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

Analogous to L.~Schwartz’ study of the space ${\mathcal{D}}^{\mathrm{\prime }}(\mathcal{E})$ of semi-regular distributions we investigate the topological properties of the space ${\mathcal{D}}^{\mathrm{\prime }}(\dot{\mathcal{B}})$ 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 $E\widehat{\otimes }F$ if $E$ and $F$ are quasi-complete and $F$ is not necessarily semi-reflexive.