International Journal of Mathematics and Mathematical Sciences
Volume 23 (2000), Issue 1, Pages 21-30
doi:10.1155/S0161171200001708
Bounded sets in the range
of an X∗∗-valued measure with
bounded variation
Departamento de Matemáticas, Escuela Politécnica Superior, Universidad de Huelva, La Rábida, Huelva 21810, Spain
Received 20 July 1998
Copyright © 2000 B. Marchena and C. Piñeiro. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.