FA7

From Exampleproblems

Assume that $c||\cdot ||_1\,$ and $||\cdot ||_2\,$ are two equivalent norms on $X\,$ , and $M \subset X\,$ . Prove that $M\,$ is compact in $(X, ||\cdot ||_1)\,$ if and only if $M\,$ is compact in $(X, ||\cdot ||_2)\,$ .

Since $M\,$ is compact in $(X,||\cdot ||_1)\,$ , every sequence $\{x_n\}\,$ in $M\,$ has a convergent subsequence $\{x_{n_k}\}\,$ that converges to $x\isin M\,$ . This means $\forall \epsilon > 0, \exists K\,$ s.t. $||x-x_{n_k}||_1<\epsilon \forall k>K\,$ .

$||\cdot ||_1\,$ and $||\cdot ||_2\,$ are equivalent norms iff $\exists a,b\,$ s.t. $a ||\cdot ||_1 \le ||\cdot ||_2 \le b ||\cdot ||_1\,$ .

So $b||\cdot ||_1 < b\epsilon\,$ . Let $\delta_\epsilon = b\epsilon\,$ .

Now, for any $\delta_\epsilon\,$ , and for all $n>N\,$ , $||x-x_n ||_2<\delta_\epsilon\,$ which means $x_n\to x\,$ , and since $x\isin M\,$ , $M\,$ is compact in $(X,||\cdot ||_2)\,$ .

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FA7

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