# FA7

From Example Problems

Assume that and are two equivalent norms on , and . Prove that is compact in if and only if is compact in .

Since is compact in , every sequence in has a convergent subsequence that converges to . This means s.t. .

and are equivalent norms iff s.t. .

So . Let .

Now, for any , and for all , which means , and since , is compact in .