Nonlinear classification of Banach spaces based on geometric structure spaces

Abstract

We introduce a nonlinear equivalence of Banach spaces based on geometric structure spaces which is weaker than that based on the structure of the Birkhoff-James orthogonality. It turns out that finite-dimensional spaces are classified isomorphically under this equivalence, and that finite-dimensional spaces with the supremum norm are identified by their geometric structure spaces. Moreover, we introduce and study quasi-smooth Banach spaces that have well-behaved geometric structure spaces. As applications, we show that the usual isomorphic equivalence is (almost) equivalent to that based on geometric structure spaces, and that the classical sequence spaces have mutually different geometric structure spaces.