Normed vector space/Metric space/Fact/Proof

Proof

1. We have ${}d{\left(u,v\right)}=\Vert {u-v}\Vert =0$ if and only if ${}u-v=0$, that is, ${}u=v$.
2. We have

   ${}{\begin{aligned}d{\left(u,v\right)}&=\Vert {u-v}\Vert \\&=\Vert {-1(v-u)}\Vert \\&=\vert {-1}\vert \cdot \Vert {v-u}\Vert \\&=d{\left(v,u\right)}.\end{aligned}}$

3. For arbitrary ${}w\in V$, we have, due to the definition of a norm

   ${}{\begin{aligned}d{\left(u,v\right)}&=\Vert {u-v}\Vert \\&\leq \Vert {u-w}\Vert +\Vert {w-v}\Vert \\&=d{\left(u,w\right)}+d{\left(w,v\right)}.\end{aligned}}$

To fact