Exercise 13, § 4

a)** Let $E$ be a Hausdorff and complete locally convex space. Suppose that there exists $(F_n)$ a decreasing sequence of closed vector subspaces in $E$ such that for every neighbourhood $V$ of 0 in $E$, there exists $n$ such that $F_n \subset V$. Show that the space $E$ is of minimal type (II, p. 85, exerc. 13). b) Let $E$ be a Fréchet space satisfying the first axiom of countability, and which is not of minimal type. Show that there exist two closed vector subspaces $M, N$ in $E$ such that $M \cap N = \{0\}$ and such that $M + N$ is not closed. (Let $(x'_n)$ denote a sequence of linearly independent continuous linear forms on $E$, forming a total set for $\sigma(E', E)$ (III, p. 19, cor. 2)); let $L_n$ be the subspace of $E$ orthogonal to the $x'_i$ for indices $i \leq 2n$; let $x_n, y_n$ be two linearly independent vectors in the complement of $L_{n+1}$ with respect to $L_n$. Let $d$ be a translation invariant distance defining the topology of $E$. Using a), show that there exists a number $\alpha > 0$ such that we can take $d(0, x_n) \geq \alpha, d(0, y_n) \geq \alpha$ and $d(x_n, y_n) \leq 1/n$. Show that if $M$ (resp. $N$) is the closed vector subspace generated by the $x_n$ (resp. $y_n$), then $M$ and $N$ have the properties required; use I, p. 19, cor. 4.) c) Let $E$ be a closed subspace of a product $\prod_{\alpha \in A} F_\alpha$ of normed spaces; show that if every closed subspace of $E$ generated by a countable family of points is of minimal type, then $E$ itself is of minimal type. (Argue by reductio ad absurdum, assuming that the projection of $E$ on each $F_\alpha$ is equal to $F_\alpha$, and that for some $\alpha$, $F_\alpha$ is infinite dimensional.) d) Let $E$ be a Hausdorff and complete locally convex space, in which every subspace generated by a countable family of points is metrizable. In order that $E$ be of minimal type, it is necessary and sufficient that for every pair of closed vector subspaces $M, N$ of $E$ such that $M \cap N = \{0\}, M + N$ is closed in $E$ (use b) and c)).