Inner product/K/Norm/Properties/Fact

Let $V$ denote a vector space over $𝕂$, endowed with an inner product $⟨-,-⟩$. Then the corresponding norm satisfies the following properties.

1. We have $\Vert v\Vert \ge 0$.
2. We have $\Vert v\Vert =0$ if and only if $v=0$.
3. For $\lambda \in 𝕂$ and $v\in V$, we have

   $\Vert \lambda v\Vert =|\lambda |\cdot \Vert v\Vert .$

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

   $\Vert v+w\Vert \le \Vert v\Vert +\Vert w\Vert .$

Proof, Write another proof