There are several possibilities to  generalize the relation of orthogonality from Euclidean to arbitrary normed spaces. Among the better known is Birkhoff-James orthogonality, which is defined, in one of the equivalent ways, as $x \perp y$ if  $y$ lies in the kernel of the supporting functional for $x$. This relation is homogeneous in both factors, but unlike Euclidean space it is not necessarily additive nor symmetric. We assign a (directed) graph to this  relation with  the nonzero vectors as the nodes  and where each pair of  orthogonal vectors forms a directed edge.

With the help of this graph one can show that Birkhoff-James orthogonality alone knows how to calculate the  dimension of  the underlying space, it knows whether the norm is smooth or not and whether it  is strictly convex or not, and actually knows everything about the norm of smooth reflexive spaces  up to (conjugate) linear isometry.

Among possible  applications we mention the study of homomorphisms of the relation (i.e. not necessarily linear mappings that preserve orthogonality).

This is a joint work with  Lj. Arambašić, A. Guterman, R. Rajić, and S. Zhilina
 

  We are looking forward to meeting you at FAMNIT-MP1. 

This Monday, November 8,  2021, from 10 am to 11 am.

Our Math Research Seminar will also be broadcasted via Zoom.

Join Zoom Meeting Here.

Everyone is welcome and encouraged to attend.