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Is there a metrizable topology on $S’$ (the set of linear continuous functionals in Schwarz’s space) whose convergent sequences are the sequences that converge pointwise?

The obvious topology with these convergent sequences is the weak* topology, which I know is not metrizable. However, there might be another topology with the same convergent sequences which is metrizable.

I know that for smooth functions, there is no such metrizable topology whose convergent sequences are the pointwise convergent sequences.

But I don’t know how prove this. Maybe it is classical fact, but I didn’t find anything.

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  • $\begingroup$ What do you mean by "pointwise convergence", precisely? I would think that is a synonym for the weak* topology. $\endgroup$ Jun 5, 2021 at 21:42
  • $\begingroup$ Yes, pointwise convergence = convergence in *-weak topology. $F_n\rightarrow F$ if and only if $\forall \phi \in S \hspace{0.5em} F_n(\phi)\rightarrow F(\phi)$ $\endgroup$
    – Dimats
    Jun 6, 2021 at 22:17
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    $\begingroup$ So what is your question then? You said you already know why the weak* topology is not metrizable. $\endgroup$ Jun 6, 2021 at 22:26
  • $\begingroup$ No, I mean that on $S’$ can be two different topologies. And sets of converging sequences in these topologies may be equal. As for arbitrary topological space. $\endgroup$
    – Dimats
    Jun 12, 2021 at 17:47
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    $\begingroup$ Ah, it makes a big difference that you are talking about pointwise convergence of only sequences (and this was not clear in your original question!). If you were talking about pointwise convergence of nets then that would uniquely determine the weak* topology. $\endgroup$ Jun 12, 2021 at 19:56

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