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For the space of all continuous functions we can have the sup norm:

$|f|=\sup|f|$

I have also seen the following norm: $|f|=\sup|f(x)|/|x|$

I don't know what this norm is called and therefore can't find any information on it. What is the distinction?

what is this norm called? is the space $C[0,1]$ with this norm still complete?

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    The norm you are thinking of (I am guessing) is the operator, or induced norm for a linear function. If $f$ is linear, we can define $\|f\| = \sup_{x \neq 0} { \|f(x)\| \over \|x\|}$.2014-11-05

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