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 Minkowski functional of $A$.

Then for each $x \in X$, $\map {\mu_A} x$ is a finite extended real number.

That is:

$\forall x \in X: \map {\mu_A} x < \infty$

Proof

Let $x \in X$.

From Characterization of Convex Absorbing Set in Vector Space:

$\exists t \in \R_{>0}: x \in t A$

where $t A$ denotes the dilation of $A$ by $t$.

Then:

$x \in t^{-1} C$

so that:

$t \in \set {t > 0 : t^{-1} x \in A}$

Then, we have:

$\map {\mu_A} x \le t < \infty$

$\blacksquare$