Equivalent norms with an extremely nonlineable set of norm attaining functionals

TL;DR

This paper constructs equivalent norms on Banach spaces where the set of norm-attaining functionals is extremely nonlineable, extending previous results and providing new geometric insights.

Contribution

It introduces a method to renorm Banach spaces so that their norm-attaining functionals form a highly nonlineable set, especially in spaces containing c0.

Findings

The constructed norms have no two-dimensional subspaces of norm-attaining functionals.

The renorming applies to Banach spaces containing c0 with a countable system of norming functionals.

The norms exhibit specific geometric properties related to the structure of norm-attaining functionals.

Abstract

We present a construction that enables one to find Banach spaces $X$ whose sets $NA(X)$ of norm attaining functionals do not contain two-dimensional subspaces and such that, consequently, $X$ does not contain proximinal subspaces of finite codimension greater than one, extending the results recently provided by Read and Rmoutil. Roughly speaking, we construct an equivalent renorming with the requested properties for every Banach space $X$ where the set $NA(X)$ for the original norm is not "too large". The construction can be applied to every Banach space containing $c_0$ and having a countable system of norming functionals, in particular, to separable Banach spaces containing $c_0$. We also provide some geometric properties of the norms we have constructed.