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In this article we obtain an optimal best approximation type result for fullydiscrete approximations of the transient Stokes problem. For the timediscretization we use the discontinuous Galerkin method and for the spatialdiscretization we use standard finite elements for the Stokes problemsatisfying the discrete inf-sup condition. The analysis uses the technique ofdiscrete maximal parabolic regularity. The results require only naturalassumptions on the data and do not assume any additional smoothness of thesolutions.

Fully discrete best approximation type estimates in  $L^{\infty}(I;L^2(\Omega)^d)$ for finite element discretizations of the  transient Stokes equations