Exercise 9, § 4

Let A be a countable set. Consider three pairs of vector spaces $(P, P'), (Q, Q'), (E, E')$, each of the six spaces being vector subspaces of $\mathbf{R}^A$ and containing the direct sum subspace $\mathbf{R}^{(A)}$; in addition, suppose that for every point $x \in P$ (resp. $x \in Q$, $x \in E$) and every point $x' \in P'$ (resp. $x' \in Q'$, $x' \in E'$), the family $(x_\alpha x'_\alpha)_{\alpha \in A}$ is summable and put $\langle x, x' \rangle = \sum_{\alpha \in A} x_\alpha x'_\alpha$; this bilinear form puts P and $P'$ (resp. Q and $Q'$, E and $E'$) in separating duality.

a) Suppose that $E = P \cap Q$, $E' \supset P' + Q'$ and $E' \neq P' + Q'$. Consider the linear mapping $u : x \mapsto (x, x)$ from E into $F = P \times Q$, which is put into separating duality with $F' = P' \times Q'$. Show that $u$ is continuous for the weak topologies $\sigma(E, E')$ and $\sigma(F, F')$ and that its image $M = u(E)$ is a closed subspace for $\sigma(F, F')$; deduce that $u'$ is a strict morphism from $F'$ into $E'$ for the topologies $\sigma(F', F)$ and $\sigma(E', E)$, and that $N = u'(F)$ is not closed in $E'$ for $\sigma(E', E)$. If $E'$ is metrizable for the topology $\tau(E', E)$, deduce that $u'$ is also a strict morphism from $F'$ into $E'$ for the topologies $\tau(F', F)$ and $\tau(E', E)$.

b) In addition, suppose that $E'$ is a Fréchet space for $\tau(E', E)$. Then $F'/M^\circ$ endowed with the quotient topology of $\tau(F', F)$ by $M^\circ$, is not semi-complete, and there exist bounded sets in $F'/M^\circ$ which are not relatively compact for $\tau(F'/M^\circ, M)$.

c) Under the same hypotheses as in b), let $x'$ be an element of $E'$ not belonging to $N = u'(F')$; for every $y \in M$, let $v(y) = \langle x, x' \rangle$, where $x \in E$ is the unique element such that $u(x) = y$. Show that the linear form $v$ on $M$ is not continuous for the topology $\sigma(F, F')$, but that its restriction to every bounded subset of $M$ is continuous for $\sigma(M, F'/M^\circ)$. Deduce that $L = v^{-1}(0)$ is a vector subspace of $F$, whose intersection with every bounded and closed subset of $F$ (for $\sigma(F, F')$) is closed for $\sigma(F, F')$, but which is not closed in $F$ for $\sigma(F, F')$.