Definition:Lp Metric/Closed Real Interval

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


Special Cases

$L^1$ Metric

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:

$\displaystyle \forall f, g \in S: d_1 \left({f, g}\right) := \int_a^b \left\vert{f \left({t}\right) - g \left({t}\right)}\right\vert \ \mathrm d t$


Then $d_1$ is the $L^1$ metric on $\left[{a \,.\,.\, b}\right]$.


$L^2$ Metric

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:

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


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


Also see


Sources