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 $\norm{ v } := d(v,0)$ is a norm on $V$ if and only if for all $x,y,z \in V$, $\lambda \in k$:

$(1): \quad d(x+z,y+z) = d(x,y)$ (homogeneity or translation invariance)
$(2): \quad d(\lambda x, \lambda y) = \norm{\lambda}_k d(x,y)$ (the enlargement property)

Proof

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

Since $d(u,v) \geq 0$ for all $u,v \in V$, $\| u \| = d(u,0) \geq 0$ for all $u \in V$.

Moreover if $\| u \| = 0$ then $d(u,0) = 0$, so $u = 0$.

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

Then, using the enlargement property of $d$:

 $\displaystyle \norm{\lambda v}$ $=$ $\displaystyle d(\lambda v, 0 )$ By the definition of $\norm{\,\cdot\,}$ $\displaystyle$ $=$ $\displaystyle d(\lambda v, \lambda \cdot 0)$ $\displaystyle$ $=$ $\displaystyle \norm{\lambda}_k d(v,0)$ By the enlargement property $\displaystyle$ $=$ $\displaystyle \norm{\lambda}_k \norm{v}$ By the definition of $\norm{\,\cdot\,}$

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

 $\displaystyle \Vert u + v \Vert$ $=$ $\displaystyle d(u + v, 0)$ By the definition of $\Vert \cdot \Vert$ $\displaystyle$ $=$ $\displaystyle d(u, -v)$ By translation invariance of $d$ $\displaystyle$ $\leq$ $\displaystyle d(u, 0) + d(0,-v)$ By the triangle inequality for metrics. $\displaystyle$ $=$ $\displaystyle d(u, 0) + d(v,0)$ By the enlargement property $\displaystyle$ $=$ $\displaystyle \Vert u \Vert + \Vert v \Vert$ By the definition of $\Vert \cdot \Vert$

$\blacksquare$