Examples of metric spaces

Much of the theory developed in Chapters 3, 4, and 5 can be extended to the vastly more general setting of metric spaces. Even if we were only interested in analysis on the real line, this would still be worthwhile. In the following chapter, we will use the abstract theory of this chapter to prove an existence and uniqueness theorem for solutions of differential equations.

9.1. Definition. A metric space is a set X with a function d : X × X → [0, ∞), such that:

• d(x, y) = 0 if and only if x = y.

• d(x, y) = d(y, x) for all x, y ∈ X.

• d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X (triangle inequality).

We call d a metric or a distance function on X. We sometimes write (X, d) for the set X with the metric d.

It turns out that all we need in order to develop such notions as convergence, completeness, and continuity is the three simple properties that define a metric. Of the three, the triangle inequality is of course the most substantial.

Examples of metric spaces abound throughout mathematics. In the remainder of this section we will explore a few of them. Be sure to verify the three defining properties of a metric if some of the details have been left out.