Suppose (E,.,.) is an inner product space and uHom(E), such that imu is non-degenerated. Then there is an orthogonal decomposition: E=imukeru. Moreover imu is non-degenerated iff keru is non-degenerated.
By exam: (imu)=u1(0)=keru and since imu is non-degenerate we conclude by lemma that E=imukeru is an orthogonal decomposition.