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Let $P$ and $Q$ be projection on a Hilbert Space. If $PQ=QP$, would you help me to prove that $% PQ$ is also a projection.

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    @beginner: Your definition of projection is incomplete. A projection is also idempotent, meaning that $P^2=P$. The condition you gave in your comment adds the condition that $P$ is an *orthogonal* projection, which is equivalent to $P^*=P$. You can show that $PQ=QP$ implies that $(PQ)^2=PQ$ and $(PQ)^*=PQ$.2012-11-05

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Using your definition of (orthogonal) projection, you want to show the following: If $x \in \mathrm{ran}(PQ)$ and $y \in \ker(PQ)$, then $\langle x, y \rangle = 0$. So take $x = PQz$ for some $z$, and calculate the quantity $ \langle PQz, y \rangle$.

You'll also want to use the following fact (or prove it, if your class has not proved it): every projection is self-adjoint.