Axiom:Norm Axioms (Vector Space)

Definition
Let $\left({R, +, \circ}\right)$ be a division ring with norm $\left\Vert{\cdot}\right\Vert_R$.

Let $V$ be a vector space over $R$, with zero $\mathbf 0_V$.

Let $\left\Vert{\cdot}\right\Vert: V \to \R_{\ge 0}$ be a norm on $V$.

The norm axioms are the following conditions on $\left\Vert{\cdot}\right\Vert$ which define $\left\Vert{\cdot}\right\Vert$ as being a norm: