Orthogonal Projection onto Closed Linear Span

Theorem
Let $H$ be a Hilbert space, and let $E = \left\{{e_1, \ldots, e_n}\right\}$ be an orthonormal subset of $H$.

Let $M = \vee E$, the closed linear span of $E$.

Then the orthogonal projection $P$ onto $M$ satisfies, $\forall h \in H$:


 * $Ph = \displaystyle \sum_{k=1}^n \left\langle{h, e_k}\right\rangle e_k$