On Cauchy-Buniakowski-Schwarz’s inequality for sequences of linear operators in Hilbert space

Let (X; (.,.)) be a real or complex Hilbert space and B(X) the Banach algebra of bounded linear operators which map X into X. We recall that a self-adjoint operator A E B(X) is positive iff (Ax, x) • 0 for all x E X. The binary relation A> • Biff A-B is a positive self-adjoin