Vector space/K/Norm/Definition

Norm

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

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

is called norm, if the following properties hold.

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

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

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

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