How is $\lvert f\rvert=\int^1_0\lvert f\rvert dx$ for $f\in R[0,1]$ where R are the regulated functions (bounded variation zero, for an epsilion I can give you a step function where the suprememum of the |difference between a step function and the function|<=epsilion, not a norm.
Yet the next exercise has me show that that $C[0,1]$ does have a norm with this definition.
I'm not sure how to do the || double thing, sorry, I hope you can work out what I mean.
This is self learning