  We discuss the structural and topological properties of a general class of
weighted $L^1$ convolutor spaces. Our theory simultaneously applies to weighted
$\mathcal{D}'_{L^1}$ spaces as well as to convolutor spaces of the
Gelfand-Shilov spaces $\mathcal{K}\{M_p\}$. In particular, we characterize the
sequences of weight functions $(M_p)_{p \in \mathbb{N}}$ for which the space of
convolutors of $\mathcal{K}\{M_p\}$ is ultrabornological, thereby generalizing
Grothendieck's classical result for the space $\mathcal{O}'_{C}$ of rapidly
decreasing distributions. Our methods lead to the first direct proof of the
completeness of the space $\mathcal{O}_{C}$ of very slowly increasing smooth
functions.
