distinguished space

Suppose that $X$ is a locally convex space and let $X^\{\backslash prime\}$ and $X^\{\backslash prime\}\_b$ denote the strong dual of $X$ (that is, the continuous dual space of $X$ endowed with the strong dual topology).

Let $X^\{\backslash prime\; \backslash prime\}$ denote the continuous dual space of $X^\{\backslash prime\}\_b$ and let $X^\{\backslash prime\; \backslash prime\}\_b$ denote the strong dual of $X^\{\backslash prime\}\_b.$

Let $X^\{\backslash prime\; \backslash prime\}\_\{\backslash sigma\}$ denote $X^\{\backslash prime\; \backslash prime\}$ endowed with the weak-* topology induced by $X^\{\backslash prime\},$ where this topology is denoted by $\backslash sigma\backslash left(X^\{\backslash prime\; \backslash prime\},\; X^\{\backslash prime\}\backslash right)$ (that is, the topology of pointwise convergence on $X^\{\backslash prime\}$).

We say that a subset $W$ of $X^\{\backslash prime\; \backslash prime\}$ is $\backslash sigma\backslash left(X^\{\backslash prime\; \backslash prime\},\; X^\{\backslash prime\}\backslash right)$-bounded if it is a bounded subset of $X^\{\backslash prime\; \backslash prime\}\_\{\backslash sigma\}$ and we call the closure of $W$ in the TVS $X^\{\backslash prime\; \backslash prime\}\_\{\backslash sigma\}$ the $\backslash sigma\backslash left(X^\{\backslash prime\; \backslash prime\},\; X^\{\backslash prime\}\backslash right)$-closure of $W$.

If $B$ is a subset of $X$ then the polar of $B$ is $B^\{\backslash circ\}\; :=\; \backslash left\backslash \{\; x^\{\backslash prime\}\; \backslash in\; X^\{\backslash prime\}\; :\; \backslash sup\_\{b\; \backslash in\; B\}\; \backslash left\backslash langle\; b,\; x^\{\backslash prime\}\; \backslash right\backslash rangle\; \backslash leq\; 1\; \backslash right\backslash \}.$

A Hausdorff locally convex space $X$ is called a distinguished space if it satisfies any of the following equivalent conditions:

1. If $W\; \backslash subseteq\; X^\{\backslash prime\; \backslash prime\}$ is a $\backslash sigma\backslash left(X^\{\backslash prime\; \backslash prime\},\; X^\{\backslash prime\}\backslash right)$-bounded subset of $X^\{\backslash prime\; \backslash prime\}$ then there exists a bounded subset $B$ of $X^\{\backslash prime\; \backslash prime\}\_b$ whose $\backslash sigma\backslash left(X^\{\backslash prime\; \backslash prime\},\; X^\{\backslash prime\}\backslash right)$-closure contains $W$.{{sfn|Khaleelulla|1982|pp=32-63}}
2. If $W\; \backslash subseteq\; X^\{\backslash prime\; \backslash prime\}$ is a $\backslash sigma\backslash left(X^\{\backslash prime\; \backslash prime\},\; X^\{\backslash prime\}\backslash right)$-bounded subset of $X^\{\backslash prime\; \backslash prime\}$ then there exists a bounded subset $B$ of $X$ such that $W$ is contained in $B^\{\backslash circ\backslash circ\}\; :=\; \backslash left\backslash \{\; x^\{\backslash prime\backslash prime\}\; \backslash in\; X^\{\backslash prime\backslash prime\}\; :\; \backslash sup\_\{x^\{\backslash prime\}\; \backslash in\; B^\{\backslash circ\}\}\; \backslash left\backslash langle\; x^\{\backslash prime\},\; x^\{\backslash prime\backslash prime\}\; \backslash right\backslash rangle\; \backslash leq\; 1\; \backslash right\backslash \},$ which is the polar (relative to the duality $\backslash left\backslash langle\; X^\{\backslash prime\},\; X^\{\backslash prime\backslash prime\}\; \backslash right\backslash rangle$) of $B^\{\backslash circ\}.${{sfn|Khaleelulla|1982|pp=32-63}}
3. The strong dual of $X$ is a barrelled space.{{sfn|Khaleelulla|1982|pp=32-63}}

If in addition $X$ is a metrizable locally convex topological vector space then this list may be extended to include:

4. (Grothendieck) The strong dual of $X$ is a bornological space.{{sfn|Khaleelulla|1982|pp=32-63}}

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 $X\_b^\{\backslash prime\}$ of a Fréchet space $X$ is distinguished if and only if $X$ is quasibarrelled.Gabriyelyan, S.S. [https://arxiv.org/pdf/1412.1497.pdf "On topological spaces and topological groups with certain local countable networks] (2014)

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

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; $l^\{1\}$ 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 non-reflexive {{em|non}}-quasibarrelled Mackey space $X$ 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.

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

{{reflist|group=note}}

{{reflist}}

• {{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}}

{{Functional analysis}}

{{Boundedness and bornology}}

{{Topological vector spaces}}

Category:Topological vector spaces