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

Bibliography

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• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.