Remark 5

Let E be a locally convex space over $\mathbf{R}$. In order that the complexified locally convex space $E_{(\mathbf{C})}$ of E be an approximation space, it is necessary and sufficient that E be one. This follows from remarks 1 and 4, since the underlying real locally convex space of $E_{(\mathbf{C})}$ is isomorphic to $E\times E$.