Norms for the characterisation of weak*-convergence of bounded sequences
type: Item
instance of: scholarly article
title: Norms for the characterisation of weak*-convergence of bounded sequences
zbMATH Open document ID: 1032.46010
published in: Mathematische Semesterberichte
publication date: 2004-03-15T00:00:00Z
review text: Let \(X\) be a separable normed space with dual space \(X'\) and let \(\sigma=(x_k)\) be a separating sequence in the unit sphere \(S_X,\) that is, the span of \(\{x_k: k\in{\mathbb N}\}\) is dense in \(X.\) The author considers on \(X',\) and this is the main idea of the paper, the norm \(\|\;\|_\sigma\) defined by \[ \|x'\|_\sigma:= \sum_{k=1}^\infty 2^{-k}|x'(x_k)| \] and satisfying \(\|x'\|_\sigma\leq \|x'\|\) (Lemma 1) where \(\|\;\|\) is the usual norm on \(X'.\) Then a bounded sequence \((x_n')\) in \(X'\) converges in the weak*-topology if and only if it is \(\|\;\|_\sigma\)-convergent (Theorem 1). This result enables the author to give elementary proofs of the Krein-Milman theorem (Theorem 2) and some variants of it (in the language of locally convex spaces). Another consequence of it is the result of Schur (Theorem 3) that in \( \ell^1\) the norm-convergence and the weak-convergence of sequences coincide. Theorem 4, a further main result based on the idea to consider \(\|\;\|_\sigma,\) says the in case of a normed space \(X\): (a) \(X\) is separable if and only if there exist a separable normed space \(Y\) and a compact linear operator \(T:Y\to X\) with dense image. (b) If \(T\) is such an operator and \((x_n')\) is a bounded sequence in \(X',\) then \((x_n')\) converges to \(x'\) in \(X'\) if and only if \((T'x_n')\) converges to \(Tx'\) in \(Y\) (where \(T'\) is the adjoint map of \(T\)). Applying Theorem 4, the author generalizes the Riemann-Lebesgue lemma (Lemma 4).
reviewed by: Johann Boos
Mathematics Subject Classification ID: 46B03
Mathematics Subject Classification ID: 46A03
Mathematics Subject Classification ID: 46A20
zbMATH DE Number: 1757426
zbMATH Keywords: separable normed space
zbMATH Keywords: Schur lemma
zbMATH Keywords: Riemann-Lebesgue lemma
zbMATH Keywords: weak* convergence
author: Folkmar Bornemann
MaRDI profile type: Publication
full work available at URL: s591-002-8143-z
OpenAlex ID: W2075632917
DOI: 10.1007/S591-002-8143-Z
description: scientific article; zbMATH DE number 1757426