Relatively weakly open subsets of the unit ball in functions spaces

For an infinite Hausdorff compact set K and for any Banach space X we show that every nonempty weak open subset relative to the unit ball of the space of X-valued functions that are continuous when X is equipped with the weak (respectively norm, weak-∗) topology has diameter 2. As consequence, we improve known results about nonexistence of denting points in these spaces. Also we characterize when every nonempty weak open subset relative to the unit ball has diameter 2, for the spaces of Bochner integrable and essentially bounded measurable X-valued functions.  2005 Elsevier Inc. All rights reserved