Orlicz spaces without extreme points

Abstract

Orlicz function and sequence spaces unit balls of which have no extreme points are completely characterized for both (the Orlicz and the Luxemburg) norms. Their subspaces of order continuous elements, with the norms induced from the whole Orlicz spaces without extreme points in their unit balls are also characterized. The well-known spaces L 1 and c 0 with unit balls without extreme points are covered by our results. Moreover, a new example of a Banach space without extreme points in its unit ball is given (see Example 1). This is the subspace ( L 1 + L ∞ ) a of order continuous elements of the space L 1 + L ∞ equipped with the norm ‖ x ‖ = a ∫ 0 1 / a x * ( t ) d t whenever 0 < a < ∞ and μ ( T ) > 1 / a .