Show that if $P$ and $Q$ are two orthogonal projections with orthogonal ranges, then $P+Q$ is also an orthogonal projection.
First I need to show $(P+Q)^\ast = P+Q$. I am thinking that since \begin{align*} ((P+Q)^\ast f , g) & = (f,(P+Q)g) \\ & = (f,Pg) + (f,Qg) \\ & = (P^\ast f,g) + (Q^\ast f,g) \\ & = (Pf,g) + (Qf,g) \\ & = ((P+Q)f,g), \end{align*} we get $(P+Q)^\ast=P+Q$.
I am not sure if what I am thinking is right since I assumed that $(P+Q)f=Pf+Qf$ is true for any bounded linear operator $P$, $Q$.
For $(P+Q)^2=P+Q$, I use $(P+Q)^2= P^2 + Q^2 + PQ +QP,$ but I cant show $PQ=0$ and $QP=0$.
Anyone can help me? Thanks.