Remark

Given a closed vector subspace M of a Hausdorff topological vector space E, it is not necessarily the case that there exists an (algebraic) complementary vector subspace to M that is closed in E (even if E is a normed space; cf. IV, p. 55, exerc. 16 (c)); a fortiori there does not necessarily exist a topological complement of M in E (cf. I, p. 26, exerc. 8). However we shall see in § 2 that when K is a non-discrete valued division ring, then every closed subspace M of E, with finite codimension, does have a topological complement in E (I, p. 14, prop. 3).