Non-Inner Product Metric Space: Understanding & Examples

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

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