Let $(V,\lVert{}\cdot{}\rVert)$ be a seminormed space. $\lVert{}\cdot{}\rVert$ is a norm if for all $v\in V$,
\[\lVert v\rVert = 0 \enspace\rimp\enspace v=0.\]$(V,\lVert{}\cdot{}\rVert)$ is then called a normed space.
If $(V,\lVert{}\cdot{}\rVert)$ is a normed space, then
\[d : V\times V\to\R, \, (x,y)\mapsto\lVert x-y\rVert\]defines a metric on $V$.
As a corollary of NOR#PROP-MT:
Let $(V,\lVert{}\cdot{}\rVert)$ be a normed space. For all $v\in V$,
\[v\neq 0 \enspace\rimp\enspace \lVert v\rVert>0.\]