The space $\dot{\mathcal{B}}'$ of distributions vanishing at infinity - duals of tensor products

TL;DR

This paper explores the topological structure and dual space representations of the space of semi-regular vanishing distributions, extending classical results to a broader class of tensor product spaces.

Contribution

It generalizes Grothendieck's duality results for tensor products to the space of semi-regular vanishing distributions, providing new insights into their duals and scalar products.

Findings

Characterization of the dual space of semi-regular vanishing distributions

Representation formulas for the duals and scalar products

Extension of Grothendieck's duality results to non semi-reflexive cases

Abstract

Analogous to L.~Schwartz' study of the space $\mathcal{D}'(\mathcal{E})$ of semi-regular distributions we investigate the topological properties of the space $\mathcal{D}'(\dot{\mathcal{B}})$ of semi-regular vanishing distributions and give representations of its dual and of the scalar product with this dual. In order to determine the dual of the space of semi-regular vanishing distributions we generalize and modify a result of A. Grothendieck on the duals of $E \hat\otimes F$ if $E$ and $F$ are quasi-complete and $F$ is not necessarily semi-reflexive.