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Let $X$ be a topological vector space or even a locally convex space such that its (vector space) topology is Lusin, i.e. there is some stronger Polish topology. Does there also exist a stronger Polish topology under which $X$ is a topological vector space or even locally convex?

Similarly, if $X$ is a Souslin vector space topology then it is known (p. 4) that there exists a stronger Souslin metrizable topology. Does there also exist a stronger Souslin metrizable vector space topology?

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    $\begingroup$ A possible obstacle is the open mapping theorem: An ultrabornological l.c.s. does not have a strictly finer Frechet space topology. One would thus need some finer complete metrizable separable topology e.g. on $s'$, the space of sequences of at most polynomial growth. $\endgroup$
    – Jochen Wengenroth
    Nov 29, 2015 at 14:34

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