Metric spaces not derived from an inner product

[kthouz]

Can somebody give me an other metric space that is not dependent on the inner product i mean which is not derived from the inner product between two vectors.

 

[quasar987]

The function d(x,y) = 0 if x=y and d(x,y)=1 if not. It's called the discrete metric.

 

[Zurtex]

I remember a particularly exotic one given as an example to me, that the details elude me right at the moment. But here's a good one:

$$m, n \in \mathbb{N}$$

$$d(m,n) = \left| m^{-1} - n^{-1} \right|$$

$$d(n,\infty) = d(\infty,n) = \frac{1}{n}$$

$$d(\infty,\infty) = 0$$