Metric Defines Norm iff it Preserves Linear Structure

Theorem
Let $\struct {k, \norm {\,\cdot\,}_k}$ be a valued field.

Let $V$ be a vector space over the valued field $\struct {k, \norm {\,\cdot\,}_k}$.

Let $d: V \times V \to k$ be a metric on $V$.

Then the function $\norm v := \map d {v, 0}$ is a norm on $V$ for all $x, y, z \in V$, $\lambda \in k$:


 * $(1): \quad \map d {x + z, y + z} = \map d {x, y}$ (homogeneity or translation invariance)


 * $(2): \quad \map d {\lambda x, \lambda y} = \norm \lambda_k \map d {x, y}$ (the enlargement property)

Proof
Suppose first that $d$ satisfies the hypotheses $(1)$ and $(2)$.

From :
 * $\forall u, v \in V: \map d {u, v} \ge 0$

Hence:
 * $\forall u \in V: \norm u = \map d {u, 0} \ge 0$

Moreover, from :
 * $\norm u = 0 \implies \map d {u, 0} = 0$

and hence:
 * $u = 0$

Now let $\lambda \in K$, $u \in V$.

Then, using the enlargement property of $d$:

Finally if $u,v \in V$, then we have