The L^p Spaces and the Riesz–Fischer Theorem

Abstract

We now depart from the study of functions to the study of function spaces. So far our interest has been in developing the Lebesgue integral. The purpose of this new chapter is to relate the Lebesgue theory of integration to functional analysis. The theory of integration developed in this book enables us to introduce certain spaces of functions that have properties which are of great importance in analysis as well as mathematical physics, in particular, quantum mechanics. These are the so-called L^p spaces of measurable functions f such that |f|^p is integrable. Aside from the intrinsic importance of these spaces, we also examine some applications of results in the previous chapters. One of the most important applications is to Fourier theory. As we remarked before, Fourier theory was a key motivation of the new theory of integration. We will present here the L² version of Fourier series, and in particular establish the Riesz-Fischer theorem which identifies the L² and l² spaces through Fourier series. We hope that this chapter will whet the reader’s appetite for further study of abstract spaces such as Banach and Hilbert spaces.