Exercise 24, § 1

Let $(E_\alpha)_{\alpha \in A}$ be a family of locally convex Hausdorff spaces, E a vector space and for every $\alpha \in A$, let $f_\alpha$ be a linear mapping from $E_\alpha$ into E. On E consider the finest locally convex topology $\mathscr{T}$ for which the functions $f_\alpha$ are continuous (II, p. 27); suppose $\mathscr{T}$ is Hausdorff and let $E'_\alpha$ denote the dual of $E_\alpha$, and $E'$ that of E endowed with $\mathscr{T}$. Show that if, for every $\alpha \in A$, the topology of $E_\alpha$ is identical with $\tau(E_\alpha, E'_\alpha)$, then the topology $\mathscr{T}$ is identical with $\tau(E, E')$.