L1 Metric is Topologically Equivalent to Supremum Metric on Bounded Continuous Real Functions

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Let $\FF$ be the set of all real functions which are also bounded on the closed interval $\closedint a b$.

Let $d: \FF \times \FF \to \R$ be the $L^1$ metric on $\closedint a b$:

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

Let $d': \FF \times \FF \to \R$ be the supremum metric on $\closedint a b$:

$\ds \forall f, g \in \FF: \map {d'} {f, g} := \sup_{x \mathop \in S} \size {\map f x - \map g x}$

Then $d$ and $d'$ are topologically equivalent metrics.


Let $U$ be an upper bound of $\set {\size {\map f x - \map g x} }$.


$\ds U \ge \sup_{x \mathop \in S} \size {\map f x - \map g x}$


$\ds \max_{x \mathop \in \closedint a b} \set {\size {\map f x - \map g x} } = \map {d'} {f, g}$


\(\ds \map d {f, g}\) \(=\) \(\ds \int_a^b \size {\map f t - \map g t} \rd t\)
\(\ds \) \(=\) \(\ds \lim_{\Delta h \mathop \to 0} \sum_{i \mathop = i}^n \size {\map f t - \map g t} \Delta h\)
\(\ds \) \(\le\) \(\ds \lim_{\Delta h \mathop \to 0} \sum_{i \mathop = i}^n \map {d'} {f, g} \Delta h\)
\(\ds \) \(=\) \(\ds \map {d'} {f, g} \int_a^b \rd t\)
\(\ds \) \(=\) \(\ds \paren {b - a} \map {d'} {f, g}\)
\(\ds \leadsto \ \ \) \(\ds \map d {f, g}\) \(\le\) \(\ds \paren {b - a} \map {d'} {f, g}\)