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 .

Yes, define a "stronger" norm (the $C^1$-norm!) $|f|_{C^1}$ to be the sum of the sup-norm of $f$ and the sup-norm of $f'$. The proof of completeness of $C^1[a,b]$ with this norm does require something, since we need to show that the derivative of a limit (in this norm) is the (sup-norm) limit of the derivative. (For example, the fundamental theorem of calculus nearly proves this.)
