  In this paper we investigate the $\mathrm{L}^\infty$-stability of fully
discrete approximations of abstract linear parabolic partial differential
equations. The method under consideration is based on an $hp$-type
discontinuous Galerkin time stepping scheme in combination with general
conforming Galerkin discretizations in space. Our main result shows that the
global-in-time maximum norm of the discrete solution is bounded by the data of
the PDE, with a constant that is robust with respect to the discretization
parameters (in particular, it is uniformly bounded with respect to the local
time steps and approximation orders).
