Show that the following properties of an endomorphism d:V-->V of a vector space with a scalar product <.|.> are equivalent:

(i) d*=d^(-1)

(ii) <d(u)|d(v)>=<u|v> for all u,v elements of V

(iii) ||d(u)||= ||u|| for all u element of V

(iv) ||d(u) - d(v)|| = ||u-v|| for all u,v element of V

Note: If V is euclidean, a map d with the above properties is orthogonal. If V is unitary, then such a d is likewise unitary.