The polar topology originates from the $S$-topology and is used in duality pairs. Due to the connection between the original topology and the weak topology, we can rephrase the original topology in terms of the polar topology, i.e. the Mackey-Arens theorem. I want to know how the whole mechanism works (the advantage of using polar topology) and what other uses there are of the polar topology.
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1$\begingroup$ Can you be more specific than "I want to know how the whole mechanism works"? Back in the old days of MO we used to have a refrain "MO is not for requests for an encyclopaedia entry", so perhaps you could single out some parts of the Mackey-Arens theorem (proof, or applications) that you find unclear? $\endgroup$– Yemon ChoiCommented Jul 26, 2016 at 11:54
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$\begingroup$ If you are wanting an example of applications of, for example, the fact that weak boundedness implies (strong) boundedness, then the proof that differentiability of $\lambda\circ f$ for all $\lambda\in V^*$ implies differentiability of $f$ is one such, where $f$ is a $V$-valued function on a (locally compact) smooth manifold, and $V$ is a quasi-complete, locally convex topological vector space. I think this is approximately due to Schwartz. Is this the sort of thing you wanted? $\endgroup$– paul garrettCommented Jul 26, 2016 at 21:24
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