Overview
Given a linearly independent set of vectors in a Hilbert space, {% (v_1,v_2, ... )_1^{\infty} %} there is an ortnormal sequence {% (u_1,u_2, ... )_1^{\infty} %} such that the span of {% u_1,u_2, ... u_k %} is the same as {% v_1,v_2, ... v_k %}
Construction
Define {% u_1 = \frac{v_1}{||v_1 ||} %} Then let
{% x_k = v_k - \sum_{i=1}^{k-1} \langle v_k, u_i \rangle u_i %}
and define
{% u_k = \frac{x_k}{|| x_k ||} %}