Exercise 7, § 3

Let E be a complete metrisable vector space over a non-discrete valued division ring K. We say that a vector subspace M of E is paracomplete if there exists on M a complete metrisable vector space structure for which the canonical injection of M in E is continuous. a) Let M, N be two paracomplete subspaces of E such that M + N and M ∩ N are closed in E. Show that M and N are closed in E. (Taking quotients by M ∩ N reduces the question to the case where M ∩ N = {0}, and we can consider then the mapping $(x, y) \mapsto x + y$ of M × N in E). b) Show that if E is the union of an increasing sequence of paracomplete subspaces, $(M_j)_{j \geq 0}$, then there exists an index j such that $M_j = E$. (Use Baire’s theorem (GT, IX, § 5.3, th. 1) and I, p. 17, th. 1).