Let be an arbitrary inner product on . Suppose is self adjointing with respect to this inner product. Prove that has real eigenvalues.
My instinct would be to do something like this: given an eigenvalue , let be a corresponding eigenvector. Then
The thing about this is that the inner product is defined on , not , and I'm having a hard time changing my thought process away from what I did above. I've been thinking maybe it should be since maybe the question would be trivial otherwise. Any thoughts?