In showing that $\displaystyle \Vert .\Vert{p}$ defines a norm on $\displaystyle L_{p} (p\geq1)$, I'm not so sure about checking the axiom

$\displaystyle \Vert f \Vert_{p}=0$ iff $\displaystyle f=0$.

A textbook mentions defining an equivalence relation on $\displaystyle L_{p}$ space but I don't really get what that's all about.