Remark

— Let $E$ be a hilbertian space and $M, N$ two closed vector subspaces of $E$. The relation $M \subset N$ is equivalent to the orthogonality of $M$ and $N^\circ$, that is to say, to the relation $p_M p_{N^\circ} = 0$ by prop. 9, c). Since we have $p_{N^\circ} = 1 - p_N$, we conclude that the relations $M \subset N$ and $p_M = p_M p_N$ are equivalent (« the three perpendicular theorem », cf. fig. 3).

!Diagram showing subspaces M, N, and projections p_M(x), p_N(x)

Fig. 3.