Some remarks concerning the spaces of multipliers and convolutors, \({\mathcal{O}_M}\) and \(\mathcal{O}^\prime_C\), of Laurent Schwartz (Q1935071)

\textit{A. Grothendieck} in [Mem. Am. Math. Soc. 16 (1955; Zbl 0064.35501)] used tensor products to prove that the spaces \(\mathcal{O}_{M}\) and \(\mathcal{O}_{C}'\) of Schwartz multipliers and convolutors on tempered distributions are bornological. In [Int. J. Math. Math. Sci. 8, 813--816 (1985; Zbl 0654.46042)], \textit{J. Kučera} claimed to get a new more constructive proof of Grothendieck's results. Kučera used a certain representation of the space \(\mathcal{O}_{M}\) as a projective limit of inductive limits of Banach spaces and claimed that certain linking maps were compact. In the article under review, the author revisits the representation of these spaces, investigates some of their topological properties and shows that the claim of Kučera was false. A sequence space representation of the space \(\mathcal{O}_{M}\), which answered a question of Grothendieck, was obtained by \textit{M. Valdivia} [Math. Z. 177, 463--478 (1981; Zbl 0472.46025)].