Until now, I had only used the Gram-Schmidt process to prove the existence of an orthonormal basis. However, I stumbled upon an exercise which uses the full theorem, that is to say :
Let $E$ be a pre-hilbertian space and $(x_k)_{k\in \mathbb{N}}$ a linearly independent sequence of vectors. Then $\exists! (e_k)_{k\in \mathbb{N}}\in E^\mathbb{N}$ such that
- $(e_k)_{k\in \mathbb{N}}$ is orthonormal
- $\forall k \in \mathbb{N}, \,\operatorname{ span}(x_0,...,x_k)=\operatorname{span}(e_0,...e_k)$
- $\forall k \in \mathbb{N},\langle e_k,x_k\rangle >0\space (\langle .,.\rangle $ being the scalar product)
I have been unable to find a proof of the full theorem, which really bothers me. If someone was so kind as to draw the sketch of a proof (especially of the uniqueness), please, bear in mind I'm an undergraduate !!