Duals and pullbacks of normed modules

Abstract

We give a general description of the dual of the pullback of a normed module. Ours is the natural generalization to the context of modules of the well-known fact that the dual of the Lebesgue-Bochner space $L^p([0,1],B)$ consists - quite roughly said - of $L^q$ maps from $[0,1]$ to the dual $B'$ of $B$ equipped with the weak$^*$ topology. In order to state our result, we study various fiberwise descriptions of a normed module that are of independent interest.