Three-space-problems and small ball properties for Fréchet spaces
type: Item
instance of: scholarly article
title: Three-space-problems and small ball properties for Fréchet spaces
zbMATH Open document ID: 1230.46003
published in: Functiones et Approximatio. Commentarii Mathematici
publication date: 2011-06-29T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.facm/1308749129
review text: \textit{L. Frerick} and \textit{A. Peris} [RACSAM, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 97, No.~2, 257--261 (2003; Zbl 1068.46001)] introduced the following concept: let \(\mathcal A\) be a family of subsets of a metric space \(X\); a subset \(D \subset X\) has the \(\mathcal A\)-small ball property if for every sequence \(\varepsilon_n > 0\) tending to 0 monotonically, there is a sequence of \(A_n \in \mathcal A\) such that \(D \subset \bigcup_{n \in \mathbb N}\{x \in X: d(x, A_n) < \varepsilon_n\}\). Several properties of Fréchet spaces are characterized in terms of \(\mathcal A\)-small ball properties. In the paper under review, the author presents a unified approach to three-space properties in the context of Fréchet spaces by using \(\mathcal A\)-small ball properties. In particular, this gives a new proof of the fact that the properties of being a Schwartz space, a quasinormable space, a Montel space and reflexivity are three-space properties. The paper is dedicated to the memory of Susanne Dierolf.
reviewed by: Vladimir Kadets
Mathematics Subject Classification ID: 46A04
Mathematics Subject Classification ID: 46A11
Mathematics Subject Classification ID: 47B07
zbMATH DE Number: 5914712
zbMATH Keywords: Fréchet spaces
zbMATH Keywords: three-space-properties
zbMATH Keywords: small ball properties
author: A. Peris
MaRDI profile type: Publication
OpenAlex ID: W1971888129
cites work: Q5457541
cites work: Metric spaces with the small ball property
cites work: Stefan Heinrich's Density Condition for Fréchet Spaces and the Characterization of the Distinguished Köthe Echelon Spaces
cites work: On the lifting of bounded sets in Fréchet spaces
cites work: Q3198332
cites work: Q4043587
cites work: On the three-space-problem for topological vector spaces
cites work: Q4652145
DOI: 10.7169/FACM/1308749129
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description: scientific article; zbMATH DE number 5914712