Orthogonal sequences in non-Archimedean locally convex spaces
type: Item
instance of: scholarly article
title: Orthogonal sequences in non-Archimedean locally convex spaces
zbMATH Open document ID: 0980.46056
DOI: 10.1016/S0019-3577(00)89076-X
author: N. de Grande-de Kimpe
author: Jerzy Kąkol
author: Cristina Pérez-García
author: Wim Schikhof
published in: Indagationes Mathematicae. New Series
publication date: 2001-06-28T00:00:00Z
reviewed by: Wim Schikhof
Mathematics Subject Classification ID: 46S10
Mathematics Subject Classification ID: 46A35
Mathematics Subject Classification ID: 46A04
zbMATH DE Number: 1611979
zbMATH Keywords: orthogonal sequence
zbMATH Keywords: Schauder base
zbMATH Keywords: basic sequence
zbMATH Keywords: non-Archimedean valued complete field
zbMATH Keywords: characterization of compactoids
MaRDI profile type: Publication
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review text: Let \(E\) be a Hausdorff locally convex space over a non-Archimedean valued complete field \(K\). A sequence \(x_1,x_2,\dots\) in \(E\) is called `orthogonal' (De Grande-De Kimpe, 1972) if there is a base \({\mathcal P}\) of continuous seminorms such that for all \(n\in\mathbb{N}\), all \(\lambda_1,\dots,\lambda_n\in K\), all \(p\in {\mathcal P}\), \(p(\sum^n_{i=1}\lambda_i x_i)= \max_i p(\lambda_ix_i)\).NEWLINENEWLINENEWLINEThe main result of the paper is the following characterization of compactoids. A bounded subset \(X\) of \(E\) is a compactoid iff each `orthogonal' sequence in \(X\) tends to \(0\). Next, this fact is being to find partial answers to the following questions.NEWLINENEWLINENEWLINE1. If \(E\) is metrizable, does it have an `orthogonal' basic sequence?NEWLINENEWLINENEWLINE2. If \(E\) is Fréchet, of countable type, does it have an `orthogonal' base?NEWLINENEWLINENEWLINENote: Recently, \textit{W. Sliwa} managed to provide full answers [`yes' on question 1, see Indeg. Math., New Ser. 11, No. 3, 463-466 (2000; review below); `no' in question 2, see ``Examples of non-Archimedean nuclear Fréchet spaces method a Schauder basis'', to appear in Indeg. Math., New Ser.].
description: scientific article; zbMATH DE number 1611979