Algebraic properties implying weak sequential completeness
type: Item
instance of: scholarly article
zbMATH Open document ID: 0987.46007
publication date: 2002-06-27T00:00:00Z
Mathematics Subject Classification ID: 46A45
Mathematics Subject Classification ID: 46A04
Mathematics Subject Classification ID: 46A20
zbMATH DE Number: 1619485
zbMATH Keywords: \(\alpha\) and \(\beta\)-duality
zbMATH Keywords: FK-space of real-vlued sequences
zbMATH Keywords: weak topology
zbMATH Keywords: sequentially complete in the weak topology
author: Charles Swartz
MaRDI profile type: Publication
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title: Algebraic properties implying weak sequential completeness
review text: Let \(F\) be an FK-space of real-valued sequences and \(W_F\) be the set of all \(t= (t_k)\in F\) for which \(t= \sum^\infty_{k=1} t_k e^{(k)}\) converges in the weak topology \(\sigma(F, F')\) of \(F\), where, as usual, \(e^{(k)}= (0,\dots,0,1,0,\dots)\) with \(1\) in the \(k\)th place and \(0\) everywhere else.NEWLINENEWLINENEWLINEIt has been established [Bennet-Kalton (1972), Ruckle (1979)] that if \(c_0\subseteq F\) then \(\ell^1\) is sequentially complete in the weak topology \(\sigma(\ell^1, W_F\cap\ell^\infty)\). The sequential completeness of certain sequence spaces under various topologies can be derived from certain algebraic properties of the sequence spaces. It seems therefore natural to ask whether the sequential completeness in the weak topology \(\sigma(\ell^1, W_F\cap \ell^\infty)\) could be guaranteed by some algebraic property of \(W_F\).NEWLINENEWLINENEWLINEIn this paper the author isolates such a property satisfied by \(W_F\cap \ell^\infty\) and gives a generalization of the Bennet-Kalton-Ruckle result.NEWLINENEWLINEFor the entire collection see [Zbl 0960.00019].
reviewed by: Constantine G. Lascarides
description: scientific article; zbMATH DE number 1619485