example of pseudometric space

Let $X={ℝ}^2$ and consider the function $d:X\times X$ to the non-negative real numbers given by

$d((x_1,x_2),(y_1,y_2))=|x_1-y_1|.$

Then $d(x,x)=|x_1-x_1|=0$, $d(x,y)=|x_1-y_1|=|y_1-x_1|=d(y,z)$ and the triangle inequality follows from the triangle inequality on ${ℝ}^1$, so $(X,d)$ satisfies the defining conditions of a pseudometric space.

Note, however, that this is not an example of a metric space, since we can have two distinct points that are distance 0 from each other, e.g.

$d((2,3),(2,5))=|2-2|=0.$

Other examples:

• Let $X$ be a set, $x_0\in X$, and let $F(X)$ be functions $X\to R$. Then $d(f,g)=|f(x_0)-g(x_0)|$ is a pseudometric on $F(X)$ [1].

• If $X$ is a vector space and $p$ is a seminorm over $X$, then $d(x,y)=p(x-y)$ is a pseudometric on $X$.

• The trivial pseudometric $d(x,y)=0$ for all $x,y\in X$ is a pseudometric.

Title	example of pseudometric space
Canonical name	ExampleOfPseudometricSpace
Date of creation	2013-03-22 14:40:24
Last modified on	2013-03-22 14:40:24
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	6
Author	mathcam (2727)
Entry type	Example
Classification	msc 54E35
Related topic	Seminorm
Related topic	VectorSpace
Related topic	MetricSpace
Related topic	Metric
Defines	trivial pseudometric