Could you please help me in solving the following problem:
Let be a bounded linear operator on a complex inner product space . If for all , show that .
Moreover, this is does not hold in the case of real inner product space.
Note that this is an intrinsically complex relation. In the real case the result fails. For example, in two-dimensional space a rotation through a right angle takes every vector to an orthogonal vector.
Edit. Sorry, didn't see drexel28's comment.