Definition:Lp Metric

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Definition

Closed Real Interval

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 $\left[{a \,.\,.\, b}\right]$.