Exercise 14, § 6

Let $E$ be a Hausdorff locally convex space and $F$ a locally convex space of minimal type (exerc. 13). *a) Show that if $M$ is a closed vector subspace of the product space $E \times F$, its projection on $E$ is closed in $E$ (use exerc. 13, e)). *b) Let $u$ be a linear mapping of $E$ in $F$. Show that if the graph of $u$ is closed in $E \times F$, then $u$ is continuous (use a). c) Suppose that, in $E$, every closed vector subspace has a topological complement (cf.* V, p. 13). Show that, in $E \times F$, every closed vector subspace $M$ has a topological complement. (If $N_1$ is the projection of $M$ on $E$ and $N_2$ a topological complement of $N_1$ in $E$, if $P_1 = M \cap F$, and $P_2$ is a topological complement of $P_1$ in $F$, show that $N_2 + P_2$ is a topological complement of $M$ in $E \times F$, using b).)