Sets of weak sequential continuity for polynomials
type: Item
instance of: scholarly article
title: Sets of weak sequential continuity for polynomials
zbMATH Open document ID: 1030.46055
DOI: 10.1016/S0019-3577(02)80012-X
published in: Indagationes Mathematicae. New Series
publication date: 2004-02-15T00:00:00Z
review text: Let \(P({}^NE)\) be the space of all \(N\)-homogeneous continuous polynomials on the Banach space \(E\). The subspace of all polynomials which are weakly sequentially continuous on \(E\) is denoted by \(P_{wsc}({}^NE)\). The authors deal with properties of the set \(C_P = \{x \in E : P\) is weakly sequentially continuous at \( x \}\), where \(P \in P({}^NE)\). In the first section several basic properties of \(C_P\) are given, the formula \[ C_{P \cdot Q} = (C_P \cap C_Q) \cup (C_P \cap P^{-1}(0)) \cup (C_Q \cap Q^{-1}(0)) \] for all \(P \in P({}^NE)\) and \(Q \in P({}^ME)\) is proved, and partial answers to the following questions are given. Question 1. Given \(P \in P({}^NE)\), does there always exist \(Q \in P({}^{N+1}E)\) such that \(C_P=C_Q\)? Question 2. Given \(P,Q \in P({}^NE)\), does there exist \(R \in P({}^{M}E)\) for some \(M\) such that \(C_R=C_P \cap C_Q\)? Question 3. Given \(P,Q \in P({}^NE)\), does there exist \(R \in P({}^{M}E)\) for some \(M\) such that \(C_R=C_P \cup C_Q\)? In the second section the authors study the set \(C_P\) for separable Banach spaces and Banach spaces with unconditional finite dimensional decompositions (FDD). In this cases the results are sharper and the answers to the above questions are more complete.
Mathematics Subject Classification ID: 46G25
zbMATH DE Number: 1893647
zbMATH Keywords: weakly sequentially continuous polynomials
zbMATH Keywords: unconditional finite dimensional decomposition (FDD)
author: Richard M. Aron
author: Verónica Dimant
reviewed by: Hans-Andreas Braunß
MaRDI profile type: Publication
cites work: Bounded weak continuity of homogeneous polynomials at the origin
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full work available at URL: s0019-3577(02)80012-x
OpenAlex ID: W2009849521
description: scientific article; zbMATH DE number 1893647