On certain subsets of Bochner Integrable Function Spaces

Abstract

One of the most important methods used in literature to introduce new properties in a Banach space E, consists in establishing some non trivial relationships between different classes of subsets of E. For instance, E is reflexive, or has finite dimension, if and only if every bounded subset is weakly relatively compact or norm relatively compact, respectively. On the other hand, Banach spaces of the type C(K) and Lp(μ) play a vital role in the general theory of Banach spaces. Their structure is so rich that many important concepts and results of the general theory have been modelled on these spaces. Also, the characterization of most important classes of subsets of these spaces, is well known. However, the situation is completely different for the analogous spaces of vector valued functions. In general, their structure is quite more involved than that of the scalar function spaces. In this talk we shall be mainly concerned with the space L1(μ,E). When E = K, most of the classes of subsets we are interested in, coincide. This is no longer true in the vectorial case, and we shall try to determine classes of Banach spaces E for which the natural extension of the characterizations of several classes of distinguished subsets of L1(μ), are valid in L1(μ,E).