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In this paper we establish best approximation type estimates for the fullydiscrete 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 fillthe gap in the error analysis of the transient Stokes problems and have anumber of applications. The analysis naturally extends to inhomogeneousparabolic problems. The best type $L^2(I;H^1(\Omega))$ error estimates seems tobe new even for scalar parabolic problems.

$L^2(I;H^1(\Omega))$ and $L^2(I;L^2(\Omega))$ best approximation type error estimates for Galerkin solutions of transient Stokes problems