Metric Defines Norm iff it Preserves Linear Structure

Theorem

Let $V$ be a vector space over a valued field $k$.

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

Then the function $\| v \| := d(v,0)$ 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)

$(2): \quad d(\lambda x, \lambda y) = | \lambda | d(x,y)$ (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 $d$:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \Vert \lambda v \Vert\)	\(=\)	\(\displaystyle d(\lambda v, 0 )\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	By the definition of $\vert \cdot \vert$
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle d(\lambda v, \lambda \cdot 0)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \vert\lambda\vert d(v,0)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	By the enlargement property
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \vert\lambda\vert \Vert v \Vert\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	By the definition of $\Vert \cdot \Vert$

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

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

$\blacksquare$