Denseness of norm-attaining mappings on Banach spaces
type: Item
instance of: scholarly article
title: Denseness of norm-attaining mappings on Banach spaces
zbMATH Open document ID: 1194.46010
DOI: 10.2977/PRIMS/4
author: Yun Sung Choi
author: Han Ju Lee
author: Hyun Gwi Song
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 2010-04-08T00:00:00Z
full work available at URL: http://www.ems-ph.org/journals/all_issues.php
review text: Summary: Let \(X\) and \(Y\) be Banach spaces. Let \(P(^nX \colon Y)\) be the space of all \(Y\)-valued continuous \(n\)-homogeneous polynomials on \(X\). We show that the set of all norm-attaining elements is dense in \(P(^nX \colon Y)\) when the set of u.s.e.\ points of the unit ball \(B_{X}\) is dense in the unit sphere \(S_{X}\). Applying strong peak points instead of u.s.e.\ points, we generalize this result to a closed subspace of \(C_b(M, Y)\), where \(M\) is a complete metric space. For complex Banach spaces \(X\) and \(Y\), let \(A_{b}(B_{X} \colon Y)\) be the Banach space of all bounded continuous \(Y\)-valued mappings \(f\) on \(B_{X}\) whose restrictions \(f|_{B_{X}^{\circ}}\) to the open unit ball are holomorphic. It follows that the set of all norm-attaining elements is dense in \(A_{b}(B_{X} \colon Y)\) if the set of all strong peak points in \(A_{b}(B_{X})\) is a norming subset for \(A_{b}(B_{X})\).
Mathematics Subject Classification ID: 46B04
Mathematics Subject Classification ID: 46G20
Mathematics Subject Classification ID: 46G25
Mathematics Subject Classification ID: 46B22
zbMATH DE Number: 5690820
zbMATH Keywords: homogeneous polynomial
zbMATH Keywords: norm-attaining element
zbMATH Keywords: uniformly strongly exposed point
zbMATH Keywords: strong peak point
Wikidata QID: Q57618282
MaRDI profile type: Publication
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OpenAlex ID: W2091443709
description: scientific article; zbMATH DE number 5690820