Minkowski Functional of Open Ball with respect to Seminorm is Seminorm

Theorem

Let $\GF \in \set {\R, \C}$.

Let $X$ be a vector space over $\GF$.

Let $p$ be a seminorm on $X$.

Let:

$B = \set {x \in X : \map p x < 1}$

be the open ball centred at ${\mathbf 0}_X$ with radius $1$ in the pseudometric induced by $p$.

Let $\mu_B$ the Minkowski functional of $B$.

Then $p = \mu_B$.

Hence the definition is valid.

Let $x \in X$.

From Seminorm Axiom $\text N 2$: Positive Homogeneity:

$\forall s > \map p x: \map p {\dfrac x s} = \dfrac 1 s \map p x < 1$

Then we have:

$s \in \set {t > 0 : t^{-1} x \in B}$

so that:

$\map {\mu_B} x \le s$

Taking the infimum over $s > \map p x$, we have:

$\map {\mu_B} x \le \map p x$

Let $\map p x = 0$.

Then:

$\map p x \le \map {\mu_B} x$

Let $\map p x \ne 0$.

Let $0 < t \le \map p x$.

Then:

$\map p {\dfrac x t} = \dfrac 1 t \map p x \ge 1$

So $t^{-1} x \notin B$.

So:

$\forall x \in X: \map {\mu_B} x \ge \map p x$

We therefore obtain:

$\forall x \in X: \map {\mu_B} x = \map p x$

$\blacksquare$