| context | $F$ … subfield of $\mathbb{C}$ |
| context | $V$ … $F$-vector space |
| definiendum | $p\in \mathrm{SemiNorm}(V)$ |
| postulate | $p:V\to \mathbb R $ |
| $v,w\in V$ | |
| postulate | $p(v+w) \le p(v)+p(w)$ |
| $\lambda\in F$ | |
| postulate | $p(\lambda\cdot v) = |\lambda|\cdot p(v)$ |
A Norm is a seminorm with the adition axiom
$p(v)=0 \implies v=0$
(which I also write as $p(!0)=0$.)
Wikipedia: Norm