The L ^p-spaces

Abstract

In this chapter we introduce and analyze an important class of Banach spaces consisting of functions, the so-called L ^p-spaces, where p is a parameter that specifies a particular space. First, in Section 5.1 we discuss some vector spaces and concepts that play a role in the analysis of the L ^p-spaces. In Section 5.2 we introduce the space L ¹($\mathbb R$), and show how to make it a normed space. Integration techniques in L ¹($\mathbb R$) are discussed in Section 5.3; in particular, we consider techniques for interchanging an integral and an infinite sum. For p∈]1,∞], the spaces L ^p($\mathbb R$) are introduced in Section 5.4; their counterparts L ^p(a, b) concerning functions defined on a subinterval ]a, b[⊂ $\mathbb R$ are discussed in Section 5.5.