Proof Outline

1. {% \Theta %} is complete
2. {% x = \sum_{i=1}^\infty \langle x, \phi_i \rangle \phi_i %}
3. {% \langle x, y \rangle = \sum_{i=1}^\infty \alpha_i \bar{\beta}_i %}
4. {% ||x||^2 = \sum_{i=1}^\infty |\langle x, \phi_i \rangle |^2 %}