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A short note on the lineability of norm-attaining functionals in subspaces of \({\ell_{\infty}}\). (English) Zbl 1336.46013

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.

MSC:

46B04 Isometric theory of Banach spaces
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[1] Acosta M.D., Aizpuru A., Aron R.M., García-Pacheco F.J.: Functionals that do not attain their norm. Bull. Belg. Math. Soc. Simon Stevin 14, 407-418 (2007) · Zbl 1144.46013
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