Let be a Hilbert Space and and be the projections on closed linear subspaces and of .
Prove that if is a projection,then its range is .
For the converse inclusion, if is a projection then
(since and ),
Multiply both sides of (3) on the left to get
Multiply both sides of (3) on the right to get
From (4), (5) and (3),
It follows from (7) that the range of is contained in the range of and in the range of , and thus in .