example of pseudometric space
Let $X={\mathbb{R}}^{2}$ and consider the function $d:X\times X$ to the nonnegative 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 ${\mathbb{R}}^{1}$, 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.
$d((2,3),(2,5))=22=0.$ 
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 $d(x,y)=p(xy)$ is a pseudometric on $X$.

•
The trivial pseudometric $d(x,y)=0$ for all $x,y\in X$ is a pseudometric.
References
 1 S. Willard, General Topology, AddisonWesley, Publishing Company, 1970.
Title  example of pseudometric space 

Canonical name  ExampleOfPseudometricSpace 
Date of creation  20130322 14:40:24 
Last modified on  20130322 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 