Corollary 2

— Let $p$ be a semi-norm on the vector space $E$. Given a point $x_0 \in E$, there exists a linear form $f$ defined over $E$, such that $f(x_0) = p(x_0)$ and that $|f(x)| \leq p(x)$ for all $x \in E$.

Apply cor. 1 to the vector subspace, $V$, generated by $x_0$ and to the linear form $\xi x_0 \mapsto \xi p(x_0)$ defined over $V$.