Editing Pseudometric space (section)

==Examples==

Any metric space is a pseudometric space. 
Pseudometrics arise naturally in [[functional analysis]]. Consider the space <math>\mathcal{F}(X)</math> of real-valued functions <math>f : X \to \R</math> together with a special point <math>x_0 \in X.</math> This point then induces a pseudometric on the space of functions, given by <math display=block>d(f,g) = \left|f(x_0) - g(x_0)\right|</math> for <math>f, g \in \mathcal{F}(X)</math>

A [[seminorm]] <math>p</math> induces the pseudometric <math>d(x, y) = p(x - y)</math>. This is a [[convex function]] of an [[affine function]] of <math>x</math> (in particular, a [[translation (geometry)|translation]]), and therefore convex in <math>x</math>. (Likewise for <math>y</math>.)

Conversely, a homogeneous, translation-invariant pseudometric induces a seminorm.

Pseudometrics also arise in the theory of [[hyperbolic manifold|hyperbolic]] [[complex manifold]]s: see [[Kobayashi metric]]. 

Every [[measure space]] <math>(\Omega,\mathcal{A},\mu)</math> can be viewed as a complete pseudometric space by defining <math display=block>d(A,B) := \mu(A \vartriangle B)</math> for all <math>A, B \in \mathcal{A},</math> where the triangle denotes [[symmetric difference]].

If <math>f : X_1 \to X_2</math> is a function and ''d''<sub>2</sub> is a pseudometric on ''X''<sub>2</sub>, then <math>d_1(x, y) := d_2(f(x), f(y))</math> gives a pseudometric on ''X''<sub>1</sub>. If ''d''<sub>2</sub> is a metric and ''f'' is [[Injective function|injective]], then ''d''<sub>1</sub> is a metric.