Exercise 13, § 6

Let $E$ be a vector space. We say that a Hausdorff locally convex topology $\mathscr{T}$ on $E$ is minimal (and that $E$, with $\mathscr{T}$, is a space of minimal type) if there exists no Hausdorff locally convex topology on $E$, that is strictly coarser than $\mathscr{T}$ (cf. II, p. 81, exerc. 1). *a) Let $\mathscr{T}$ be a minimal topology on $E$, and let $E'$ be the dual of $E$ (when $E$ has the topology $\mathscr{T}$); show that $\mathscr{T} = \sigma(E, E')$ and $E = {E'}^*$ (note that there cannot be an everywhere dense hyperplane in $E'$ for the topology $\sigma(E', E)$ using the cor. 3 of II, p. 43). Deduce that spaces of minimal type are products of lines. *b) Show that in a Hausdorff locally convex space $F$, every subspace $E$ of minimal type has a topological complement, and in particular is closed (use a) and the Hahn-Banach th. for extending the identity mapping of $E$ in itself to a mapping of $F$ in $E$. *c) Let $u$ be a continuous linear mapping of a space $E$ of minimal type in a Hausdorff locally convex space $F$. Show that $u(E)$ is closed in $F$ and that $u$ is a strict morphism of $E$ in $F$ (use b) and the definition of a space of minimal type). *d) Let $F$ be a Hausdorff locally convex space and $M$ be a closed vector subspace of $F$. Show that, if there exists a complement $N$ of $M$ in $F$ that is a subspace of minimal type, then $N$ is a topological complement of $M$ in $F$ (use c). *e) Let $M$ be a subspace of minimal type in a Hausdorff locally convex space $F$; show that, for every closed vector subspace $N$ of $F$ the sum $M + N$ is closed in $F$ (consider the quotient space $F/N$ and use c)). If further $N$ is of minimal type, then $M + N$ is of minimal type.