Orthogonal Projection onto Closed Linear Span

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

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

Then the orthogonal projection $P$ onto $M$ satisfies:


 * $\forall h \in H: P h = \ds \sum_{k \mathop = 1}^n \innerprod h {e_k} e_k$