Metric Spaces

Inner product spaces as well as normed spaces have a distance function, namely the norm of the difference,

$$\Vert f-g \Vert \equiv d(f,g) ~~~~,$$

between two vectors $f$ and $g$ . This norm of the difference is called the distance between $f$ and $g$ . Applying this formula to the three pairs of points of a generic triangle, one obtains the triangle inequality

$$\Vert f-h\Vert\le\Vert f-g\Vert +\Vert g-h\Vert$$

i.e., $d(f,h)\le d(f,g)+d(g,h)$ .

The importance of a distance function and the triangle inequality is that it can also be applied to certain nonlinear spaces, which have no zero element (``origin''). Such spaces are called metric spaces. More precisely we have the following definition.

By a metric space is meant a pair $(X,d)$ consisting of set $X$ and a distance function $d$ , i.e., a single-valued, nonnegative, real function $d(f,g)$ defined for $f,g\in X$ which has the following three properties.

1. Positive definiteness: $d(f,g)\ge 0$ for all $f$ and $g$ in $X$ , and $d(f,g)=0$ if and only if $f=g$ .
2. Symmetry: $d(f,g)=d(g,f)$ .
3. Triangle inequality: $d(f,h)\le d(f,g)+d(g,h)$ .

The distance function $d(~~,~~)$ is called the metric of the metric space. All inner product spaces are metric spaces with

$$d(f,g)=\Vert f-g\Vert ~~.$$

All normed linear spaces are metric spaces with

$$d(f,g)=\Vert f-g\Vert\,.$$

However not all metric spaces are normed linear spaces.

Figure 1.4: Hierarchy of linear and nonlinear spaces

Example 1: The two dimensional surface of a sphere

$$X=\{ (x,y,z)\colon x^2+y^2+z^2-1\}~~(\equiv S^2)$$

is not a vector space. It is, however, a metric space whose distance function is the (shortest) length of the great circle passing between a pair of points.

Exercise 13.1 (DISTANCE FUNCTIONS AS METRICS)

Show that (a) the Hamming distance, (b) the Pythagorean distance, and (c) the Chebyshev distance each satisfy the triangle inequality.