Complex Separation Theorem (Real Parts)

Let $X$ be a complex vector space and let $\Omega_1,\Omega_2\subset X$ be nonempty convex sets . Assume core(Ω₁) $\neq\emptyset$ and $\operatorname{core}(\Omega_1)\cap\Omega_2=\emptyset$.

Theorem: There exists a nonzero complex-linear functional $F$ on $X$ such that

Context: View $X$ as a real vector space and apply the real separation theorem to obtain a real linear functional $f$. Then form a complex functional $F$ whose real part is $f$.