I am always confused between whether metric induces norm or norm induces metric. Can someone clarify this relation a bit for me? Was there any intuition on which direction being true?
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1Are you familiar with the definition of induce? – user642796 Sep 11 '14 at 02:53
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A induces B if B can be obtained from A. We can obtain a metric from a norm. We usually can't obtain a norm from a metric. – Sep 11 '14 at 03:01
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Also see this – BigM Sep 11 '14 at 05:01
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In a vector space, a norm can induce a metric. For example, suppose that $u,v\in\mathbb{R}^n$ and let $||\cdot||$ be the Euclidean norm. Then one can induce a metric $d(u,v):\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}$ given by
$$ d(u,v)=||u-v||. $$
Riemann1337
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