Norm-attaining functionals and proximinal subspaces
Functional Analysis
2015-03-23 v1
Abstract
G. Godefroy asked whether, on any Banach space, the set of norm-attaining functionals contains a 2-dimensional linear subspace. We prove that a recent construction due to C.J. Read provides an example of a space which does not have this property. This is done through a study of the relation between the following two sentences where X is a Banach space and Y is a closed subspace of finite codimension in X: (A) Y is proximinal in X. (B) The annihilator of Y consists of norm-attaining functionals. We prove that these are equivalent if X is the Read's space. Moreover, we prove that any non-reflexive Banach space X with any given closed subspace Y of finite codimension at least 2 admits an equivalent norm such that (B) is true and (A) is false.
Cite
@article{arxiv.1503.06112,
title = {Norm-attaining functionals and proximinal subspaces},
author = {Martin Rmoutil},
journal= {arXiv preprint arXiv:1503.06112},
year = {2015}
}