Inner Product Spaces

Abstract

To motivate the subject matter of this book, we begin this chapter by listing five basic problems that arise in various applications of “least-squares” approximation. While these problems seem to be quite different on the surface, we will later see that the first three (respectively the fourth and fifth) are special cases of the general problem of best approximation in an inner product space by elements of a finite-dimensional subspace (respectively convex set). In this latter formulation, the problem has a rather simple geometric interpretation: A certain vector must be orthogonal to the linear subspace.