Some remarks concerning the spaces of multipliers and convolutors, \({\mathcal{O}_M}\) and \(\mathcal{O}^\prime_C\), of Laurent Schwartz
type: Item
instance of: scholarly article
title: Some remarks concerning the spaces of multipliers and convolutors, \({\mathcal{O}_M}\) and \(\mathcal{O}^\prime_C\), of Laurent Schwartz
zbMATH Open document ID: 1276.46029
author: Julian Larcher
published in: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas. RACSAM
publication date: 2013-01-30T00:00:00Z
review text: \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)].
reviewed by: María Carmen Gómez-Collado
Mathematics Subject Classification ID: 46F05
Mathematics Subject Classification ID: 46A13
Mathematics Subject Classification ID: 46A11
Mathematics Subject Classification ID: 46F10
zbMATH DE Number: 6132874
zbMATH Keywords: tempered distributions
zbMATH Keywords: projective limit
zbMATH Keywords: inductive limit
zbMATH Keywords: compact embeddings
zbMATH Keywords: ultrabornological
zbMATH Keywords: Schwartz space
zbMATH Keywords: multipliers
zbMATH Keywords: convolutors
MaRDI profile type: Publication
full work available at URL: s13398-012-0059-5
OpenAlex ID: W1966986029
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DOI: 10.1007/S13398-012-0059-5
description: scientific article; zbMATH DE number 6132874