Exercise 28, § 6

Let $E, F$ be two commutative topological groups with operators, $u : E \to F, v : F \to E$ two continuous linear mappings. Now prove that if $u \circ v \circ u = u, u^{-1}(0)$ and $v(u(E))$ are topological direct summands in $E, u(E)$ and $v^{-1}(u^{-1}(0))$ are topological direct summands in $F$, and that the restriction of $u$ to $v(u(E))$ is an isomorphism of $v(u(E))$ onto $u(E)$.