# Collection of all norms such that a Space is Complete

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?

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