Lemma 4

Let E be a real Banach space and $u\in \mathscr{L}(E_{(\mathbf{C})})$ a nonzero endomorphism of $E_{(\mathbf{C})}$. Let C be a total convex cone in E. There exists $x\in C$ such that $u(x)\not = 0$.

Indeed, if $u$ vanishes on C, then $u$ vanishes on $C-C$, hence on E, and therefore on $E_{(\mathbf{C})}$.