I'm confused about the common proof of the equivalence of the norms in Banach spaces. It was very clear and nice answered here equivalent norms in Banach spaces of infinite dimension, but i still don't understand, why we can use identity map?
More precisely: Let $X$ be a Banach space and $i: (X, \|\cdot\|_2) \rightarrow (X, \|\cdot\|_1), x \mapsto x$ be the identity map on $X$ (i.e. $i(x)=x$ for all $x \in X$). It is given, that $\|x\|_2 \le C \|x\|_1, C > 0$. Then $i$ is continuous, since $\|i(x)\|_1=\|x\|_2 \le C \|x\|_1$ (by assumption).
The only question is: why it does not means, that $\|x\|_1=\|i(x)\|_1=\|x\|_2$?