Polynomial sequential continuity on \(C(K,E)\) spaces
type: Item
instance of: scholarly article
title: Polynomial sequential continuity on \(C(K,E)\) spaces
zbMATH Open document ID: 1031.46050
DOI: 10.1016/S0022-247X(03)00161-6
author: Ignacio Villanueva
author: Fernando Bombal
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-07-30T00:00:00Z
review text: Let \(E\) be a Banach space. A sequence \((x^n) \subseteq E\) is said to be \(\tau_{\leq N}\) convergent to \(x \in E\) if \(P(x^n)\) tends to \(P(x)\) for all \(P \in P(^kE)\) and \(k \leq N\). The authors consider properties like the following one. The space \(E\) has property \(P_{N,M}\) if \(T(x_1^n,\dots, x_M^n)\) tends to \(T(x_1,\dots,x_M)\) for all \(T \in L_s(^ME)\) and all sequences \((x_j^n)\) with \(\tau_{\leq N}\)-\(\lim x_j^n=x_j\), \(j=1,\dots,M\), where \(N\in\mathbb{N}\cup\{\infty\}\), \(M\in\mathbb{N}\). In this paper several relations among the properties \(P_{N,M}\), \(SP_{N,M}\), \(Q_{N,M}\), and \(SQ_{N,M}\) and relations to the polynomial Schur property and polynomial Dunford-Pettis property are given. Roughly speaking, the main results show that a space \(E\) has one of the above properties iff every (or some) \(C(K,E)\) has this property too, where \(K\) is a scattered compact Hausdorff space. In this way polynomial sequential convergence is connected with pointwise polynomial sequential convergence but the results are different from those obtained in the case of weak sequential convergence.
Mathematics Subject Classification ID: 46G25
zbMATH DE Number: 1956921
zbMATH Keywords: polynomial sequential continuity
zbMATH Keywords: \(C(K,E)\) spaces
Wikidata QID: Q62471004
reviewed by: Hans-Andreas Braunß
MaRDI profile type: Publication
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full work available at URL: s0022-247x(03)00161-6
OpenAlex ID: W1995879545
description: scientific article; zbMATH DE number 1956921