Exercise 8, § 3

Let E be a Banach space over a non-discrete valued division ring K. We say that a vector subspace M of E is strongly paracomplete if there exists a norm $\|x\|_M$ on M for which M is a Banach space and the canonical injection of M in E is continuous. a) Show that if M and N are two strongly paracomplete subspaces of E, then M + N and M ∩ N are also strongly paracomplete subspaces. (On M + N, consider the norm $\|x\|_{M+N} = \inf(\|u\|_M + \|v\|_N)$, where the lower bound is taken over all pairs $(u, v)$ such that $x = u + v,\ u \in M$ and $v \in N$.) b) Let M, N be two strongly paracomplete subspaces of E such that N and M + N are closed. Show that $\overline{M} = M + (\overline{M} \cap \overline{N})$ and $\overline{M \cap N} = \overline{M} \cap \overline{N}$ (use exerc. 7, a)).