Theorem 1 (Banach)

Let E and F be two metrisable vector spaces over a non-discrete valued division ring K, and let u be a continuous linear mapping of E in F. Suppose that E is complete. Then the following conditions are equivalent: (i) u is a strict surjective morphism. (ii) F is complete and u is surjective. (iii) The image of u is not meagre in F (GT, IX, § 5.2). (iv) For every neighbourhood V of 0 in E, the set $\overline{u(V)}$ is a neighbourhood of 0 in F.

Firstly (i) implies (ii), for let u be a strict surjective morphism and N be the kernel of u. Then u induces an isomorphism of E/N on F. But E is metrisable and complete, hence E/N is complete (GT, IX, § 3.1, prop. 4), therefore F is complete.

Next (ii) implies (iii). Let F be complete and u be surjective. The image of u is precisely F and therefore not meagre in F from Baire’s theorem (GT, IX, § 5.3).

The following lemma shows that (iii) implies (iv).