Internal functionals and bundle duals

Abstract

If π:E→X is a bundle of Banach spaces, X compact Hausdorff, a fibered space π*:E*→X can be constructed whose stalks are the duals of the stalks of the given bundle and whose sections can be identified with the “functionals” studied by Seda in [1] and [2] or elements of the “internal dual” Mod(Γ(π),C(X)) studied by Gierz in [3]. If the given bundle is separable and norm continuous, then the fibered space π*:E*→X is actually a full bundle of locally convex topological vector spaces (Theorem 3). In the second portion of the paper two results are stated, both of them corollaries of theorems by Gierz, concerning functionals for bundles of Banach spaces which arise, in turn, from “fields of topological spaces.”

Keywords

• bundles of Banach spaces
• fields of topological spaces
• internal functionals
• internal duals.