# Difference between metric space and normed space

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?

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!