Proposition 11

— Let $p \in \mathscr{L}(E)$. In order that p is the orthoprojector from E onto a closed vector subspace of E, it is necessary and sufficient that $p^2 = p = p^*$.

Suppose $p^2 = p$. Let M be the image of p and N its kernel. E is the topological direct sum of M and N. In order that p is an orthoprojector, it is necessary and sufficient that M is orthogonal to N, that is to say that we have $\langle p(x)|y - p(y) \rangle = 0$ for all x, y in E. This latter relation is equivalent to $p = p^*p$, and implies that $p^* = (p^*p)^* = p^*p = p$; conversely if $p^* = p$, we have $p = p^2 = p^*p$.