Let H be a Hilbert space and M a closed subspace of H. Corresponding to any vector x element H, there is a unique vector m_0 element M such that left bracketing bar x - m_0 right bracketing bar <= left bracketing bar x - m right bracketing bar for all m element M. Furthermore, a necessary and sufficient condition that m_0 element M be the unique minimizing vector is that x - m_0 be orthogonal to M. 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.