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Citing from Wikipedia,
A Hausdorff topological space is a Lusin space if some stronger topology makes it into a Polish space.
Is there a (previously studied) analogous concept of a Hausdorff (locally convex) topological vector space with the property that some stronger topology makes it into a separable Hilbert space? Clearly any such space is a Lusin space. Is the converse also true? I suspect not, but perhaps someone has already answered that question.
The closed graph theorem implies that nice comparable topologies on a vector space coincide. More precisely, If $(X,\sigma)$ is a barrelled locally convex space (e.g., a Fréchet space) and $\tau$ is a finer Fréchet space topology on $X$, then $\sigma=\tau$.
This implies that,e.g., the space $\ell^1$ is a Lusin space but it does not have a finer Hilbert space topology.
On the other hand, the Anderson-Kadets theorem implies that all infinite dimensional separable Fréchet spaces are homeomorphic (via an, of course, non-linear homeomorphism).
