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

## 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 \ \$ $\ds \map d {f, g}$ $\le$ $\ds \paren {b - a} \map {d'} {f, g}$

$\blacksquare$