ε-weakly precompact sets in Banach spaces

A bounded subset M of a Banach space X is said to be epsilon-weakly precompact, for a given epsilon >= 0, if every sequence (x(n))(n is an element of N) in M admits a subsequence (x(nk))(k is an element of N) such that lim sup(k ->infinity) x*(x(nk)) - lim(k ->infinity) inf x*(x(nk)) <= epsilon for all x* is an element of B-X*. In this paper we discuss several aspects of epsilon-weakly precompact sets. On the one hand, we give quantitative versions of the following known results: (a) the absolutely convex hull of a weakly precompact set is weakly precompact (Stegall), and (b) for any probability measure mu, the set of all Bochner mu-integrable functions taking values in a weakly precompact subset of X is weakly precompact in L-1(mu, X) (Bourgain, Maurey, Pisier). On the other hand, we introduce a Banach space property related to the one considered by Kampoukos and Mercourakis when studying subspaces of strongly weakly compactly generated spaces. We say that a Banach space X has property RMw if there is a family {M-n,M-p : n, p is an element of N} of subsets of X such that: (i) M-n,M-p is 1/p -weakly precompact for all n, p is an element of N, and (ii) for each weakly precompact set C subset of X and for each p is an element of N there is n is an element of N such that C subset of M-n,M-p. All subspaces of strongly weakly precompactly generated spaces have property RMw. Among other things, we study the three-space problem and the stability under unconditional sums of property RMw.