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 $\closedint a b$.
Let $p \in \R_{\ge 1}$.
Let the real-valued function $d: S \times S \to \R$ be defined as:
- $\ds \forall f, g \in S: \map d {f, g} := \paren {\int_a^b \size {\map f t - \map g t}^p \rd t}^{\frac 1 p}$
Then $d$ is the $L^p$ metric on $\closedint a b$.