Minkowski Functional of Convex Absorbing Set is Finite

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

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

Let $A \subseteq X$ be a convex absorbing set.

Let $\mu_A$ be the Minkowki functional of $A$.

Then $\map {\mu_A} x < \infty$ for each $x \in X$.

Proof
Let $x \in X$.

From the definition of an absorbing set, there exists $t > 0$ such that $x \in t A$.

Then:


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

so that:


 * $\map {p_A} x \le t < \infty$