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Section D.2 Metrics

Let $X$ be a set. A function $d:X\times X\to\R$ is called a metric on $X$ if, for all $x,y,z\in X\text{,}$ the following hold:

$d(x,y)\ge 0\text{.}$
$d(x,y)=0$ if and only if $x=y\text{.}$
$d(x,y)=d(y,x)\text{.}$
$d(x,z)\le d(x,y)+d(y,z)\text{.}$

For $x,y\in\R^n\text{,}$ define the Euclidean distance by $d(x,y)=\|x-y\|\text{.}$

Proposition D.2.1.

The Euclidean distance is a metric on $\R^n\text{.}$

Proof.

Since norms are nonnegative, $d(x,y)=\|x-y\|\ge0\text{.}$ Also $d(x,y)=0$ if and only if $\|x-y\|=0\text{,}$ which happens if and only if $x-y=0\text{,}$ that is, $x=y\text{.}$

Symmetry follows from $d(x,y)=\|x-y\|=\|-(y-x)\|=|-1|\,\|y-x\|=\|y-x\|=d(y,x)\text{.}$

For the triangle inequality, $d(x,z)=\|x-z\|=\|(x-y)+(y-z)\| \le \|x-y\|+\|y-z\| = d(x,y)+d(y,z),$ by Proposition D.1.2. Therefore $d$ is a metric.

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