I find this problem interesting since it gives another characterization of finite dim. normed spaces.

It is a standard result that every finite dim. normed space (over ) is complete. Prove the converse:

If is a vector space over such that for every norm we have that is complete then

I include a hint and the solution for those that want them.

Hint

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Solution

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One thing that also came to mind, but haven't given much thought (not posting it as a question but as recreation) if is such that every two norms are comparable, does it follow ? (The weaker statement using that two norms are equivalent follows directly from the posed problem).