Projection Theorem

Let H be a Hilbert space and M a closed subspace of H. Corresponding to any vector x in H, there is a unique vector m_0 in M such that

|x-m_0|<=|x-m|

for all m in M. Furthermore, a necessary and sufficient condition that m_0 in M be the unique minimizing vector is that x-m_0 be orthogonal to M (Luenberger 1997, p. 51).

This theorem can be viewed as a formalization of the result that the closest point on a plane to a point not on the plane can be found by dropping a perpendicular.