Symmetric sequence spaces, bases, and applications
type: Item
instance of: scholarly article
title: Symmetric sequence spaces, bases, and applications
zbMATH Open document ID: 0607.46005
DOI: 10.1016/0022-247X(86)90345-8
author: P. K. Kamthan
author: M. Gupta
author: William H. Ruckle
published in: Journal of Mathematical Analysis and Applications
publication date: 1986-01-01T00:00:00Z
review text: The purpose of this paper is three-fold, namely (i) to prove results on symmetric sequence spaces which are not normal, (ii) relate the symmetric character of a Schauder base (S.b.) in a locally convex space (l.c. TVS) with the symmetry of the associated sequence space (a.s.s.); and (iii) look for the applications of these results. For instance, we prove Theorem: If \(\lambda\) is a symmetric sequence space such that \(\phi \subset \lambda \subset c_ 0\), then \(\lambda^ x=\lambda^{\beta}.\) Theorem: An S.b. for a barrelled space is symmetric if and only if the a.s.s. is symmetric. Theorem: Let \(\{x_ n;f_ n\}\) be an s-symmetric s.b. for an l.c. TVS (X,T) such that \(f_ n(x_ 0)\not\in \ell^{\infty}\) for some \(x_ 0\) in x. Then (X,T) is metrizable and if in addition, (X,T) is also sequentially complete, then (X,T) is a Fréchet space and \(\{x_ n\}\) is bounded in (X,T). Besides, there are many other results related to (i), (ii), (iii); and counter-examples in support of assumption of various hypotheses.
Mathematics Subject Classification ID: 46A45
Mathematics Subject Classification ID: 46A35
Mathematics Subject Classification ID: 46A04
zbMATH DE Number: 3981803
zbMATH Keywords: symmetric sequence spaces which are not normal
zbMATH Keywords: Schauder base
zbMATH Keywords: barrelled space
zbMATH Keywords: Fréchet space
MaRDI profile type: Publication
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full work available at URL: 0022-247x(86)90345-8
OpenAlex ID: W2050709246
description: scientific article; zbMATH DE number 3981803