by the generalized Schwarz inequality [2, p. 262]. Now (2) follows immediately from (1) and (3), using (U*y, U*y) = (y, y). If aC is finite dimensional, the bound is attained for x U*y =u-+v, where u and v are unit eigenvectors of A corresponding to eigenvalues m and M. In the general case, the bound need not be attained; but a sequence xn= U*yn=un+vn, where !1un|1=!fVnfl, (Em+i/n Em-o) n = Un, (EM+o EM-ln)Vn = Vn ({ Ex } being the spectral resolution for A) shows on calculation that the bound is best possible.