Gram-Schmidt Orthogonalization/Corollary 1

Corollary to Gram-Schmidt Orthogonalization
Let $H$ be a Hilbert space. Let $m \in \N$ be a natural number.

Let $S = \set {h_n: n \le m}$ be a finite linearly independent subset of $H$.

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


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

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