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Recall that for a Hausdorff locally convex space $X$ the Mackey topology $\tau (X^*,X)$ is the topology in its topological dual $X^*$ of uniform convergence on all weakly compact absolutely convex subsets of $X$.

Does this definition amount to saying the following?

Let $\mathcal{C}$ be the family of absolutely convex and $\sigma (X,X^{*})$-compact subset of $X$ let $\{x_n^*\}_n\subset X^*$ be a sequence and $x^*\in X^*$ then :

$$ x_n^*\overset{\tau (X^*,X)}{\longrightarrow} x^* \Longleftrightarrow \forall K\in \mathcal{C}~:~\text{$x_n^* \longrightarrow x^*$ uniformly on $K$} $$

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  • $\begingroup$ Yes, in the sense that the definition implies that this is what it means for a sequence to converge. $\endgroup$
    – user131781
    Commented Mar 5, 2020 at 8:46
  • $\begingroup$ But no in the sense that the definition also says something similar about convergence of nets/filters which is (in general) not equivalent to a statement about convergence of sequences alone. $\endgroup$
    – Johannes Hahn
    Commented Mar 5, 2020 at 13:34
  • $\begingroup$ @johannes_Hahn I did not understand the definition enactment of Mackey topology, can you explain to me what do they mean by : " the Mackey topology on $ X^*$ is the topology in its topological dual $X^*$ of uniform convergence on all weakly compact absolutely convex subsets of $X$". $\endgroup$
    – Wer Wer
    Commented Mar 5, 2020 at 14:00

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