Minkowski Functional of Balanced Convex Absorbing Set in Vector Space is Seminorm

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

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

Let $A \subseteq X$ be a set that is balanced, convex and absorbing.

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

Then $\mu_A$ is a seminorm.

Case 1: $\GF = \R$
From Balanced Set in Real Vector Space is Symmetric, $A$ is symmetric.

Hence by Minkowski Functional of Symmetric Convex Absorbing Set in Real Vector Space is Seminorm, $\mu_A$ is a seminorm in this case.

Case 2: $\GF = \C$
From Minkowski Functional of Convex Absorbing Set is Positive Homogeneous, $\mu_A$ is a sublinear functional.

Hence we have:


 * $\map {\mu_A} {x + y} \le \map {\mu_A} x + \map {\mu_A} y$ for all $x, y \in X$

and hence.

We also have:


 * $\map {\mu_A} {r x} = r \map {\mu_A} x$ for all $r \in \hointr 0 \infty$ and $x \in X$.

Towards, we want to show that:


 * $\map {\mu_A} {r x} = \map {\mu_A} x$ for all $r \in \C$ with $\cmod r = 1$.

From the definition of a balanced set, we have:


 * $r A \subseteq A$

and since $\cmod {r^{-1} } = 1$, we have:


 * $r^{-1} A \subseteq A$

Then we have:


 * $r A \subseteq A \subseteq r A$

so that:


 * $A = r A$

Then we have:

Now take $r \in \C \setminus \hointl 0 \infty$ and $x \in X$.

We have:

proving.