Definition:L2 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:
 * $\ds \forall f, g \in S: \map d {f, g} := \paren {\int_a^b \paren {\map f t - \map g t}^2 \rd t}^{\frac 1 2}$

Then $d$ is the $L^2$ metric on $\closedint a b$.

Also see

 * $L^2$ Metric on Closed Real Interval is Metric