Lebesgue spaces

Abstract

The Lebesgue spaces L ^p(Ω) play a central role in many applications of functional analysis. This chapter focuses upon their basic properties, as well as certain attendant issues that will be important later. Notable among these are the semicontinuity of integral functionals, and the existence of measurable selections. We begin by identifying a geometric property of the norm which, when present, turns out to have a surprising consequence. A normed space X is said to be uniformly convex if it satisfies the following property:

$$\forall ~ \varepsilon > 0\,, ~ \exists ~ \delta\, >\, 0 \mbox{ such that } x \in\, B\,,\: y \in\, B\,, \,\: \|\, x - y\, \| >\, \varepsilon \:\: \Longrightarrow \:\: \Big\| \, \frac {x + y}{2}\, \Big\| ~ <\, 1 - \delta\,.$$

In geometric terms, this is a way of saying that the unit ball is curved. The property depends upon the choice of the norm on X, even among equivalent norms, as one can see even in ${\mathbb{R}}^{\,2}$. Yet it implies an intrinsic property independent of the choice of norm: a uniformly convex Banach space is reflexive.