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What is the difference between a normed linear space and a metric space?

When I was going through definition I saw that a metric space is defined on a set. But a normed space is defined on a vector space.

Is there any metric space where the metric is defined on a set but not on a vector space?

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The only rules a metric has to satisfy are positivity, symmetry, and the triangle inequality. Given any set, finite or infinite, you can define the trivial metric: $d(a,b) = 1$ if $a\neq b$, $d(a,b) = 0$ if $a = b$. E.g. this metric can be used on the set $S = \{a, b, c\}$ to create a metric space, but $S$ is clearly not a vectorspace!

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  • $\begingroup$ But what defining these separate abstractions buys us? How are metric spaces you gave above are useful? $\endgroup$ – meguli Aug 10 '17 at 22:15
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Take a normed space and remove the origin. Now you have a metric space which is not a normed space. Alternatively, consider any non-contractible Riemannian manifold. For example, take a sphere with its standard round metric. This will be a metric space which is certainly not a normed space.

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