Restoration of the norm and order in finite-dimensional spaces \(X\) and \(Y\) from the space \(L(X, Y)\) of operators (Q1905317)

The paper contains a parallel discussion of two problems: from the norm in the finite-dimensional space \(L(X_0, Y_0)\) of linear operators when can we restore uniquely, up to permutation of spaces and taking the conjugate, the norm in the initial spaces \(X_0\) and \(Y_0\) and also a similar problem concerning order rather than norm. In spite of the externally large distance between these problems, one can solve them simultaneously with the aid of the same method, based on the investigation of the analytic properties of the set of extreme points of the unit ball in the first case and the set of extreme rays of the positive elements in the second case. A similar technique has been already used by the author and also by Bosznay and Garay for solving some problems of the geometry of normed spaces.