Question:

Let

be any norm on

and let

. Prove that

is compact. Hint: It suffices to show that

is closed and bounded with respect to the Euclidean metric.

-------------------------------------------------------------------

I don't think I'd have any difficulty doing this problem. I know that a subset of

is compact if and only if it's closed and bounded, but I don't get why the hint says that it's sufficient to show that it's closed and bounded ONLY under the Euclidean metric. Isn't it possible that there may be other norms on

such that the set is not closed or not bounded with respect to the other norm?