Theorem 1 (Hahn-Banach)

— Let V be a vector subspace of E, a complex vector space, and let f be a (complex) linear form on V and p a semi-norm on E such that $|f(y)| \leq p(y)$ for all $y \in V$. Then there exists a linear form $f_1$ on E extending f and such that $|f_1(x)| \leq p(x)$ for all $x \in E$.

For $g = \Re f$ is a real linear form defined in V and satisfying $|\cdot(y)| \leq p(y)$ at every point of V; therefore there exists a real linear form $g_1$ in E extending g and such that $|g_1(x)| \leq p(x)$ for all $x \in E$ (II, p. 23, cor. 1). Let $f_1(x) = g_1(x) - ig_1(ix)$ be the complex linear form on E of which $g_1$ is the real part (II, p. 61). For all real $\vartheta$

since $p$ is a semi-norm on the complex space E; this implies the relation $|f_1(x)| \leq p(x)$, and the theorem is proved.