Distinguished space

Suppose that X is a locally convex space and let ${\displaystyle X^{\prime }}$ and ${\displaystyle X_{b}^{\prime }}$ denote the strong dual of X (i.e. the continuous dual space of 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 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 W in the TVS ${\displaystyle X_{\sigma }^{\prime \prime }}$ the ${\displaystyle \sigma \left(X^{\prime \prime },X^{\prime }\right)}$-closure of W. If B is a subset of X then the polar of 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 TVS X is called a distinguished space if it satisfies any of the following equivalent conditions:

1. If W ⊆ ${\displaystyle X^{\prime \prime }}$ is a ${\displaystyle \sigma \left(X^{\prime \prime },X^{\prime }\right)}$-bounded subset of ${\displaystyle X^{\prime \prime }}$ then there exists a bounded subset B of ${\displaystyle X_{b}^{\prime \prime }}$ whose ${\displaystyle \sigma \left(X^{\prime \prime },X^{\prime }\right)}$-closure contains W.[1]
2. If W ⊆ ${\displaystyle X^{\prime \prime }}$ is a ${\displaystyle \sigma \left(X^{\prime \prime },X^{\prime }\right)}$-bounded subset of ${\displaystyle X^{\prime \prime }}$ then there exists a bounded subset B of X such that 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 X is a barrelled space.[1]

If in addition X is a metrizable locally convex space then we may add to this list:

4. The strong dual of X is a bornological space.[1]

Every normed space and semi-reflexive space is a distinguished space.[2]

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; ${\displaystyle l^{1}}$ is such a space.[3] The strong dual of a distinguished Fréchet space is not necessarily metrizable.[1] There exists a distinguished semi-reflexive non-reflexive non-quasibarrelled Mackey space X whose strong dual is a non-reflexive Banach space.[1] There exist H-spaces that are not distinguished spaces.[1]

• Montel space

• Bourbaki, Nicolas (1950). "Sur certains espaces vectoriels topologiques". Annales de l'Institut Fourier (in French). 2: 5–16 (1951). MR 0042609.
• Robertson, Alex P.; Robertson, Wendy J. (1964). Topological vector spaces. Cambridge Tracts in Mathematics. 53. Cambridge University Press. pp. 65–75.
• Husain, Taqdir (1978). Barrelledness in topological and ordered vector spaces. Berlin New York: Springer-Verlag. ISBN 3-540-09096-7. OCLC 4493665.CS1 maint: ref=harv (link)
• Jarhow, Hans (1981). Locally convex spaces. Teubner. ISBN 978-3-322-90561-1.CS1 maint: ref=harv (link)
• Khaleelulla, S. M. (1982). Written at Berlin Heidelberg. Counterexamples in topological vector spaces. GTM. 936. Berlin New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.CS1 maint: ref=harv (link)
• Schaefer, Helmut H. (1971). Topological vector spaces. GTM. 3. New York: Springer-Verlag. p. 60. ISBN 0-387-98726-6.
• Schaefer, Helmut H. (1999). Topological Vector Spaces. GTM. 3. New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.CS1 maint: ref=harv (link)
• Treves, Francois (2006). Topological vector spaces, distributions and kernels. Mineola, N.Y: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.CS1 maint: ref=harv (link)