The form we prefers says that every bounded self-adjoint operator is a multiplication operator. This means that given a bounded self-adjoint operator on a Hilbert space , we can always find a measure on a measure space and a unitary operator so that
for some bounded real-valued measurable function on .
In the case of real, the Hilbert space becomes a Euclidean space with an inner product. So an operator in corresponds with an orthogonal matrix . And an orthogonal matrix can be diagonalized, i.e. there exists and such that