Bounded sets in the range of \(X^{**}\)-valued measure with bounded variation (Q1974402)

The authors begin by recalling that if \(X\) is an infinite dimensional Banach space, then its unit ball cannot lie in the range of an \(X''\) valued vector measure of bounded variation. They then use this observation to motivate the study of two general problems. First, they give a characterization of the bounded linear operators \(T\) from a Banach space \(X\) into a Banach space \(Y\) which carry the unit ball of \(X\) into a subset which lies in the range of a \(Y\) valued (respectively, \(Y''\) valued) vector measure of bounded variation. Next, the authors characterize those closed, bounded, absolutely convex subsets of \(X\) which lie in the range of an \(X''\) valued vector measure of bounded variation.