Least squares in general vector spaces revisited

Approximation theory and the theory of optimization provide algebraic theorems characterizing the global minima of a quadratic functional on a linear variety in abstract vector spaces. Surprisingly, little use has been made of these results in statistics. Estimating equations for M-estimators and optimality results in best or minimax estimation are usually derived by more or less unhandy techniques of calculus. This even applies to results that could be gained without effort from algebraic theorems. The purpose of the present paper is to recall an elementary vector space minimum theorem and to exhibit the ease of its use.