Corollary 1

— Let $p$ be a semi-norm on the vector space $E$. Let $V$ be a vector subspace of $E$ and $f$ a linear form on $V$ such that $|f(y)| \leq p(y)$ for all $y \in V$. Then there exists a linear form $h$ defined on $E$ which is an extension of $f$ and is such that $|h(x)| \leq p(x)$ for $x \in E$.

For a semi-norm $q$ and a linear form $g$ on $E$, the relation $g \leq q$ is the same as $|g| \leq q$. The corollary follows from th. 1.