In a textbook I am reading the trace on operators of a Hilbert space is considered. Since in the book it is generally not assumed that the Hilbert spaces are separable, I wonder how the trace is even defined on a non-separable space.
One possibility would be if an operator is non-zero only on a separable subset of the entire space, and it is of trace class in a separable Hilbertspace containing this subset you could define the trace to be the trace of the restriction onto this space. That this is well defined seems an easy enough exercise.
But this seems an extremely restrictive and unimaginative definition, is there another way?