Example 3

Let E be a separated locally convex space over $\mathbf{R}$, and let $E_{(\mathbf{C})}$ be its complexification. Let $u$ be an endomorphism of E. In order that $u$ be a Riesz endomorphism, it is necessary and sufficient that its complexification $u_{(\mathbf{C})}$ be a Riesz endomorphism of $E_{(\mathbf{C})}$.