Local geometry of L1 ∩ L and L1 + L


Extreme points of the unit sphere S(L1 + L∞) of L1 + L∞ under the classical norm used in the interpolation theory were characterized in [8] and [11], while extreme points of S(L1 ∩ L∞) under the classical norm were characterized in [7]. In this paper extreme points of the unit sphere of L1 + L∞ and L1 ∩ L∞ under the "dual" norms are characterized. Moreover, all the extreme points in L1 ∩ L∞ and L1 + L∞ (under both kinds of norms) are examined if they are exposed, H-points, or strongly exposed. Smooth points in both these spaces for both the norms are also characterized. Finally, it is proved that in general the spaces Lp + Lq and Lp ∩ Lq are not isometric to Orlicz spaces, provided that 1 < p < q < + ∞.

Publication Title

Archiv der Mathematik