Spaceability in norm-attaining sets
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}}
scientific article; zbMATH DE number 6667689