We know that the space of real-valued continuous functions over a compact subset of $\mathbb{R}$ (call is $X$) is complete with respect to the supremum norm, but not complete when equipped with the $L^2$ norm. My question is how would one find the set of all norms such that that $X$ is complete? From this question (Complete Inequivalent Norms ) we know the cardinality of this set - but is there no way to describe this set?

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    $\begingroup$ Is there a reason you would want to describe this set? It is huge and complicated and not a particularly useful set. $\endgroup$ – Eric Wofsey Sep 9 '18 at 0:32
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    $\begingroup$ I was just curious about the set. $\endgroup$ – Sean Nemetz Sep 9 '18 at 0:56

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