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.

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

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

1. If ${\displaystyle W\subseteq X^{\prime \prime }}${\displaystyle W\subseteq X^{\prime \prime }} is a ${\displaystyle \sigma \left(X^{\prime \prime },X^{\prime }\right)}${\displaystyle \sigma \left(X^{\prime \prime },X^{\prime }\right)}-bounded subset of ${\displaystyle X^{\prime \prime }}${\displaystyle X^{\prime \prime }} then there exists a bounded subset ${\displaystyle B}${\displaystyle B} of ${\displaystyle X_{b}^{\prime \prime }}${\displaystyle X_{b}^{\prime \prime }} whose ${\displaystyle \sigma \left(X^{\prime \prime },X^{\prime }\right)}${\displaystyle \sigma \left(X^{\prime \prime },X^{\prime }\right)}-closure contains ${\displaystyle W}${\displaystyle W}.^[1]
2. If ${\displaystyle W\subseteq X^{\prime \prime }}${\displaystyle W\subseteq X^{\prime \prime }} is a ${\displaystyle \sigma \left(X^{\prime \prime },X^{\prime }\right)}${\displaystyle \sigma \left(X^{\prime \prime },X^{\prime }\right)}-bounded subset of ${\displaystyle X^{\prime \prime }}${\displaystyle X^{\prime \prime }} then there exists a bounded subset ${\displaystyle B}${\displaystyle B} of ${\displaystyle X}${\displaystyle X} such that ${\displaystyle W}${\displaystyle W} is contained in ${\displaystyle B^{\circ \circ }:=\left\{x^{\prime \prime }\in X^{\prime \prime }:\sup _{x^{\prime }\in B^{\circ }}\left\langle x^{\prime },x^{\prime \prime }\right\rangle \leq 1\right\},}${\displaystyle 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\},} which is the polar (relative to the duality ${\displaystyle \left\langle X^{\prime },X^{\prime \prime }\right\rangle }${\displaystyle \left\langle X^{\prime },X^{\prime \prime }\right\rangle }) of ${\displaystyle B^{\circ }.}${\displaystyle B^{\circ }.}^[1]
3. The strong dual of ${\displaystyle X}${\displaystyle X} is a barrelled space.^[1]

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

4. (Grothendieck) The strong dual of ${\displaystyle X}${\displaystyle X} is a bornological space.^[1]

All normed spaces and semi-reflexive spaces are distinguished spaces.^[2] LF spaces are distinguished spaces.

The strong dual space ${\displaystyle X_{b}^{\prime }}${\displaystyle X_{b}^{\prime }} of a Fréchet space ${\displaystyle X}${\displaystyle X} is distinguished if and only if ${\displaystyle X}${\displaystyle X} is quasibarrelled.^[3]

Every locally convex distinguished space is an H-space.^[2]

There exist distinguished Banach spaces spaces that are not semi-reflexive.^[1] The strong dual of a distinguished Banach space is not necessarily separable; ${\displaystyle l^{1}}${\displaystyle l^{1}} is such a space.^[4] The strong dual space of a distinguished Fréchet space is not necessarily metrizable.^[1] There exists a distinguished semi-reflexive non-reflexive non-quasibarrelled Mackey space ${\displaystyle X}${\displaystyle X} whose strong dual is a non-reflexive Banach space.^[1] There exist H-spaces that are not distinguished spaces.^[1]

Fréchet Montel spaces are distinguished spaces.

• Montel space – Barrelled space where closed and bounded subsets are compact
• Semi-reflexive space

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• Topological vector spaces

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