Let $L_p([a,b];\mathbb{R}) = \{f:[a,b]\to\mathbb{R} | f$ is Riemann-integrable $\}$. I want to prove that $$ d_p(f,g) = \left(\int_a^b |f(x)-g(x)|^p dx\right)^{1/p} $$ defines a metric in $L_p([a,b],\mathbb{R}).$
Given $f,g$ integrable I easily see that $d_p(f,g)$ exists and that $d_p(f,f)=0$ and $d_p(f,g) = d_p(g,f)$. I also proved the triangle inequality by proving the Hölder inequality for positive real functions. I'm just stuck in the condition $f\neq g \to d_p(f,g)>0$. For simplicity I'm assuming $f \neq 0$ and trying to prove $\int |f|^p \neq 0$.
I know that $f\ge 0$ implies $\int f \ge 0$ but I don't see how to get the strict inequality. Any help or hint will be appreciated.
You can't prove it because it is not true. If$$f(x)=\begin{cases}1&\text{ if }x=a\\0&\text{ otherwise,}\end{cases}$$then $d_p(f,0)=0$, but $f\neq0$. The function $d_p$ is a pseudometric, but not a metric.