We now arrive finally at the main desired result for this section:
Theorem: The projections of any vector onto any orthogonal basis set for can be summed to reconstruct exactly.
Proof: Let denote any orthogonal basis set for . Then since is in the space spanned by these vectors, we have
(using the linearity of the projection operator which follows from linearity of the inner product in its first argument). Since the basis vectors are orthogonal, the projection of onto is zero for :
We therefore obtain
Therefore, the sum of projections onto the vectors , , is just the linear combination of the which forms :
by Eq. (5.3).