Am I wrong ? (2)

Question

Let $X=C[0,1]$ be the space of real continous functions on $[0,1]$. $X$ is a Banach space with the two norms $$|f|_\infty=\sup_{s\in[0,1]}|f(s)|$$ and $$|f|_2=\left(\int_0^1|f(s)|^2\right)^\frac{1}{2}$$ The canonical map $(X,|.|_\infty)\to (X,|.|_2):f\mapsto f$ is a bounded bijective operator since $$|f|_2\leq |f|_\infty.$$ Then by the "Bounded inverse theorem" the inverse of this map is also continuous and we have a constant $c>0$ such that $$|f|_\infty\leq c|f|_2.$$ I feel something is wrong in this.

Answer 1

The problem is that $X$ is not a Banach space when equipped with the second norm (the $L^2$ norm). The continuous functions are dense in $L^2$. They aren't closed, so they aren't a Banach space.