Best Approximations Computed from Formula

Theorem

If $W$ is in finite dimension, and $\{{\alpha }_1,\dots ,{\alpha }_n\}$ is any Orthogonal Basis for $W$. Then, $\alpha ={\sum }_{i=1}^n\frac{⟨\alpha |{\alpha }_k⟩}{||{\alpha }_k|{|}^2}\cdot {\alpha }_k$ is the best approximation