Past exam of the theoretical physics II course of the University of Cambridge / 2012 / Paper 1 / 1 / c

In a finite-dimensional basis, interchange the two finite sums:This cyclic property of the trace does not assert $AB=BA$ or permit arbitrary reordering of three factors. For infinite-dimensional linear operators, the traces and rearrangements need hypotheses, for example one trace-class operator and one bounded operator; an arbitrary pair of unbounded linear operators is not covered.