Proposition 1

— Let A be an open non-empty convex set in a complex topological vector space E and M be a non-empty (complex) linear variety that does not meet A. Then there exists a closed complex hyperplane H that contains M and does not meet A.

We can suppose that $0 \in M$. Then there exists a closed real hyperplane $H_0$ containing M and not meeting A (II, p. 36; th. 1). As $M = iM$, the closed complex hyperplane $H = H_0 \cap (iH_0)$ has the properties required.