Proof. We use induction on n. For n = 1, the result is immediate, First, observe there exist linear transformations Ui (i = 1, 2): X + X such that (U,x, y) = &(x, y). For define&: X -R by Liz(y) = #Q(x, y). Then L,, is a linear functional and since X is an inner product space, there exists Z< E X with L,,(y) = (zi, y) by the canonical isomorphism between X and its dual. Define the transformatio...