Definition:L1 Metric/Closed Real Interval

Definition
Let $S$ be the set of all real functions which are continuous on the closed interval $\closedint a b$.

Let the real-valued function $d: S \times S \to \R$ be defined as:
 * $\ds \forall f, g \in S: \map {d_1} {f, g} := \int_a^b \size {\map f t - \map g t} \rd t$

Then $d_1$ is the $L^1$ metric on $\closedint a b$.

Also see

 * $L^1$ Metric on Closed Real Interval is Metric