Weak convergence and weak compactness in the space of integrable functions with respect to a vector measure.

We consider weak convergence and weak compactness in the space L¹(m) of real valued integrable functions with respect to a Banach space valued measure m equipped with its natural norm. We give necessary and sufficient conditions for a sequence in L¹(m) to be weak Cauchy, and we give necessary and sufficient conditions for a subset of L¹(m) to be conditionally sequentially weakly compact. [ABSTRACT FROM AUTHOR]