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.
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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?