Corollary

— In a complex locally convex space E, every closed complex linear variety M is the intersection of the closed complex hyperplanes which contain it.

In fact, for all $x \notin M$, there exists a convex open neighbourhood $V$ of $x$ that does not meet $M$, and thus there exists a closed complex hyperplane $H$ containing $M$ and not meeting $V$; a fortiori $H$ does not contain $x$.