Definition:L1 Metric/Closed Real Interval

Definition
Let $S$ be the set of all real functions which are continuous on the closed interval $\left[{a \,.\,.\, b}\right]$.

Let the real-valued function $d: S \times S \to \R$ be defined as:
 * $\displaystyle \forall f, g \in S: d \left({f, g}\right) := \int_a^b \left\vert{f \left({t}\right) - g \left({t}\right)}\right\vert \ \mathrm d t$

Then $d$ is the $L^1$ metric on $\left[{a \,.\,.\, b}\right]$.

Also see

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