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For Euclidean space $\mathbb{R}^n$, it is a well-known fact that the Schwartz space $\mathcal{S}(\mathbb{R}^n)$ is a Montel space. Moreover, the space of tempered distributions $\mathcal{S}'(\mathbb{R}^n)$ with the strong dual topology is a Montel space too. That is, they have the Heine-Borel properties.

Now, let $M(\mathcal{S}')$ be the vector space of complex-valued measures on $\mathcal{S}'(\mathbb{R}^n)$. Then, my question is

Is there some topology on $M(\mathcal{S}')$ that makes it a topological vector space with the Heine-Borel property?

I introduced $M(\mathcal{S}')$ to have a vector space structure, but my interest is mostly on a sequence of probability measures $\{ \mu_n \}$ on $\mathcal{S}'$ and sufficient conditions to extract a subsequence convergent in some sense.

Perhaps, the generating funcitonal may play a role, I am not sure...

Could anyone provide any information or reference?

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    $\begingroup$ This doesn't answer your precise question, but tightness of $(\mu_n)_n$ (i.e. for all $\epsilon > 0$ there is a compact (in the weak (!) topology; so for example the Banach-Alaoglu theorem can provide this) set $K_\epsilon$ such that $\sup_n \mu_n(K_\epsilon) \geq 1-\epsilon$) would be a sufficient condition to extract a convergent (where convergence here means convergence in distribution where one tests against bounded functions $\mathcal{S}' \to \mathbb{R}$ which are continuous when $\mathcal{S}'$ is equipped with the strong (!) topology) subsequence, no? $\endgroup$
    – unwissen
    Commented Jun 29 at 11:08
  • $\begingroup$ @unwissen By strong topology on $\mathcal{S}'$, you mean the strong dual topology? Anyway...I do recall running into the notion of tightness before...perhaps could you recommend any reference relevant for $\mathcal{S}'$? $\endgroup$
    – Isaac
    Commented Jun 29 at 12:55
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    $\begingroup$ To be honest, I'm not an expert on this and just wanted to know why you didn't mention the (for Polish spaces very standard) "tightness method" in your question, but my internet search brought me to arxiv.org/abs/1706.09326, where maybe Corollary 2.4 and the reference right before Lemma 5.2 may be of interest. $\endgroup$
    – unwissen
    Commented Jun 29 at 13:16
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    $\begingroup$ @unwissen Thank you very much for reminding me of the keywords I forgot some time ago! $\endgroup$
    – Isaac
    Commented Jun 29 at 13:25

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