Bivariational bounds

Complementary bivariational bounds are derived on the quantity $\langle $$\phi $, $g$$\rangle $ associated with the linear equation $A$$\phi $ = $f$ in a Hilbert space, where the operator $A$ is self-adjoint. The vector $g$ is arbitrary, and variational bounds on $\langle $$\phi $, $f$$\rangle $ are taken as the starting-point. Possible applications, including point-wise bounds on $\phi $ are briefly discussed.