 Solution

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