  In this paper we establish best approximation type estimates for the fully
discrete Galerkin solutions of transient Stokes problem in
$L^2(I;L^2(\Omega)^d)$ and $L^2(I;H^1(\Omega)^d)$ norms. These estimates fill
the gap in the error analysis of the transient Stokes problems and have a
number of applications. The analysis naturally extends to inhomogeneous
parabolic problems. The best type $L^2(I;H^1(\Omega))$ error estimates seems to
be new even for scalar parabolic problems.
