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

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

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

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

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

Sufficient conditions

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

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

Properties

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

Examples

There exist distinguished Banach spaces spaces that are not semi-reflexive.^[1] The strong dual of a distinguished Banach space is not necessarily separable; ${\displaystyle l^{1}}$ 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 ${\displaystyle X}$ 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.

See also

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

References

1. Khaleelulla 1982, pp. 32–63.
2. Khaleelulla 1982, pp. 28–63.
3. Gabriyelyan, S.S. "On topological spaces and topological groups with certain local countable networks (2014)
4. Khaleelulla 1982, pp. 32–630.

Bibliography

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• Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.
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