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?
Why do we have the notions of both 'norm' and 'metric'?
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metric-spaces
norm
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11The 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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6In 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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1And 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