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In the case of vectors in euclidean space, for instance, we can express one in terms of the other--i.e. length is distance from zero, distance is the length of the vector difference. Does this break down somewhere?

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    The notion of a norm only makes sense on a vector space, while metrics can be defined on every set (except, perhaps, on the empty set -- depending on your conventions).2011-08-06
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    In a vector space (over $\mathbb{R}$, say), you can talk about "positive homogeneous translation-invariant metrics". "Norm" just sounds a bit nicer.2011-08-06
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    And once you have a $\text{norm}$ on a vector space, you have a metric on the vector space, namely $d(x,y) = ||x-y||$.2011-08-06

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