Suavidad, convexidad, dualidad y renormamientos

Abstract

The goal of this Bachelor thesis is to study some specifical geometric properties, which are related to the convexity and smoothness of Banach spaces. More precisely we will go over the definitions of strict convexity, uniform convexity, smoothness and uniform smoothness. We will study some duality relationships between strict convexity and smoothness, namely that the strict convexity (smoothness) of the dual norm implies the smoothness (strict convexity) of the norm. We show an example proving that the converse of this duality results do not hold. However, we will prove that there is a complete duality relationship between uniform convexity and uniform smoothness through Lindenstrauss’ formula. With regard to renorming theory, it will be also showed that every separable space can be renormed in order to be strictly convex. We will go also over Enflo’s Theorem: a space is superreflexive if and only if it admits an uniform convex renorming. In the last chapter we improve the above results. In fact, almost all renormings of a separable space are strictly convex and almost all renomings of a superreflexive space are uniformly convex.