View source for Distinguished space

{{Short description|TVS whose strong dual is barralled}}
{{one source|date=June 2020}}

In [[functional analysis]] and related areas of [[mathematics]], '''distinguished spaces''' are [[topological vector spaces]] (TVSs) having the property that [[weak-* topology|weak-*]] bounded subsets of their biduals (that is, the [[strong dual space]] of their strong dual space) are contained in the weak-* [[Closure (topology)|closure]] of some bounded subset of the bidual.

==Definition==

Suppose that <math>X</math> is a [[locally convex space]] and let <math>X^{\prime}</math> and <math>X^{\prime}_b</math> denote the [[strong dual]] of <math>X</math> (that is, the [[continuous dual space]] of <math>X</math> endowed with the [[strong dual|strong dual topology]]). 
Let <math>X^{\prime \prime}</math> denote the continuous dual space of <math>X^{\prime}_b</math> and let <math>X^{\prime \prime}_b</math> denote the strong dual of <math>X^{\prime}_b.</math> 
Let <math>X^{\prime \prime}_{\sigma}</math> denote <math>X^{\prime \prime}</math> endowed with the [[weak-* topology]] induced by <math>X^{\prime},</math> where this topology is denoted by <math>\sigma\left(X^{\prime \prime}, X^{\prime}\right)</math> (that is, the topology of pointwise convergence on <math>X^{\prime}</math>). 
We say that a subset <math>W</math> of <math>X^{\prime \prime}</math> is <math>\sigma\left(X^{\prime \prime}, X^{\prime}\right)</math>-bounded if it is a bounded subset of <math>X^{\prime \prime}_{\sigma}</math> and we call the closure of <math>W</math> in the TVS <math>X^{\prime \prime}_{\sigma}</math> the <math>\sigma\left(X^{\prime \prime}, X^{\prime}\right)</math>-closure of <math>W</math>. 
If <math>B</math> is a subset of <math>X</math> then the [[Polar set|polar]] of <math>B</math> is <math>B^{\circ} := \left\{ x^{\prime} \in X^{\prime} : \sup_{b \in B} \left\langle b, x^{\prime} \right\rangle \leq 1 \right\}.</math> 

A [[Hausdorff space|Hausdorff]] [[Locally convex topological vector space|locally convex space]] <math>X</math> is called a '''distinguished space''' if it satisfies any of the following equivalent conditions: 

<ol>
<li>If <math>W \subseteq X^{\prime \prime}</math> is a <math>\sigma\left(X^{\prime \prime}, X^{\prime}\right)</math>-bounded subset of <math>X^{\prime \prime}</math> then there exists a bounded subset <math>B</math> of <math>X^{\prime \prime}_b</math> whose <math>\sigma\left(X^{\prime \prime}, X^{\prime}\right)</math>-closure contains <math>W</math>.{{sfn|Khaleelulla|1982|pp=32-63}}</li>
<li>If <math>W \subseteq X^{\prime \prime}</math> is a <math>\sigma\left(X^{\prime \prime}, X^{\prime}\right)</math>-bounded subset of <math>X^{\prime \prime}</math> then there exists a bounded subset <math>B</math> of <math>X</math> such that <math>W</math> is contained in <math>B^{\circ\circ} := \left\{ x^{\prime\prime} \in X^{\prime\prime} : \sup_{x^{\prime} \in B^{\circ}} \left\langle x^{\prime}, x^{\prime\prime} \right\rangle \leq 1 \right\},</math> which is the [[polar set|polar]] (relative to the [[dual system|duality]] <math>\left\langle X^{\prime}, X^{\prime\prime} \right\rangle</math>) of <math>B^{\circ}.</math>{{sfn|Khaleelulla|1982|pp=32-63}}</li>
<li>The [[Strong dual space|strong dual]] of <math>X</math> is a [[barrelled space]].{{sfn|Khaleelulla|1982|pp=32-63}}</li>
</ol>

If in addition <math>X</math> is a [[Metrizable topological vector space|metrizable]] [[locally convex topological vector space]] then this list may be extended to include: 

<ol start=4>
<li>([[Alexander Grothendieck|Grothendieck]]) The strong dual of <math>X</math> is a [[bornological space]].{{sfn|Khaleelulla|1982|pp=32-63}}</li>
</ol>

==Sufficient conditions==

All [[normed space]]s and [[semi-reflexive space]]s are distinguished spaces.{{sfn|Khaleelulla|1982|pp=28-63}} 
[[LF space]]s are distinguished spaces.  

The [[strong dual space]] <math>X_b^{\prime}</math> of a [[Fréchet space]] <math>X</math> is distinguished if and only if <math>X</math> is [[Quasibarrelled space|quasibarrelled]].<ref name="Gabriyelyan 2014">Gabriyelyan, S.S. [https://arxiv.org/pdf/1412.1497.pdf "On topological spaces and topological groups with certain local countable networks] (2014)</ref>  

==Properties==

Every locally convex distinguished space is an [[H-space]].{{sfn|Khaleelulla|1982|pp=28-63}}  

==Examples==

There exist distinguished [[Banach space]]s spaces that are not [[semi-reflexive]].{{sfn|Khaleelulla|1982|pp=32-63}} 
The [[strong dual]] of a distinguished Banach space is not necessarily [[Separable space|separable]]; [[Lp space|<math>l^{1}</math>]] is such a space.{{sfn|Khaleelulla|1982|pp=32-630}} 
The [[strong dual space]] of a distinguished [[Fréchet space]] is not necessarily [[metrizable]].{{sfn|Khaleelulla|1982|pp=32-63}}
There exists a distinguished [[Semi-reflexive space|semi-reflexive]] <em>non</em>-[[Reflexive space|reflexive]] {{em|non}}-[[Quasibarrelled space|quasibarrelled]] [[Mackey space]] <math>X</math> whose strong dual is a non-reflexive Banach space.{{sfn|Khaleelulla|1982|pp=32-63}}  
There exist [[H-space]]s that are not distinguished spaces.{{sfn|Khaleelulla|1982|pp=32-63}}  

Fréchet [[Montel space]]s are distinguished spaces. 

==See also==

* {{annotated link|Montel space}}
* {{annotated link|Semi-reflexive space}}

==References==

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

==Bibliography==

* {{cite journal|last = Bourbaki|first = Nicolas|authorlink = Nicolas Bourbaki|journal = [[Annales de l'Institut Fourier]]|language = French|mr = 0042609|pages = 5–16 (1951)|title = Sur certains espaces vectoriels topologiques|url = http://www.numdam.org/item?id=AIF_1950__2__5_0|volume = 2|year = 1950| doi = 10.5802/aif.16|doi-access = free}}
* {{Robertson Topological Vector Spaces}} <!--{{sfn|Robertson|Robertson|1980|p=}}-->
* {{Husain Khaleelulla Barrelledness in Topological and Ordered Vector Spaces}} <!--{{sfn|Husain|Khaleelulla|1978|p=}}-->
* {{Jarchow Locally Convex Spaces}} <!--{{sfn|Jarchow|1981|p=}}-->
* {{Khaleelulla Counterexamples in Topological Vector Spaces}} <!--{{sfn|Khaleelulla|1982|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=}}-->
* {{Trèves François Topological vector spaces, distributions and kernels}} <!--{{sfn|Trèves|2006|p=}}-->

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

[[Category:Topological vector spaces]]