Weak property \((Y_0)\) and regularity of inductive limits
type: Item
instance of: scholarly article
title: Weak property \((Y_0)\) and regularity of inductive limits
zbMATH Open document ID: 0968.46002
published in: Journal of Mathematical Analysis and Applications
publication date: 2000-11-27T00:00:00Z
review text: Let \((E_n, t_n)_{n\in\mathbb{N}}\) be an inductive sequence of locally convex spaces, let \((E,t)\) be its inductive limit and assume that \((E,t)\) is Hausdorff. Then \((E,t)\) is said (a) to be regular, if each bounded set in \((E,t)\) is already contained and bounded in some space \((E_m, t_m)\), (b) to satisfy the weak property \((Y_0)\) if for each series \(\sum^\infty_{k=1} x_k\) in \((E,t)\) which is weakly unconditionally Cauchy (w.u.c.) there is \(m\in\mathbb{N}\) such that \(\sum^\infty_{k=1} x_k\) is w.u.c. in \((E_m, t_m)\). The main result of the paper is that \((E, t)\) is regular if and only if \((E, t)\) has the weak property \((Y_0)\). If all the spaces \((E_n, t_n)_{n\in\mathbb{N}}\) are sequentially complete then these properties are also equivalent to the property that for each w.u.c. series \(\sum^\infty_{k=1} x_k\) in \((E,t)\) there is \(m\in\mathbb{N}\) such that for each null-sequence \((\xi_k)_{k\in\mathbb{N}}\) of scalars, the sequence \(\sum^\infty_{k=1} \xi_kx_k\) converges in \((E_m,t_m)\). Further equivalences are derived under additional hypotheses like: No space \((E_n, t_n)\) contains a copy of \(c_0\) or all \((E_n,t_n)\) are strictly webbed spaces.
reviewed by: Reinhold Meise
Mathematics Subject Classification ID: 46A13
Mathematics Subject Classification ID: 46A30
zbMATH DE Number: 1496299
zbMATH Keywords: regularity properties
zbMATH Keywords: inductive limit
zbMATH Keywords: weak property \((Y_0)\)
zbMATH Keywords: strictly webbed spaces
author: Jinghui Qiu
MaRDI profile type: Publication
full work available at URL: jmaa.2000.6793
OpenAlex ID: W1985476965
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DOI: 10.1006/JMAA.2000.6793
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description: scientific article; zbMATH DE number 1496299