Gram-Schmidt Orthogonalization

Theorem
Let $\struct {V, \innerprod \cdot \cdot}$ be a inner product space.

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

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


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

where $\span$ denotes linear span.

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