Weak compactness in Orlicz spaces of vector functions. (Spanish: Compacidad débil en espacios de Orlicz de funciones vectoriales).

Abstract

There is a well-known characterisation of the weakly compact subsets of L 1 -spaces—usually known as the theorem of Dunford. Using the fact that if 1<p<∞ , then L p ⊆L 1 while the dual of L 1 is dense in that of L q (we are assuming that the underlying measure space is a probability space), this immediately provides characterisations of the weakly compact subsets of L p -spaces or, more generally, of suitable Orlicz spaces. Since these L p -spaces are semireflexive, the weakly compact sets are just the closed, bounded ones. However, the same method can be applied to Bochner L p -spaces or Orlicz spaces (even in the absence of the Radon-Nikodým property in which case the dual spaces are not identifiable with the expected Bochner spaces). The authors use this fact to show that if the weakly compact subsets of such Bochner Orlicz spaces can be characterised in the "natural'' way, i.e. by carrying over the descriptions in the scalar case to the vector-valued case, then the same is true in L 1 . They then deduce from known results in the latter case that this is only the case when the functions take their values in a Banach space which, along with its dual, has the Radon-Nikodým property.