Remarks on independent sequences and dimension in topological linear spaces
type: Item
instance of: scholarly article
title: Remarks on independent sequences and dimension in topological linear spaces
zbMATH Open document ID: 0926.46001
published in: Collectanea Mathematica
publication date: 1998-09-10T00:00:00Z
full work available at URL: https://eudml.org/doc/41117
review text: Most of the paper has its origin in the theorem that a linearly independent sequence in a Banach space has an \(\omega\)-independent subsequence, which is due essentially to P. Erdős and E. G. Straus (1953). This theorem, clearly, extends to topological linear spaces that admit a continuous norm. In the class of locally convex \(F\)-spaces a converse holds, as shown by V. M. Kadets (1993). We generalize this equivalence to the class of metrizable locally convex spaces. In this connection, we establish some properties of the space \(\varphi\) of scalar sequences with finite support equipped with the product topology. In particular, we prove some results on the containment of \(\varphi\) in topological linear spaces, generalizing the corresponding results on the containment of \(\omega\) \((=\mathbb{R}^{\mathbb{N}})\) in \(F\)-spaces due to C. Bessaga, A. Pełczyński and S. Rolewicz (1957, 1959). Moreover, we strengthen the Erdős-Straus theorem to the effect that a linearly independent sequence in a dual Banach space has a subsequence which is \(\omega\)-independent with respect to the weak* topology. (The author has recently noticed that if the predual space is separable, then this last result reduces to the Erdős-Straus theorem via a standard argument; see, e.g. \textit{M. A. Rieffel} [Trans. Am. Math. Soc. 131, 466-487 (1968; Zbl 0169.46803), Corollary 5.4]). We also present two different proofs of the theorem that the algebraic dimension of a closed convex subset of an \(F\)-space is either finite or at least \(2^{\aleph_0}\).
Mathematics Subject Classification ID: 46A03
Mathematics Subject Classification ID: 46A16
Mathematics Subject Classification ID: 46B99
Mathematics Subject Classification ID: 52A07
Mathematics Subject Classification ID: 46A04
zbMATH DE Number: 1186311
zbMATH Keywords: topological linear space
zbMATH Keywords: \(F\)-space
zbMATH Keywords: weak* topology
zbMATH Keywords: \(\omega\)-independence
zbMATH Keywords: subsequence
zbMATH Keywords: algebraic dimension
zbMATH Keywords: closed convex set
zbMATH Keywords: metrizable locally convex spaces
zbMATH Keywords: Erdős-Straus theorem
zbMATH Keywords: linearly independent sequence
zbMATH Keywords: dual Banach space
author: Zbigniew Lipecki
reviewed by: Zbigniew Lipecki
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1186311