The Hilbert Space L ²

Abstract

In Chapter 5, we introduced the general L ^p($\mathbb R$)-spaces. Among the L ^p($\mathbb R$)- spaces, the case p = 2 has a very special status: L ² $\mathbb R$ is a Hilbert space, and in fact the only L ^p($\mathbb R$)-space with that property. The space L ²($\mathbb R$)is discussed in Section 6.1. As a continuation and specialization of the previous sections on operators, some fundamental operators on L ²($\mathbb R$) are considered in Section 6.2. The considered operators will play important roles in the later chapters on the Fourier transform and wavelets. Section 6.3 deals with the Hilbert space L ²(a, b); in particular it is shown that the polynomials form a dense subspace of L ²(a, b). Section 6.4 discusses Fourier expansions in the framework of the Hilbert space L ²(-π, π).