Duals and pullbacks of normed modules

TL;DR

This paper generalizes the duality of pullback normed modules, extending classical Lebesgue-Bochner space duality to a broader module context with fiberwise descriptions.

Contribution

It provides a comprehensive description of the dual of pullback normed modules, generalizing known duality results to a more abstract module setting.

Findings

Dual of pullback normed modules characterized

Fiberwise descriptions of normed modules developed

Extension of Lebesgue-Bochner duality to 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.