if we change the norm, is it possible to make it complete ? There are many examples about $C^1[0,1]$ that is not complete under supremum norm. if we change the norm, is it possible to make it complete ? Are there any examples about this ? 
Thank you for your help .

Instead of considering :
$$||f||:=\sup_{x\in[0,1]}|f(x)| $$
Choose :
$$||f||_1:=\sup_{x\in[0,1]}|f(x)|+\sup_{x\in[0,1]}|f'(x)| $$
Using this I think that if you take $(f_n)$ to be a Cauchy sequence for this norm you must have that both $(f_n)$ and $(f_n')$ are Cauchy sequence for the usual supremum norm so $f_n\rightarrow_n f$ and $f_n'\rightarrow g$ then you just have to show that $f'=g$.
