2015/03/20 by Martin Rmoutil, Rmoutil, Martin
Mathematics · #46B03 #46B10 #46B20 #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA)
paper · pdf · doi:10.48550/arxiv.1503.06112
openalex publication_date 2015/03/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.