bornological spaces that are not ultrabornological.

<ul>
<li>Every ultrabornological space <math>X</math> is the [[inductive limit]] of a family of [[Nuclear space|nuclear]] [[Fréchet  space]]s, spanning <math>X.</math></li>
<li>Every ultrabornological space <math>X</math> is the inductive limit of a family of nuclear [[DF-space]]s, spanning <math>X.</math></li>
</ul>

== Examples and sufficient conditions ==

The finite product of locally convex ultrabornological spaces is ultrabornological.{{sfn|Narici|Beckenstein|2011|pp=441-457}} Inductive limits of ultrabornological spaces are ultrabornological.  

Every Hausdorff sequentially complete [[bornological space]] is ultrabornological.{{sfn|Narici|Beckenstein|2011|pp=441-457}}  Thus every [[Complete topological vector space|complete]] Hausdorff [[bornological space]] is ultrabornological.  In particular, every [[Fréchet space]] is ultrabornological.{{sfn|Narici|Beckenstein|2011|pp=441-457}}  

The [[strong dual space]] of a [[Complete topological vector space|complete]] [[Schwartz topological vector space|Schwartz space]] is ultrabornological.  

Every Hausdorff [[bornological space]] that is [[quasi-complete]] is ultrabornological.{{citation needed|date=September 2020}}  

;Counter-examples

There exist [[ultrabarrelled space]]s that are not ultrabornological.  
There exist ultrabornological spaces  that are not ultrabarrelled.

== See also ==

* {{annotated link|Bounded linear operator}}
* {{annotated link|Bounded set (topological vector space)}}
* {{annotated link|Bornological space}}
* {{annotated link|Bornology}}
* {{annotated link|Locally convex topological vector space}}
* {{annotated link|Space of linear maps}}
* {{annotated link|Topological vector space}}
* {{annotated link|Vector bornology}}

== External links ==

* [http://www.numdam.org/article/AIF_1974__24_3_57_0.pdf Some characterizations of ultrabornological spaces]

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

* {{cite book
| last = Hogbe-Nlend
| first = Henri
| title = Bornologies and functional analysis
| publisher = North-Holland Publishing Co.
| location = Amsterdam
| year = 1977
| pages = xii+144
| isbn = 0-7204-0712-5
| mr = 0500064
}}
* {{Edwards Functional Analysis Theory and Applications}} <!-- {{sfn|Edwards|1995|p=}} -->
* {{Grothendieck Produits Tensoriels Topologiques et Espaces Nucléaires}} <!-- {{sfn|Grothendieck|1966|p=}} -->
* {{Grothendieck Topological Vector Spaces}} <!-- {{sfn|Grothendieck|1973|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|edition=1}} <!-- {{sfn|Wilansky|2013|p=}} -->

{{Functional analysis}}
{{Boundedness and bornology}}
{{Topological vector spaces}}

[[Category:Topological vector spaces]]