Birkhoff–James classification of norm’s properties

Abstract

AbstractFor an arbitrary normed space $$\mathcal {X}$$ X over a field $$\mathbb {F}\in \{ \mathbb {R}, \mathbb {C}\},$$ F ∈ { R , C } , we define the directed graph $$\Gamma (\mathcal {X})$$ Γ ( X ) induced by Birkhoff–James orthogonality on the projective space $$\mathbb P(\mathcal {X}),$$ P ( X ) , and also its nonprojective counterpart $$\Gamma _0(\mathcal {X}).$$ Γ 0 ( X ) . We show that, in finite-dimensional normed spaces, $$\Gamma (\mathcal {X})$$ 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 $$\Gamma _0({\mathcal {R}})$$ Γ 0 ( R ) of a (real or complex) Radon plane $${\mathcal {R}}$$ R is isomorphic to the graph $$\Gamma _0(\mathbb {F}^2, {\Vert \cdot \Vert }_2)$$ Γ 0 ( F 2 , ‖ · ‖ 2 ) of the two-dimensional Hilbert space and construct examples of such nonsmooth Radon planes.