Tensor sequences and inductive limits with local partition of unity
type: Item
instance of: scholarly article
title: Tensor sequences and inductive limits with local partition of unity
zbMATH Open document ID: 0576.46053
DOI: 10.1007/BF01171493
author: Ralf Hollstein
published in: Manuscripta Mathematica
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155085
review text: In his previous article [J. Reine Angew. Math. 319, 38-62 (1980; Zbl 0426.46053)], the author proved that a locally convex (l.c.) space E with the countable neighborhood property (c.n.p.) is an \(\epsilon\)-space if and only if \[ (+)\quad E\otimes_{\epsilon}(\sum^{\infty}_{i=1}A_ i(F_ i))\cong \sum^{\infty}_{i=1}Id\otimes A_ i(E\otimes_{\epsilon}F_ i) \] holds for all countable l.c. hulls \(F=\sum^{\infty}_{i=1}A_ i(F_ i)\) and thus characterized the spaces E for which the \(\epsilon\)-tensor product commutes with each countable l.c. hull F. Here, the author complements his discussion by characterizing those l.c. hulls F for which \((+)\) holds for all l.c. spaces E with c.n.p. In fact, this is true if and only if \(F=\sum^{\infty}_{i=1}A_ i(F_ i)\) ''admits a local partition of unity'' (a property much weaker than the existence of a partition of unity in the sense of M. de Wilde); e.g., \((+)\) holds if N(A) is an \(\epsilon\)- space or F is a \(\pi\)-space or if all \(F_ i\) are hilbertisable. As a consequence, one obtains conditions under which the \(\epsilon\)-tensor product of two bornologicalDF)-spaces is again a bornological (DF)-space; e.g., \(E\otimes_{\epsilon}F\) is a bornological (DF)-space if E is normed and F a bornological \(\pi\)-(DF)-space or if both E and F are bornological \(\pi\)-(DF)-spaces. Grothendieck's problem whether the \(\epsilon\)-tensor product of two (DF)-spaces must always be a (DF)-space remains open, but the author makes some interesting remarks on this problem and mentions several other open questions in this area. Since a l.c. hull \(F=\sum^{\infty}_{i=1}A_{\alpha}(F_{\alpha})\) has a local partition of unity if and only if \(0\to N(A)\to \oplus_{\alpha}F_{\alpha}\to F\to 0\) is a ''\(\otimes\)-sequence'', the author's work is closely related to \textit{W. Kaballo, D. Vogt} [Manuscripta Math. 32, 1-27 (1980; Zbl 0456.46058)], and he establishes two new conditions which imply that a topologically exact sequence \(0\to H\to G\to Q\…
Mathematics Subject Classification ID: 46M05
Mathematics Subject Classification ID: 46M40
Mathematics Subject Classification ID: 46A13
Mathematics Subject Classification ID: 46A11
Mathematics Subject Classification ID: 46A08
Mathematics Subject Classification ID: 46F05
Mathematics Subject Classification ID: 46A45
Mathematics Subject Classification ID: 46B25
Mathematics Subject Classification ID: 46A04
zbMATH DE Number: 3921293
zbMATH Keywords: countable neighborhood property
zbMATH Keywords: \(\epsilon \) -space
zbMATH Keywords: \(\epsilon \) -tensor product
zbMATH Keywords: local partition of unity
zbMATH Keywords: bornologicalDF)-spaces
zbMATH Keywords: topologically exact sequence
zbMATH Keywords: \(\otimes \)-sequence
zbMATH Keywords: \(\pi \) -spaces
reviewed by: Klaus Dieter Bierstedt
MaRDI profile type: Publication
cites work: Lokaltopologische Vektorräume
cites work: Q4080199
cites work: Q4741203
cites work: Tensor products and spaces of vector-valued continuous functions
cites work: Inductive limits and partitions of unity
cites work: Q3924699
cites work: Charakterisierung lokalkonvexer Räume mit Hilfe von Tensornormtopologien
cites work: (DCF)-Räume und lokalkonvexe Tensorprodukte
cites work: Inductive limits and ε-tensor products.
cites work: Extension and lifting of continuous linear mappings in locally convex spaces
cites work: Q3918541
cites work: Lifting-Probleme für Vektorfunktionen und Tensorprodukt-Sequenzen
cites work: Induktive und projektive Limiten mit Zerlegung der Einheit
cites work: Q5569793
cites work: Q3207571
cites work: Absolutely summing operators in $ℒ_{p}$-spaces and their applications
cites work: Q4183861
cites work: Produits tensoriels d'espaces de Banach et classes d'applications linéaires
cites work: Q3951046
description: scientific article; zbMATH DE number 3921293