# completeness under different metric

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.

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 a) You can get proper formatting for the norm symbols by using \lVert\cdot\rVert to produce $\lVert\cdot\rVert$. b) The unusual notation in part $3$ is unnecessarily confusing; $\lVert f\rVert_2$ is usually defined to be the square root of what you wrote. – joriki Jun 13 '12 at 12:43 In fact what you wrote isn't a norm, since it doesn't satisfy the triangle inequality. Note also that you're not distinguishing clearly between norms and metrics. – joriki Jun 13 '12 at 12:55 @joriki You can also write \| that is "backslash pine" – AD. Jun 13 '12 at 13:15 for continuous function dont essential sup norm and sup norm coincide? – clark Jun 13 '12 at 13:35 @clark Sorry, I'd already deleted that comment because I realized it was misguided, as you point out. – joriki Jun 13 '12 at 13:37
for the second case the norm you asked is equivalent to the sup norm so the normed space it is induced also complete $|\!|f|\!| _{\infty} \leq |\!| f |\!|_{\infty , \frac{1}{2}} =|\!|f|\!| _{\infty}+ |f(\frac{1}{2}) | \leq 2|\!|f|\!| _{\infty }$