= Solution
{c}
A <norm> on a real <vector space> $X$ is a map $\|\cdot\|:X\to[0,\infty)$ such that, for $x,y\in X$ and $\lambda\in\mathbb R$,
$$
\|x\|=0\iff x=0,\qquad \|\lambda x\|=|\lambda|\|x\|,\qquad \|x+y\|\leq\|x\|+\|y\|.
$$
The last property is the <triangle inequality>; the middle property is <absolute homogeneity of a norm>. The induced <metric> is
$$
\boxed{d(x,y)=\|x-y\|.}
$$
Positivity and separation follow from the <norm> axioms, symmetry from <absolute homogeneity of a norm> with $\lambda=-1$, and the <triangle inequality> from $x-z=(x-y)+(y-z)$.
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