Let $\mathbb{F}$ be either $\mathbb{R}$ or $\mathbb{C}$ and consider the vector space $C[0,1]$, the collection of continuous functions $f\colon[0,1]\to\mathbb{F}$. I want to show that $\|\cdot\|_2$ is a norm on $C[0,1]$. I have proven all the properties of a norm, but I still need to show that in that $\|f\|=0\implies f=0$.
EDIT: The norm $\|\cdot\|_2$ is assumed to be given by a Riemann integral.
If $\|f\|_2^2=0$, then $\int_0^1|f(x)|^2~\mathrm{d}x=0.$ I was wondering if the fact that $|f(x)|$ is nonnegative is sufficient to conclude that $|f(x)|=0$ (which ensures that $f(x)=0$ for all $x\in[0,1]$). Or do you need the fact that each $f$ is continuous?