Exercise 6, § 4

Let E be a vector space with its finest locally convex topology. a) Show that every vector subspace of E is closed, and that, if M, N are two subspaces that are vectorial complements in E, then E is the direct topological sum of M and N. If $(e_i)_{i \in I}$ is a basis for E, then E is the direct topological sum of the subspaces $\mathbf{R} e_i$. b) Let F be a locally convex space whose topology is also the finest locally convex topology. Show that every linear mapping of E in F is a strict morphism.