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 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.