In reading I came across the claim that the following is a metric.
For the space $X$ of all integrable functions on the interval [$0,1$] , for $f, g \in X$, the following equation defines a metric:
$\rho(f, g) = \int_0^1 |(f(x) - g(x))| dx$
It is clear to me that $\rho(f, g) = \rho(g, f)$ and also $\rho(f, g) = 0$ if and only $ f = g$. However, does the triangle inequality hold?