Category:Definitions/Lp Metrics

This category contains definitions related to $L^p$ metrics.
Related results can be found in Category:Lp Metrics.

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 $\closedint a b$.