Definition:Lp 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 $p \in \R_{\ge 1}$.

Let the real-valued function $d: S \times S \to \R$ be defined as:
 * $\displaystyle \forall f, g \in S: d \left({f, g}\right) := \left({\int_a^b \left\vert{f \left({t}\right) - g \left({t}\right)}\right\vert^p \ \mathrm d t}\right)^{\frac 1 p}$

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

Also see

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