A Banach space whose set of norm-attaining functionals is algebraically trivial Martín Suárez, Miguel The author has been supported by MICIU/AEI/10.13039/501100011033 and ERDF/EU through the grant PID2021-122126NB-C31, and by “Maria de Maeztu” Excellence Unit IMAG, reference CEX2020-001105-M funded by MICIU/AEI/10.13039/501100011033. We construct a Banach space X for which the set of norm-attaining functionals NA(X, R) does not contain any non-trivial cone. Even more, given two linearly independent norm-attaining functionals on X, no other element of the segment between them attains its norm. Equivalently, the intersection of NA(X, R) with a two-dimensional subspace of X∗ is contained in the union of two lines. In terms of proximinality, we show that for every closed subspace M of X of codimension two, at most four elements of the unit sphere of X/M have a representative of norm-one. We further relate this example with an open problem on normattaining operators. 2025-02-17T13:33:52Z 2025-02-17T13:33:52Z 2024 journal article Published version: M. Martín / Journal of Functional Analysis 288 (2025) 110815. https://doi.org/10.1016/j.jfa.2024.110815 https://hdl.handle.net/10481/102416 10.1016/j.jfa.2024.110815 eng http://creativecommons.org/licenses/by-nc-nd/4.0/ open access Attribution-NonCommercial-NoDerivatives 4.0 Internacional Elsevier