completeness under different metric

Question

would any one tell me whether $C[0,1]$ is complete under these metrics

1.sup norm i mean $\|f\|_{\infty}$

2.$\|f\|_{\infty,1/2}=\|f\|_{\infty}+|f(1/2)|$

3.$\|f\|_{2}=\int_0^1|f|^2dx$

Under supnorm I know it is complete,I am not sure about the other two.

Answer 1

for the second case the norm you asked is equivalent to the sup norm so the normed space it is induced also complete $ |\!|f|\!| _{\infty} \leq |\!| f |\!|_{\infty , \frac{1}{2}} =|\!|f|\!| _{\infty}+ |f(\frac{1}{2}) | \leq 2|\!|f|\!| _{\infty } $