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:
Let $(V, \lVert {}\cdot{} \rVert)$ be a normed space. For all $v \in V$,
\[v \neq 0 \enspace\rimp\enspace \lVert v \rVert > 0.\]