TITLE: $C[0,1]$ is complete w.r.t. which norm(s)
QUESTION [9 upvotes]: $C[0,1]$ is complete w.r.t. which norm(s)

$\displaystyle\|f\|_\infty=\sup_{t\in[0,1]}|f(t)|$
$\displaystyle\|f\|_1=\int_0^1|f(t)| \, dt$
$\displaystyle\|f\|_\infty^{0,1}=\|f\|_\infty+|f(0)|+|f(1)|$
$\displaystyle\|f\|=\sqrt{\int_0^1|f(t)| \, dt}$

I know 1 is true. But I don't know how to check completeness of norm base on know complete norm. How can I solve it?

REPLY [3 votes]: Norm $1$ and  $3$ are equivalent, so $C[0,1]$ is also complete in the norm $3$.
$||f||_{\infty}^{0,1}=||f||_{\infty}+|f(0)|+|f(1)|\le 3||f||_{\infty}$