Exercise 12, § 6

*a) Let E be a vector space and $(E_\alpha)_{\alpha \in A}$ be an increasing directed family of subspaces of E, whose union is E; each $E_\alpha$ is supposed to carry a locally convex topology $\mathscr{T}_\alpha$ such that for $\alpha \leq \beta$ the canonical injection $E_\alpha \to E_\beta$ is continuous. Let $\mathscr{T}$ be the topology on E, which is the inductive limit of the $\mathscr{T}_\alpha$ (II, p. 29, Example II); show that the dual $E'$ of E (for $\mathscr{T}$) with the topology $\sigma(E', E)$ can be canonically identified with the projective limit of the duals $E'_\alpha$, with topology $\sigma(E'_\alpha, E_\alpha)$. b) Let $(X_\alpha, \phi_{\alpha\beta})$ be a projective system of non-empty sets corresponding to a directed set of indices A, such that $\phi_{\alpha\beta}$ are surjective and that $\lim X_\alpha = \varnothing$ (S, III, § 7, exerc. 4). Put $F_\alpha = \mathbf{R}^{(X_\alpha)}$ and denote by $f_{\alpha\beta} : F_\beta \to F_\alpha$ for $\alpha \leq \beta$ the linear mapping deduced canonically from $\phi_{\alpha\beta}$

(A, II, § 1.11, cor. 1). If, we give to each $F_\alpha$ the topology which is the direct sum topology of its factors, the dual $E_\alpha = F'_\alpha$ of $F_\alpha$, with the weak topology $\sigma(E_\alpha, F_\alpha)$, can be identified with the product space $\mathbf{R}^{x_\alpha}$, and $f'_{\alpha\beta}$ is an isomorphism of $E_\alpha$ on a closed subspace of $E_\beta$, having a topological complement in $E_\beta$. Show that on $E = \lim_{\leftarrow} E_\alpha$ (for the $f'_{\alpha\beta}$) the topology which is the inductive limit of those of $E_\alpha$ is the coarsest topology (therefore non-Hausdorff) (using $a$) and noting that $\lim_{\leftarrow} F_\alpha = \{0\}$.