Projection: Suppose $X$ is a normed vector space, we define a projection $P$ as a linear and continuous mapping from X to X, such that $P^2 = P$.
im$P$: the image of $P$
commutative: $PQ=QP$
The question is: If $P$ and $Q$ are commutative projections, $P + Q − PQ$ projects onto $\text{im} P + \text{im}Q$.(Here by "onto", I mean "surjective".)
First prove that $P+Q-PQ $ is a projection: \begin{align} (P+Q-PQ)^2 &=P^2+Q^2+P^2Q^2+2PQ-2P^2Q-2PQ^2\\ &=P+Q+PQ+2PQ-2PQ-2PQ\\ &=P+Q-PQ \end{align} For every $ x,y\in X $ we have \begin{align} (P+Q-PQ)(Px+Qy) &=Px+PQy+PQx+Qy-PQx-PQy\\ &=Px+Qy \end{align} thus proving $\newcommand\Im{\operatorname{Im}}\Im (P+Q-PQ)=\Im P+\Im Q $.