7  Metric Spaces

7.1 Definition of a Metric Space

A metric space \(\mathcal{M}\) is a set \(M\) equipped with a map \(d:M\times M\rightarrow\mathbb{R}\) (the metric). The metric satisfies the following axioms:

• [M1]: \(d\left(x,y\right)\ge0\), and \(d\left(x,y\right)=0\Longleftrightarrow x=y\) (positivity)

• [M2]: \(d\left(x,y\right)=d\left(y,x\right)\) (symmetry)

• [M3]: \(d\left(x,z\right)\le d\left(x,y\right)+d\left(y,z\right)\) (subadditivity / triangle inequality).

Equivalently, a metric space is an ordered pair \(\mathcal{M}=\left(M,d\right)\) obeying these axioms.

Note: a metric space need not be a vector space, and the map (metric) need not be linear.

7.2 Examples of Metric Spaces

The norm is one example of a metric. Hence, all normed vector spaces \(\mathcal{N}\) are metric spaces \(\mathcal{M}\): \(\mathcal{N}\subset\mathcal{M}\).

Specifically, some cases of interest are:

• \(\mathcal{M}=\left(\left(M,\left\{ +,\cdot\right\} \right),d\right)\): \(M=\mathbb{R}\), \(d\left(x,y\right)=|x-y|\) (real numbers, with their distance defined with the Euclidean metric in the usual way)

• \(\mathcal{M}=\left(\left(M,\left\{ +,\cdot\right\} \right),d\right)\): \(M=V\), \(d\left(|x\rangle,|y\rangle\right)=\left\Vert |x\rangle-|y\rangle\right\Vert\) (a normed vector space with the metric induced by the norm).