A short note on the lineability of norm-attaining functionals in subspaces of \({\ell_{\infty}}\) (Q895806)

Let \(E\) be a Banach space. Recall that a continuous (linear) functional \( f:E\rightarrow \mathbb{K}\) is norm attaining when there exists a vector \(x\) in \(E\) such that \(\left\| x\right\| =1\) and \(f(x)=\left\| f\right\| .\) Denote the set of norm attaining functionals on \(E\) by \(NA(E).\) In this paper, the authors show that there are closed infinite-dimensional subspaces \(E\) of \(l_{\infty }\) such that \(NA(E)\) is not lineable. On the other hand, they show that for any closed subspace \(E\) of \( l_{\infty }\) containing \(c_{0},\) the set \(NA(E)\) is lineable.