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example of pseudometric space
Let $X=\mathbb{R}^{2}$ and consider the function $d:X\times X$ to the nonnegative real numbers given by
$\displaystyle d((x_{1},x_{2}),(y_{1},y_{2}))=x_{1}y_{1}.$ 
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.
$\displaystyle d((2,3),(2,5))=22=0.$ 
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.
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pseudometric d(f,g) triangle inequality
Example of pseudometric pasted from the pseudometric page:
Let X be a set, x0X , and let F(X) be functions XR . Then d(fg)=f(x0)âˆ’g(x0) is a pseudometric on F(X).
How do you prove the triangle inequality for this pseudometric?