Is the absolute value function a metric?

Question

I was trying to find out more information about absolute value, and I came upon the fact that AV satisfies a whole set of properties that usually defines a distance function or metric. But in the Wikipedia article on metrics, there's no mention of the AV function, so I'm a bit confused now. Is it some sort of metric subspace instead?

P.S.: I'm not at all well-versed on metrics, so I'd appreciate simple answers :)

Answer 1

The absolute value $x\mapsto |x|$ is not a metric but a norm on $\mathbb R$ (or $\mathbb C$), viewed as a one-dimensional vector space. However, from any norm you can derive a metric in a standard way.

In the case of the absolute value, this gives the well-known metric $d(x,y)=|x-y|$.

Answer 2

Yes, for $\Bbb R$ (or $\Bbb C$ for that matter), the map $d(x,y):=|x-y|$ is a metric.

Answer 3

No,the absolute value function $|\cdot|:\mathbb R\to[0,\infty)$ is not a metric.

A metric on a set $X$ is a function $d:X\times X\to[0,\infty)$ defined on the cartesian square of the set $X$ satisfying certain properties, and the absolute value function is not of that form.

What it is true is that out of the absolute value you can construct a metric: indeed, the function $$(x,y)\in\mathbb R\times\mathbb R\longmapsto|x-y|\in[0,\infty)$$ is a metric.