Some extremal properties of section spaces of Banach bundles and their duals

Abstract

When X is a compact Hausdorff space and E is a real Banach space there is a considerable literature on extremal properties of the space C(X,E) of continuous E-valued functions on X. What happens if the Banach spaces in which the functions on X take their values vary over X? In this paper, we obtain some extremal results on the section space Γ(π) and its dual Γ(π)* of a real Banach bundle π:ℰ→X (with possibly varying fibers), and point out the difficulties in arriving at totally satisfactory results.