Gram-Schmidt Orthogonalization

Theorem
Let $H$ be a Hilbert space, and let $S = \left\{{h_n: n \in \N}\right\}$ be a linearly independent subset of $H$.

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


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

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

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