Contribution to the geometry of the besicovith- Orlicz space of almost periodic functions

Abstract

In this thesis we are interested in the geometry of Banach spaces. In particular, the one of the Besicovitch-Orliczspaces of almost periodic functions a.p.

a.p is the closure of the set of generalized trigonometric polynomials relative to the Luxemburg norm and Φis a convex function with properties similar to those of the power function. Some geometric properties of these spaces, such as uniform and strict convexity, uniform non-squareness, β-propertyhave been already studied.

Our main objective is to characterize the extreme points of the unit ball of a.p for the bothnorms: the Luxemburg norm, and the Orlicznorm (equal to the Amemiya norm). The results obtained depend closely on the strict convexity and the structural affine intervals of the functionΦ. Thanks to these results, were found the sufficient conditions for the strict convexity of a.p in the case of the Luxemburg norm. Furthermore, some properties of the set of points where the infimum is atteined in Amemya norm are also obtained.