I am reading "Introduction to Set Theory and Topology" (in Japanese) by Kazuo Matsuzaka.
Let $S$ be a nonempty set.
Let $d:S\times S\to\mathbb{R}$ be a function such that $d(x,y)=1$ if $x\neq y$ and $d(x,y)=0$ if $x=y$.
Then, $(S,d)$ is a metric space.
This is the only example of a metric space with a metric which is not induced from a norm I know.
Is there an interesting metric space with a metric which is not induced from a norm?
The following is the definition of a norm:
Let $E$ be a vector space. A norm on $E$ is a fucntion $v\mapsto|v|$ from $E$ into $\mathbb{R}$ satisfying the following axioms:
N1. We have $|v|\geq 0$ and $|v|=0$ if and only if $v=0$.
N2. If $a\in\mathbb{R}$ and $v\in E$, then $|av|=|a||v|$.
N3. For all $v,w\in E$ we have $|v+w|\leq |v|+|w|.$
