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

TL;DR

This paper constructs a Banach space with a highly restricted set of norm-attaining functionals, showing that no non-trivial cones are contained within, and explores implications for norm-attaining operators.

Contribution

It introduces a Banach space where the set of norm-attaining functionals is algebraically trivial, providing new insights into the structure of such spaces.

Findings

The set of norm-attaining functionals contains no non-trivial cone.

Between two linearly independent norm-attaining functionals, no other segment attains its norm.

At most four elements in the unit sphere of certain quotient spaces attain norm-one.

Abstract

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