Problem: $\displaystyle P_1$ and $\displaystyle P_2$ are two self-adjoint projections. Is it true that $\displaystyle \mbox{R} (P_1) \subseteq \mbox{R} (P_2) \Longrightarrow P_1 \leq P_2$? (Note that R() indicates the range).

I believe it is not true, but I cannot find any counterexamples or way to disprove.

One fact that might be useful to consider is that for a self-adjoint projection $\displaystyle \mbox{R}(P) = \mbox{N}(P)^\perp $