Metric Spaces Not Based on Inner Product

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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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Let (X,d) be a metric space, where X = {A, B}, with A and B two points in the plane, and d the distance between them. It is easily verified that d is a metrix function on X.