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Norm

Set

context $F$ … subfield of $\mathbb{C}$
context $V$ … $F$-vector space
definiendum $p\in \mathrm{Norm}(V)$
postulate $p:V\to \mathbb R $
$v,w\in V$
postulate $p(v+w) \le p(v)+p(w)$
postulate $p(v)=0 \implies v=0$
$\lambda\in F$
postulate $p(\lambda\cdot v) = |\lambda|\cdot p(v)$

Discussion

$ p(v)\ge 0 $

The last axiom $\ p(v)=0 \implies v=0\ $ isn't part of seminorm.

Reference

Wikipedia: Norm

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