On two classes of (F)-spaces
type: Item
instance of: scholarly article
title: On two classes of (F)-spaces
zbMATH Open document ID: 0621.46001
DOI: 10.1007/BF01275578
author: Dietmar Vogt
published in: Archiv der Mathematik
publication date: 1985-01-01T00:00:00Z
review text: Let \(\| \cdot \|_ 1\leq \| \cdot \|_ 2\leq...\leq \| \cdot \|_ n\leq..\). be a fundamental system of seminorms on a Fréchet space E. \(\| y\|_ n^*=\sup \{| y(x)|:\| x\|_ n\leq 1\}\), \(x\in E\), \(y\in E'\), \(n=1,2,..\). The following conditions of \textit{D. Vogt} are important in the theory of nuclear Fréchet spaces: (DN) There exist \(k_ 0\in N\) such that for every \(k\in N\) there are \(n\in N\) and \(C>0\) with \(\| x\|^ 2_ k\leq C\| x\|_{k_ 0}\| x\|_ n\) for all \(x\in E.\) (\(\Omega)\) For any \(p\in N\) there is \(q\in N\) such that for every \(k\in N\) there are \(m\in N\) and \(C>0\) with \(\| y\|_ q^{*1+m}\leq \| y\|^*_ k\| y\|_ p^{*m}\) for all \(y\in E'\) [Math. Z. 155, 109-117 (1977; Zbl 0337.46015), Stud. Math. 67, 225-240 (1980; Zbl 0464.46010)]. Let \(K:=R\) or C. If M is a set and a(.) a function in M such that a(t)\(\geq 1\), \(t\in M\), then \[ \Lambda^{\infty}=\Lambda^{\infty}(M,a)=\{x\in K^ M:\| x\|_ k=\sup_{t}| x(t)| a^ k(t)<\infty \text{ for all } k\} \] \[ \Lambda^ 1=\Lambda^ 1(M,a)=\{x\in K^ M:\| x\|_ k=\sum_{t}| x(t)| a^ k(t)<\infty \text{ for all } k\}. \] (s) is the Fréchet space of rapidly decreasing sequences. The author proves the following results: (a) E has property (DN) if and only if E is isomorphic to a subspace of a space \(\Lambda^{\infty}.\) (b) The class (DN) is the smallest class of Fréchet spaces, which contains the Banach spaces and (s) and which is closed under subspaces and \({\hat \otimes}_{\epsilon}.\) (c) The following are equivalent: (1) E has property (DN) and E' is strictly separable in its weak*-topology; (2) E is isomorphic to a subspace of \(\ell^{\infty}{\hat \otimes}_{\epsilon}(s)\); (3) is isomorphic to a subspace of \({\mathcal B}(R)\); (4) E is isomorphic to a subspace of some \(\Lambda^{\infty}(N,a).\) (c) E has property \((\Omega)\) if and only if E is isomorphic to a quotient of a space \(\Lambda^ 1.\) (d) The class \((\Omega)\) is the smallest class of Fréchet spaces, which contains the Banach spaces…
Mathematics Subject Classification ID: 46A04
Mathematics Subject Classification ID: 46A13
Mathematics Subject Classification ID: 46A11
Mathematics Subject Classification ID: 46A45
zbMATH DE Number: 4006956
zbMATH Keywords: nuclear Fréchet spaces
zbMATH Keywords: Fréchet space of rapidly decreasing sequences
reviewed by: Manuel Valdivia
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01275578
OpenAlex ID: W2063630128
description: scientific article; zbMATH DE number 4006956