Consider the set of $\displaystyle C[0,1]$, the set of all continuous functions on [0,1], and define $\displaystyle p(f,g)= \int ^1 _0 \mid f-g \mid dx $.

Prove that if $\displaystyle p(f,g)=0$, then f = g.

If $\displaystyle p(f,g)=0$, then I have $\displaystyle \int ^1 _0 \mid f(x)-g(x) \mid dx =0$.

But now, what can I do with the absolute values? Thanks!