Spaceability in norm-attaining sets
type: Item
instance of: scholarly article
title: Spaceability in norm-attaining sets
zbMATH Open document ID: 1366.46032
DOI: 10.1215/17358787-3750182
author: Javier Falcó
author: Domingo García
author: Manuel Maestre
author: Pilar Rueda
published in: Banach Journal of Mathematical Analysis
publication date: 2016-12-21T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.bjma/1478746988
review text: The authors prove several interesting results concerning lineability and spaceability in norm-attaining sets. Recall that a subset \(A\) of a topological vector space \(E\) is called lineable if there is an infinite-dimensional subspace \(V\) of \(E\) such that \[ V\subset A \cup \{0\}. \] If \(V\) is also closed, then \(A\) is spaceable. Among other nice results, the authors prove the following: {\parindent = 0.6 cm \begin{itemize}\item[--] There exist two Banach spaces \(X\) and \(Y\) such that the set of compact linear operators from \(X\) to \(Y\) which cannot be approximated by norm attaining operators is spaceable. \item[--] For every natural number \(n\), the set of norm-attaining \(n\)-linear forms on \(c_{0}\) is lineable but not spaceable. \item[--] If \(X\) and \(Y\) are Banach spaces and \(n\) is a natural number, then the transpose of every compact \(n\)-homogeneous polynomial from \(X\) to \(Y\) is norm-attaining. \end{itemize}} As a consequence of these previous result they conclude the following: {\parindent = 0.6 cm \begin{itemize}\item[--] Given two infinite-dimensional Banach spaces \(X\) and \(Y\), the set of all \(n\)-homogeneous continuous polynomials from \(X\) to\(\;Y\) whose transpose is norm-attaining is spaceable. \end{itemize}}
reviewed by: Daniel Pellegrino
Mathematics Subject Classification ID: 46G25
Mathematics Subject Classification ID: 46B04
zbMATH DE Number: 6667689
zbMATH Keywords: norm-attaining
zbMATH Keywords: multilinear mappings
zbMATH Keywords: Arens extension
zbMATH Keywords: Banach space
MaRDI profile type: Publication
OpenAlex ID: W2552796637
Recommended article: The attainable spaces and its analogues
Recommended article: Spaceability of sets of \(p\)-compact maps
Recommended article: Q5866278
Recommended article: Spaceability on some classes of Banach spaces
Recommended article: On the completeness of normed spaces
Recommended article: On \(n\)-normed spaces
Recommended article: Spaceability of subsets of the disc algebra
Recommended article: Spaceability in Banach and quasi-Banach sequence spaces
Recommended article: Q2963564
Recommended article: Norming sets and compactness
description: scientific article; zbMATH DE number 6667689