Bounded sets in the range of an X∗∗-valued measure with bounded variation

Abstract

Let X be a Banach space and A⊂X an absolutely convex, closed, and bounded set. We give some sufficient and necessary conditions in order that A lies in the range of a measure valued in the bidual space X∗∗ and having bounded variation. Among other results, we prove that X∗ is a G. T.-space if and only if A lies inside the range of some X∗∗-valued measure with bounded variation whenever XA is isomorphic to a Hilbert space.

Keywords

• Banach spaces
• Banach discs
• vector measure
• bounded variation.