Let $X$ be a topological vector space. We know that the weak topology $\sigma(X,X^*)$ is the weakest locally convex topology in $X$ that make every $f \in X^*$ continuous. Consequently, if $\tau$ is the topology of $X$, $\sigma(X,X^*)\subset \tau.$ Does something similar happen with the strong topology $\beta(X,X^*)?$
When we consider the definition of reflexive space, we topologize $X^*$ with $\beta(X^*,X),$ but I'd like to know the reason of that.
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4$\begingroup$ I think that if you check out the topics "Mackey topology" and "Mackey-Arens theorem", this will provide the desired information. $\endgroup$– terceiraCommented Nov 13, 2022 at 20:18
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1$\begingroup$ The strong toology (also used in the non-reflexive situation) does not have the property xou mention in genersl. There are seversl reasons for its usage, one being thst it is the natural generalisation of the dual norm on the dual of a Banach space. $\endgroup$– terceiraCommented Nov 13, 2022 at 20:22
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