Falcó, Javier; García, Domingo; Maestre, Manuel; Rueda, Pilar Spaceability in norm-attaining sets. (English) Zbl 1366.46032 Banach J. Math. Anal. 11, No. 1, 90-107 (2017). 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: = 0.6 cm – 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. – For every natural number \(n\), the set of norm-attaining \(n\)-linear forms on \(c_{0}\) is lineable but not spaceable. – 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. As a consequence of these previous result they conclude the following: = 0.6 cm – 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. Reviewer: Daniel Pellegrino (João Pessoa) Cited in 4 Documents MSC: 46G25 (Spaces of) multilinear mappings, polynomials 46B04 Isometric theory of Banach spaces Keywords:norm-attaining; multilinear mappings; Arens extension; Banach space PDFBibTeX XMLCite \textit{J. Falcó} et al., Banach J. Math. Anal. 11, No. 1, 90--107 (2017; Zbl 1366.46032) Full Text: DOI Euclid