Proposition 4

Let E, F be two commutative topological groups with operators, and let u be a continuous linear mapping of E into F. In order that there should exist a continuous linear mapping v of F into E such that $v \circ u$ is the identity mapping of E onto itself (in which case u is said to be left invertible and v is said to be a left inverse of u) it is necessary and sufficient that u is a (topological) isomorphism of E onto u(E), and that u(E) has a topological complement in F.

The conditions are sufficient, for if they are satisfied we obtain a left inverse v of u by taking the composition of the isomorphism of u(E) onto E which is the inverse of u, with a continuous projector of F onto u(E).

The conditions are necessary. For the relation $v(u(x)) = x$ shows that $\overline{u}^1(o) = \{ o \}$; u is therefore a bijection of E onto u(E), and since the restriction of v to u(E) is continuous, it follows that u is an isomorphism of E onto u(E). On the other hand, if we put $q = u \circ v$, then q is a continuous linear mapping of F onto u(E) such that $q^2 = q$, which proves (Corollary to Proposition 2) that u(E) has a topological complement in F.