Distinguished space

In functional analysis and related areas of mathematics, '''distinguished spaces''' are topological vector spaces (TVSs) having the property that weak-* bounded subsets of their biduals (that is, the strong dual space of their strong dual space) are contained in the weak-* 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 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 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 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 (relative to the 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 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 locally convex topological vector space then this list may be extended to include:

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

==Sufficient conditions==

All normed spaces and semi-reflexive spaces are distinguished spaces.{{sfn|Khaleelulla|1982|pp=28-63}} LF spaces 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.<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 spaces spaces that are not semi-reflexive.{{sfn|Khaleelulla|1982|pp=32-63}} The strong dual of a distinguished Banach space is not necessarily separable; <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 <em>non</em>-reflexive {{em|non}}-quasibarrelled Mackey space <math>X</math> whose strong dual is a non-reflexive Banach space.{{sfn|Khaleelulla|1982|pp=32-63}} There exist H-spaces that are not distinguished spaces.{{sfn|Khaleelulla|1982|pp=32-63}}

Fréchet Montel spaces are distinguished spaces.

==See also==

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

==References==

==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}}  * {{Husain Khaleelulla Barrelledness in Topological and Ordered Vector Spaces}}  * {{Jarchow Locally Convex Spaces}}  * {{Khaleelulla Counterexamples in Topological Vector Spaces}}  * {{Narici Beckenstein Topological Vector Spaces|edition=2}}  * {{Schaefer Wolff Topological Vector Spaces|edition=2}}  * {{Trèves François Topological vector spaces, distributions and kernels}}

Category:Topological vector spaces