DE LA VALLÉE POUSSIN'S THEOREM AND WEAKLY COMPACT SETS IN ORLICZ SPACES

Abstract

The classical theorem of Dunford and Pettis identifies the bounded, uniformly integrable subsets of L¹(μ) with the relatively weakly compact sets. Another characterization of uniform integrability is given in a theorem of De La Vallée Poussin which states that a subset K of L¹ (μ) is bounded and uniformly integrable if and only if there is an N-function F so that sup{f F(f)dμ: f ε K} < ∞. De La Vallée Poussin's theorem is the focal point of the fmt part of this paper as well as the driving force for the results in the second part. We refine and improve this theorem in several directions. The theorem of De La Vallée Poussin does not, for instance, specify just how well the function F can be chosen. It gives little additional information in case the set in question is relatively norm compact in L¹ (μ). Finally it gives no information on the structure of the set in the corresponding Band space of F-integrable functions. More specifically we establish the fact that a subset K of L¹ is relatively compact if and only if there is an N-function F ε δ' so that K is relatively compact in L*_F. Furthermore we prove that a subset K of L¹ is relatively weakly compact if and only if there is an N-function F ε δ' so that K is relatively weakly compact in L*_F. We then go on to show that a large class of non-reflexive Orlicz spaces has the weak Band-Saks property, by establishing a result for these spaces, very similar to the Dunford-Pettis theorem for L¹.