Let V be a finite dimensional complex inner product space with inner product < , >. Let U be unitary with respect to this inner product. If ( , ) is another inner product, is U also unitary with respect to ( , )? The definition of unitary I'm working with is the one that says: U is unitary if <Uv, Uw> = <v w>, i.e. it preserves inner products. Now it is easy to show that U is unitary with respect to < , > if and only if U'U = 1, where U' is the adjoint and 1 is the identity transformations. But by replacing < , > with ( , ), the prior statement says that U is also unitary with respect to ( , ). Am I missing something?