Exercise 3, § 2

Let $E$ be a complex Hausdorff locally convex space, $E_0$ the underlying real locally convex space to $E$, and let $E'$ and $E'_0$ be the duals of $E$ and $E_0$ respectively. Show that the canonical $\mathbf{R}$-linear mapping $f \mapsto \mathscr{R}f$ from $E'$ onto $E'_0$ is a homeomorphism for the $\mathfrak{S}$-topologies on $E'$ and $E'_0$, where $\mathfrak{S}$ is an arbitrary set of bounded subsets of $E$. From this deduce the definition of the canonical $\mathbf{R}$-linear mapping from the bidual $E''$ onto the bidual $E''_0$, which is a homeomorphism for the weak topologies $\sigma(E'', E')$ and $\sigma(E''_0, E'_0)$, as also for the strong topologies $\beta(E'', E')$ and $\beta(E''_0, E'_0)$; by this mapping, $E$ (considered embedded in $E''$) transforms into $E_0$ (considered embedded in $E''_0$).