Birkhoff-James orthogonality in normed vector spaces

Abstract: Birkhoff-James orthogonality generalizes the classical orthogonality from Hilbert to general normed spaces. It can be a useful tool for finding the best approximation of a vector within a given subspace. However, unlike the classical one, it is in general not symmetric. In fact (in dimensions greater than 2) it is symmetric if and only if the norm is induced by the inner product. This early classification of inner-product spaces goes back to James (for real Banach spaces) with an aid of Bohnenblust (for complex ones). One can visualize this relation as an infinite directed graph (called ortho-digraph), where vertices are all the vectors from a normed space (or all points in its projectivisation) and two vertices x, y form a directed edge if x is Birkhoff-James orthogonal to y .We will show that this digraph contains a lot of information about the normed space: It knows how to calculate the dimension of the underlying space, knows if the norm is rotund or smooth, knows how to find smooth points and in some special cases even knows if the underlying field is real or complex. At least for smooth spaces in can even completely characterize them, modulo (conjugate) linear isometry.

geometric topologynumber theoryoperator algebrasrepresentation theory

Audience: researchers in the topic

( slides | video )

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Noncommutative geometry in NYC

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