L1 Metric is Topologically Equivalent to Supremum Metric on Bounded Continuous Real Functions
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Theorem
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.
Proof
Let $U$ be an upper bound of $\set {\size {\map f x - \map g x} }$.
Then:
- $\ds U \ge \sup_{x \mathop \in S} \size {\map f x - \map g x}$
Hence:
- $\ds \max_{x \mathop \in \closedint a b} \set {\size {\map f x - \map g x} } = \map {d'} {f, g}$
Then:
\(\ds \map d {f, g}\) | \(=\) | \(\ds \int_a^b \size {\map f t - \map g t} \rd t\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{\Delta h \mathop \to 0} \sum_{i \mathop = i}^n \size {\map f t - \map g t} \Delta h\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \lim_{\Delta h \mathop \to 0} \sum_{i \mathop = i}^n \map {d'} {f, g} \Delta h\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map {d'} {f, g} \int_a^b \rd t\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {b - a} \map {d'} {f, g}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map d {f, g}\) | \(\le\) | \(\ds \paren {b - a} \map {d'} {f, g}\) |
$\blacksquare$
Sources
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 2$: Metric Spaces: Exercise $6$