Nonlinear equivalence of Banach spaces based on Birkhoff-James orthogonality, II

Abstract

We introduce the notion of geometric structure spaces of Banach spaces equipped with closure space structure. It is shown that the nonlinear equivalence of Banach spaces based on Birkhoff-James orthogonality induces homeomorphisms between geometric structure spaces. As applications, we prove that spaces of continuous functions are isomorphic with respect to the structure of Birkhoff-James orthogonality if and only if they are isometrically isomorphic, and that classical sequence spaces c0 and ℓp are completely classified from the viewpoint of Birkhoff-James orthogonality.