Applying Hilbert spaces to the 'line of best fit' problem

Abstract

Although it is possible to study various mathematical applications without explicit use of Hilbert space terminology and techniques, there are great advantages to be gained from Hilbert space formulation. Concepts with which we are familiar with in two- and threc- dimensional Euclidean geometry, in particular orthogonality and projections, can also be appropriately extended to infinite-dimensional Hilbert Spaces. One Hilbert Space application which we shall discuss further on today is the application of Hilbert Spaces to the general linear model. The properties which identify a Hilbert Space are its completeness and its inner product space properties. With reference to Cauchy Sequences, we hence define the property of completeness in the Hilbert Space context as follows.