Computing the Distance Between Frames and Between Subspaces of a Hilbert Space

Nontrivial interband effect: applications to magnetic susceptibility, nonlinear optics, and topological degeneracy pressure

“…Mathematically, the difference between the subspaces is reduced to that between the projection operators onto the subspaces. By using the projection operators, one can define the Hilbert-Schmidt inner product and the chordal distance between subspaces [27]:…”