Metrics

Why

We want to talk about a set with a prescribed quantitative degree of closeness (or distance) between its elements.

Definition

The correspondences which serve as a degree of closeness, or measure of distance, must satisfy our previously developed (see Distance) notion of distance.

A function on ordered pairs which does not depend on the order of the elements so considered is symmetric. A function into the real numbers which takes only nonnegative values is nonnegative. A repeated pair is an ordered pair of the same element twice. A function which satisfies a triangle inequality for any three elements is triangularly transitive.

A metric (or distance function) is a function on ordered pairs of elements of a set which is symmetric, non-negative, zero only on repeated pairs, and triangularly transitive. A metric space is an ordered pair whose first coordinate is a nonempty set and whose second coordinate is a metric.

In a metric space, we say that one pair of objects is closer together if the metric of the first pair is smaller than the metric value of the second pair.

Notice that a set can be made into different metric spaces by using different metrics.

Notation

Let $A$ be a set. We commonly denote a metric by the letter $d$, as a mnemonic for “distance.” Let $d: A \times A \to \R $. Then $d$ is a metric if:

1. it is non-negative, which we tend to denote by

   \[ d(a, b) \geq 0 \quad \forall a,b \in A. \]

2. it is $0$ only on repeated pairs, which we tend to denote by

   \[ d(a, b) = 0 \iff a = b, \quad \forall a,b \in A. \]

3. it is symmetric, which we tend to denote by:

   \[ d(a, b) = d(b, a), \quad \forall a,b \in A. \]

4. it is triangularly transitive, which we tend to denote by

   \[ d(a, b) \leq d(a, c) + d(c, b), \quad \forall a,b,c \in A. \]

Examples

$\R $ with the absolute value distance is a metric space. As is $\R ^2$ and $\R ^3$ with the Euclidean distance. $\R ^n$ with Euclidean metric is an example of a metric space for which the objects ($n$-dimensional tuples of real numbers) are impossible to visualize.