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:


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

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

Then $p = \mu_B$.

Proof
Let $x \in X$.

Then for $s > \map p x$, we have:


 * $\ds \map p {\frac x s} = \frac 1 s \map p x < 1$

from.

Then we have:


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

so that:


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

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


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

Now if $x \in X$ has $\map p x = 0$, we have:


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

Now take $\map p x \ne 0$.

Then, if $0 < t \le \map p x$, we have:


 * $\ds \map p {\frac x t} = \frac 1 t \map p x \ge 1$

So $t^{-1} x \not \in B$.

So, we must have $\map {\mu_B} x \ge \map p x$ for each $x \in X$.

We therefore obtain:


 * $\map {\mu_B} x = \map p x$ for each $x \in X$.