Definition:Lp 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 $p \in \R_{\ge 1}$.

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 $\closedint a b$.

Special Cases

$L^1$ 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:

$\ds \forall f, g \in S: \map {d_1} {f, g} := \int_a^b \size {\map f t - \map g t} \rd t$

Then $d_1$ is the $L^1$ metric on $\closedint a b$.

$L^2$ 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:

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

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

Also see

• $L^p$ Metric on Closed Real Interval is Metric

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.2$: Examples: Example $2.2.18$