A Banach space whose set of norm-attaining functionals is algebraically trivial

Abstract

We construct a Banach space $X$ for which the set of norm-attaining functionals $\mathrm{NA}(X,R)$ 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 $\mathrm{NA}(X,R)$ 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 norm-attaining operators.