Structure of spaces of \(C^{\infty}-functions\) on nuclear spaces
type: Item
instance of: scholarly article
title: Structure of spaces of \(C^{\infty}-functions\) on nuclear spaces
zbMATH Open document ID: 0543.46027
DOI: 10.1007/BF02760667
published in: Israel Journal of Mathematics
publication date: 1983-01-01T00:00:00Z
review text: Let E be a real nuclear locally convex space. The authors prove that the space \({\mathcal E}_{ub}(E)\) of \(C^{\infty}\)-functions of uniform bounded type on E (the inductive limit, when V ranges over a base of convex balanced O-neighbourhoods of E, of the spaces \({\mathcal E}_ b(E_ V)\) of \(C^{\infty}\) functions on \(E_ V\) which are bounded with all their derivatives on each bounded subset of \(E_ V)\), coincides with the inductive limit of the space of Nachbin-Dineen \({\mathcal E}_{Nbc}(E_ V).\) If E is a real nuclear bornological vector space, they prove that the space \({\mathcal E}(E)\) coincides with the projective limit of the spaces \({\mathcal E}_{Nbc}(E_ B)\), where B is a closed convex balanced bounded subset of E. As a consequence, a version of the Paley-Wiener-Schwartz theorem is obtained.
reviewed by: P.Raboin
Mathematics Subject Classification ID: 46G20
Mathematics Subject Classification ID: 46A08
Mathematics Subject Classification ID: 46A13
Mathematics Subject Classification ID: 46A11
Mathematics Subject Classification ID: 46M40
zbMATH DE Number: 3863868
zbMATH Keywords: spaces of \(C^{\infty}\)-functions on nuclear spaces
zbMATH Keywords: inductive limit
zbMATH Keywords: projective limit
zbMATH Keywords: Paley-Wiener-Schwartz theorem
author: J.-F. Colombeau
author: Otilia W. Paques
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02760667
OpenAlex ID: W2317772661
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description: scientific article; zbMATH DE number 3863868