# L1 Metric on Closed Real Interval is Metric

## Theorem

Let $S$ be the set of all real functions which are continuous on the closed interval $\left[{a \,.\,.\, b}\right]$.

Let $d: S \times S \to \R$ be the $L^1$ metric on $\left[{a \,.\,.\, b}\right]$:

$\displaystyle \forall f, g \in S: d \left({f, g}\right) := \int_a^b \left\vert{f \left({t}\right) - g \left({t}\right)}\right\vert \ \mathrm d t$

Then $d$ is a metric.

## Proof

### Proof of $M1$

 $\displaystyle d \left({f, f}\right)$ $=$ $\displaystyle \int_a^b \left\vert{f \left({t}\right) - f \left({t}\right)}\right\vert \ \mathrm d t$ Definition of $d$ $\displaystyle$ $=$ $\displaystyle \int_a^b 0 \ \mathrm d t$ Definition of Absolute Value $\displaystyle$ $=$ $\displaystyle 0$ Definite Integral of Constant

So axiom $M1$ holds for $d$.

$\Box$

### Proof of $M2$

 $\displaystyle \left\vert{f \left({t}\right) - g \left({t}\right)}\right\vert + \left\vert{g \left({t}\right) - h \left({t}\right)}\right\vert$ $\ge$ $\displaystyle \left\vert{f \left({t}\right) - h \left({t}\right)}\right\vert$ Triangle Inequality for Real Numbers $\displaystyle \implies \ \$ $\displaystyle \int_a^b \left\vert{f \left({t}\right) - g \left({t}\right)}\right\vert \ \mathrm d t + \int_a^b \left\vert{g \left({t}\right) - h \left({t}\right)}\right\vert \ \mathrm d t$ $\ge$ $\displaystyle \int_a^b \left\vert{f \left({t}\right) - h \left({t}\right)}\right\vert \ \mathrm d t$ Relative Sizes of Definite Integrals $\displaystyle \implies \ \$ $\displaystyle d \left({f, g}\right) + d \left({g, h}\right)$ $\ge$ $\displaystyle d \left({f, h}\right)$ Definition of $d$

So axiom $M2$ holds for $d$.

$\Box$

### Proof of $M3$

 $\displaystyle d \left({f, g}\right)$ $=$ $\displaystyle \int_a^b \left\vert{f \left({t}\right) - g \left({t}\right)}\right\vert \ \mathrm d t$ Definition of $d$ $\displaystyle$ $=$ $\displaystyle \int_a^b \left\vert{g \left({t}\right) - f \left({t}\right)}\right\vert \ \mathrm d t$ Definition of Absolute Value $\displaystyle$ $=$ $\displaystyle d \left({g, f}\right)$ Definition of $d$

So axiom $M3$ holds for $d$.

$\Box$

### Proof of $M4$

 $\, \displaystyle \forall t \in \left[{a \,.\,.\, b}\right]: \,$ $\displaystyle \left\vert{f \left({t}\right) - g \left({t}\right)}\right\vert$ $\ge$ $\displaystyle 0$ Definition of Absolute Value $\displaystyle \implies \ \$ $\displaystyle \int_a^b \left\vert{f \left({t}\right) - g \left({t}\right)}\right\vert \ \mathrm d t$ $\ge$ $\displaystyle 0$ Sign of Function Matches Sign of Definite Integral
$d \left({f, g}\right) = 0 \implies f = g$

on $\left[{a \,.\,.\, b}\right]$.

So axiom $M4$ holds for $d$.

$\blacksquare$