Vector space/K/Norm/Definition

Let ${\displaystyle {}V}$ be a ${\displaystyle {}{\mathbb {K} }}$-vector space. A mapping

${\displaystyle \Vert {-}\Vert \colon V\longrightarrow \mathbb {R} ,v\longmapsto \Vert {v}\Vert ,}$

is called norm if the following properties hold.

1. We have ${\displaystyle {}\Vert {v}\Vert \geq 0}$ for all ${\displaystyle {}v\in V}$.
2. We have ${\displaystyle {}\Vert {v}\Vert =0}$ if and only if ${\displaystyle {}v=0}$.
3. For ${\displaystyle {}\lambda \in {\mathbb {K} }}$ and ${\displaystyle {}v\in V}$, we have

   ${\displaystyle {}\Vert {\lambda v}\Vert =\vert {\lambda }\vert \cdot \Vert {v}\Vert \,.}$

4. For ${\displaystyle {}v,w\in V}$, we have

   ${\displaystyle {}\Vert {v+w}\Vert \leq \Vert {v}\Vert +\Vert {w}\Vert \,.}$