The L ^p spaces

Abstract

So far in this book we have considered only pointwise, or almost everywhere, convergence of sequences (and nets) of measurable functions. In many problems where Lebesgue integration occurs, several different kinds of convergence appear naturally. In most cases, different kinds of convergence are associated with different vector spaces of (equivalence classes of) measurable functions, and it is the purpose of this chapter to describe some of these spaces and discuss the corresponding types of convergence. One of the most basic requirements for a mode of convergence is that the usual operations of addition and multiplication by scalars preserve convergence. Because of this, we will be dealing with vector spaces endowed with a topology that is compatible with the linear structure. We start therefore with a brief discussion of the concept of a topological vector space . The discussion is carried out for complex vector spaces, but the analogous concepts make sense for real vector spaces as well.