Proposition 3

*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 $u \circ v$ is the identity mapping of $F$ (in which case $u$ is said to be right invertible and $v$ is said to be a right inverse of $u$) it is necessary and sufficient that $u$ is a strict morphism (§ 2, no. 8) of $E$ onto $F$ and that $\overline{u}(0)$ has a topological complement in $E$.*

The conditions are necessary. For we have then $u(v(F)) = F$ and a fortiori $u(E) = F$; furthermore, if $p = v \circ u$, then $p$ is a continuous linear mapping of $E$ into itself such that $p^2 = p$; therefore (Corollary to Proposition 2) $p(E) = v(u(E)) = v(F)$ has a topological complement $\overline{p}(0)$ in $E$; but since $u(p(x))) = u(x)$ by hypothesis, we have $\overline{u}(0) = \overline{p}(0)$. Finally, the bijective mapping of $E/\overline{u}(0)$ onto $F$, associated with $u$, is the composition of the bijective mapping of $E/\overline{u}(0)$ onto $v(F)$, associated with $p$, and the restriction of $u$ to $v(F)$; since $v$ is continuous, both these mappings are isomorphisms, and therefore $u$ is a strict morphism of $E$ onto $F$.

The conditions are sufficient. For if $\varphi$ is the canonical homomorphism of E onto $E/\overline{u}^1(o)$, to say that $\overline{u}^1(o)$ has a topological complement M in E is to say that the restriction of $\varphi$ to M is an isomorphism of M onto $E/\overline{u}^1(o)$. Since on the other hand $u = w \circ \varphi$, where w is an isomorphism of $E/\overline{u}^1(o)$ onto F, we see that the restriction of u to M is an isomorphism of M onto F, and the inverse isomorphism v is therefore such that $u \circ v$ is the identity mapping of F onto itself.