# Different norms for space $C[0,1]$

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 $\|f\| = \sup_{x \neq 0} { \|f(x)\| \over \|x\|}$. –  copper.hat Nov 5 at 5:34

$||f||:=\sup_{x\in(0,1]}\frac{|f(x)|}{|x|}$ is not a norm because $||f||$ may be infinite, for example consider $f(x)=\sqrt x$ (also $x=0$ hat to be excluded from the definition in the first place).
However, if you consider the subspace $V\subset C([0,1])$ of continuous functions for which the result is finite, then we have the map $V\to C([0,1])$, $f\mapsto(x\mapsto xf(x))$, which is an isometric embedding.