Corollary

Let $E$ be a commutative topological group with operators, and let $M$ be a stable subgroup of $E$. Then the following conditions are equivalent:

a) $M$ has a topological complement in $E$.

b) There is a continuous projector $p$ of $E$ into $E$ such that $p(E) = M$.

c) The identity mapping of $M$ can be extended to a continuous linear mapping of $E$ onto $M$.

It follows from Proposition 2 that a) implies b), and it is clear that b) implies c). Finally, if $p$ is a continuous linear mapping of $E$ onto $M$ which extends the identity mapping of $M$, then $p$ is a continuous projector, and the projectors $p$ and $1 - p$ are associated with the direct sum decomposition $E = M + N$, where $N = \overline{p}(0)$.