Inner product/K/Norm/Properties/Fact

Let ${}V$ denote a vector space over ${}{\mathbb {K} }$, endowed with an inner product ${}\left\langle -,-\right\rangle$. Then the corresponding norm satisfies the following properties.

1. We have ${}\Vert {v}\Vert \geq 0$.
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

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

Proof, Write another proof