Equivalent norms with an extremely nonlineable set of norm attaining functionals

Abstract

We present a construction that enables one to nd 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 nite codimension greater than one, extending the results recently provided by Read [27] and Rmoutil [28]. 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 c0 and having a countable system of norming functionals, in particular, to separable Banach spaces containing c0. We also provide some geometric properties of the norms we have constructed.