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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 ${\displaystyle W}$.
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 ${\displaystyle B^{\circ }.}$
3. The strong dual of ${\displaystyle X}$ is a barrelled space.

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.

Sufficient conditions

All normed spaces and semi-reflexive spaces are distinguished spaces. 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.

Properties

Every locally convex distinguished space is an H-space.

Examples

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

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

• Bourbaki, Nicolas (1950). "Sur certains espaces vectoriels topologiques". Annales de l'Institut Fourier (in French). 2: 5–16 (1951). doi:10.5802/aif.16. MR 0042609.
• Robertson, Alex P.; Robertson, Wendy J. (1980). Topological Vector Spaces. Cambridge Tracts in Mathematics. Vol. 53. Cambridge England: Cambridge University Press. ISBN 978-0-521-29882-7. OCLC 589250.
• Husain, Taqdir; Khaleelulla, S. M. (1978). Barrelledness in Topological and Ordered Vector Spaces. Lecture Notes in Mathematics. Vol. 692. Berlin, New York, Heidelberg: Springer-Verlag. ISBN 978-3-540-09096-0. OCLC 4493665.
• Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.
• Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. Vol. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
• Trèves, François (2006) . Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.

• Topological vector spaces