Birkhoff–James classification of norm’s properties

Abstract For an arbitrary normed space X X over a field F \ {R, C\}, F ∈ R, C, we define the directed graph (X) Γ (X) induced by Birkhoff–James orthogonality on the projective space P (X), P (X), and also its nonprojective counterpart ₀ (X). Γ 0 (X). We show that, in finite-dimensional normed spaces, (X) Γ (X) carries all the information about the dimension, smooth points, and norm’s maximal faces. It also allows to determine whether the norm is a supremum norm or not, and thus classifies finite-dimensional abelian C^* C ∗ -algebras among other normed spaces. We further establish the necessary and sufficient conditions under which the graph ₀ ({R}) Γ 0 (R) of a (real or complex) Radon plane {R} R is isomorphic to the graph ₀ (F², ₂) Γ 0 (F 2, ‖ · ‖ 2) of the two-dimensional Hilbert space and construct examples of such nonsmooth Radon planes.