Exercise 14, § 4

a) Let $E$ be a Fréchet space on which there exists no continuous norm. Show that there exists a closed vector subspace of minimal type in $E$ (II, p. 85, exerc. 13), which is infinite dimensional, and consequently, has a topological complement in $E$. (There exists a strictly decreasing fundamental sequence $(V_n)$ of convex, balanced neighbourhoods of 0 in $E$, and a sequence $(x_n)$ of points of $E$ such that $x_n \notin V_{n+1}$ and such that the line passing through $x_n$ is contained in $V_n$, the $x_n$ being linearly independent; show that the required space is the closed vector subspace generated by the $x_n$.)

b) Let E be a Fréchet space whose topology cannot be defined by a single norm, but by an increasing sequence $(p_n)$ of norms. Let $V_n$ be the neighbourhood defined by $p_n(x) \leq 1$, and let $A'_n = V_n^\circ$ be its polar in $E'$; we can assume that $A'_{n+1}$ is not contained in the vector subspace of $E'$ generated by $A'_n$ (IV, p. 49, exerc. 12, c)). Let $(x'_n)$ be a sequence of points of $E'$ such that $x'_n \in A'_n$ and that $x'_n$ does not belong to the vector subspace generated by $A'_{n-1}$; show that the vector subspace $M'$ generated by the sequence $(x'_n)$ is weakly closed in $E'$ (cf. IV, p. 25, cor. 2) and does not have a topological complement in $E'$ for the topology $\sigma(E', E)$. (Observe that if there existed a weakly continuous projector $u'$ from $E'$ onto $M'$, then $u'(A'_1)$ would be contained in one of the sets $M' \cap A'_n$ by Baire's theorem, and derive a contradiction, since $A'_1$ is weakly total in $E'$.) Deduce that the subspace ${M'}^\circ$ of $E$ does not have a topological complement in $E$.

c) Let E be a Fréchet space whose topology cannot be defined by a single norm. Show that if E is not isomorphic to a product of Banach spaces, then there exists a closed vector subspace in E which has no topological complement. (Argue by reductio ad absurdum; let $(p_n)_{n \geq 1}$ be an increasing sequence of semi-norms on E, defining the topology of E; let $F_n = p_n^{-1}(0)$ and let $E_{n+1}$ be a topological complement of $F_{n+1}$ with respect to $F_n$ (we put $E = F_0$); using b), show that $E_{n+1}$ is a Banach space and that E is isomorphic to the product of the $E_n (n \geq 1)$; for this use I, p. 17, th. 1.)