The instationary Stokes and Navier-Stokes equations are considered in a simultaneously space-time variational saddle point formulation, so involving both velocities ${\bf u}$ and pressure $p$. For the instationary Stokes problem, it is shown that the corresponding operator is a boundedly invertible linear mapping between $H_1$ and $H_2'$, both Hilbert spaces $H_1$ and $H_2$ being Cartesian products of (intersections of) Bochner spaces, ...