A function $d : X \times X \to R_{0}^{+}$ over some set $X$ is a metric if it satisfies the following properties:

Non-degeneracy: $d (x, y) = 0 ⟺ x = y$
Triangle Inequality: $\forall x, y, z \in X, d (x, z) \leq d (x, y) + d (y, z)$
Positivity: $\forall x, y \in X, d (x, y) \geq 0$
Commutativity: $\forall x, y \in X, d (x, y) = d (y, x)$

Sufficient conditions for a metric

One can shrink these conditions to two:

Non-degeneracy (as above)
(Modified) Triangular Inequality: $\forall x, y, z \in X, d (x, z) \leq d (x, y) + d (z, y)$

From those two, one can prove the other conditions:

$1 \land 2 ⟹ 3$ : $0 = d (x, x) \leq d (x, y) + d (x, y) □$
$1 \land 2 ⟹ 4$ : $d (x, y) \leq d (x, x) + d (y, x)$ and $d (y, x) \leq d (y, y) + d (y, x) □$

Examples of metrics

The initial motivation for metric spaces comes from $R^{n}$ with the Euclidean metric

d_{E} (u, v) = i = 1 \sum n (u_{i} - v_{i})^{2}

An example of a metric borne out of this general definition is the trivial metric:

d_{t} (x, y) = {0 if x = y 1 if x \neq = y

Consider $X = C ([0, 1])$ the set of all continuous functions over $[0, 1]$ , and define the supremum norm as

d_{\infty} (f, g) = x \in [0, 1] sup ∣ f (x) - g (x) ∣