# Definition:Metric Space/Triangle Inequality

## Definition

Let $M = \struct {A, d}$ be a metric space, satisfying the metric space axioms:

 $(M1)$ $:$ $\displaystyle \forall x \in A:$ $\displaystyle \map d {x, x} = 0$ $(M2)$ $:$ $\displaystyle \forall x, y, z \in A:$ $\displaystyle \map d {x, y} + \map d {y, z} \ge \map d {x, z}$ $(M3)$ $:$ $\displaystyle \forall x, y \in A:$ $\displaystyle \map d {x, y} = \map d {y, x}$ $(M4)$ $:$ $\displaystyle \forall x, y \in A:$ $\displaystyle x \ne y \implies \map d {x, y} > 0$

Axiom $M2$ is referred to as the triangle inequality, as it is a generalization of the Triangle Inequality which holds on the real number line and complex plane.

## Also see

$\forall x, y, z \in X: \size {\map d {x, z} - \map d {y, z} } \le \map d {x, y}$