International Journal of Mathematics and Mathematical Sciences
Volume 7 (1984), Issue 4, Pages 689-695
doi:10.1155/S0161171284000715
Internal functionals and bundle duals
1Department of Mathematics, Duke University, Durham 27706, North Carolina, USA
2Department of Mathematics, Trinity College, Hartford 06106, Connecticut, USA
Received 8 February 1983; Revised 1 August 1984
Copyright © 1984 Joseph W. Kitchen and David A. Robbins. 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
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.”