Chapter 3 Lp spaces

Abstract

This chapter describes the structure of the L p -spaces and their subspaces. The L p -spaces have provided much fodder for the general theory of Banach spaces, because they appeared early in the theory, and the study of these spaces has motivated the definitions of many properties of more general Banach spaces. For example, with its usual norm L p is a Banach lattice under the pointwise almost everywhere ordering. The spaces naturally occur as interpolation spaces and are the simplest of the rearrangement invariant spaces. The study of the structure of the finite dimensional subspaces of L p paved the way for much of the extraordinary development of the local theory of Banach spaces in 1980. In the investigations of other Banach spaces and operators, the existence and classification of operators from, into, or factoring through L p -spaces provide fundamental information on the structure.