TITLE: How many well-behaved norms are there on $C[0,1]$?
QUESTION [7 upvotes]: 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.

REPLY [5 votes]: The hypothesis implies that for all $t\in I=[0,1]$ the linear functional $\mathrm{ev}_t:X\to\Bbb R, ~f\mapsto f(t)$ is continuous. Also, for any $f\in X$, $$\sup_{t\in I}~|\mathrm{ev}_t(f)|=||f||_{\infty}<+\infty$$ Since $X$ is a Banach space, the uniform boundedness principle applies, and $$ C=\sup_{t\in I}~||\mathrm{ev}_t||_{X'}<+\infty$$ that is, for any continuous function $f$ and any $t\in I$, $|f(t)|\leq C||f||_X$ thus $$||\cdot||_{\infty}\leq C||\cdot||_X$$ As you noted, the open mapping theorem then tells us that both norms are equivalent.