Let $\Vert \cdot \Vert$ a norm on $X=C([0,1])$ s.t.
1. $X$ is complete w.r.t. $\Vert \cdot \Vert$;
2. convergence in $\Vert \cdot \Vert$ implies pointwise convegence, i.e. $ \Vert x_n - x \Vert \to 0 \Rightarrow \forall t \in [0,1], \quad x_n(t) \to x(t). $
Is $\Vert \cdot \Vert$ equivalent to the usual $\sup$-norm, $\Vert \cdot \Vert_{\infty}$?
By open mapping and 1 (completeness), we just need to prove only one inequality of the two we need to prove equivalence (indeed, the other would follow by open mapping).
Anyway, I think the answer is affirmative; am I right? I do not know how to prove it. Any ideas, please? Thanks.