Norm

Definition

A norm on $V$ is a map $\Vert \cdot \Vert :V\to [0,\infty ⟩$ such that

1. $\Vert u+v\Vert \le \Vert u\Vert +\Vert v\Vert ,\forall u,v\in V$
2. $\Vert c\times v\Vert =|c|\times \Vert v\Vert ,\forall v\in V,\forall c\in \mathbb{C}$
3. $\Vert v\Vert =0⟹v=0$

Think of $\Vert v\Vert$ as the length of $v$.

Norm of 0

$\Vert 0\Vert =\Vert 0\times u\Vert =\Vert c\times u\Vert =\Vert 0\Vert \times \Vert u\Vert =0\times \Vert u\Vert =0$