Gram-Schmidt Orthogonalization

Theorem
Let $H$ be a Hilbert space.

Let $S = \set {h_n: n \in \N}$ be a linearly independent subset of $H$.

Then there exists an orthonormal subset $E = \set {e_n: n \in \N}$ of $H$ such that:


 * $\forall k \in \N: \operatorname{span} \set {h_n: 0 \le n \le k} = \operatorname{span} \set {e_n: 0 \le n \le k}$

where $\operatorname{span}$ denotes linear span.

Corollary
The theorem also holds for finite sets $S$: