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 a start, is a projection, with range . So if then , , and hence . Thus if is a projection then its range contains .
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),
.
Therefore
.
It follows from (7) that the range of is contained in the range of and in the range of , and thus in .