Chapter 12 Special Banach lattices and their applications

Abstract

This chapter discusses certain Banach lattices of importance in analysis, particularly, the Lorentz and Orlicz spaces. Special Banach lattices arise naturally in probability theory and in many areas of analysis, for example, in interpolation theory, in Fourier analysis, and in functional analysis in the theory of absolutely summing operators. The theory of stable Banach spaces initiated by Krivine and Maurey provides the machinery for extending Aldous' theorem to a large class of Banach and quasi-Banach spaces. One area of analysis in which Orlicz and Lorentz norms arise frequently is probability theory. Indeed, a useful way of studying the integrability of a random series is to determine its closed linear span in an appropriate function space. Banach discussed the question of the linear dimension of the classical Banach spaces—that is, the question of the existence (or nonexistence) of an isometric or an isomorphic embedding from one space into another.