  We consider fully discrete time-space approximations of abstract linear
parabolic partial differential equations (PDEs) consisting of an $hp$-version
discontinuous Galerkin (DG) time stepping scheme in conjunction with standard
(conforming) Galerkin discretizations in space. We derive abstract computable a
posteriori error bounds resulting, for instance, in concrete bounds in
$L_{\infty}(I;L_2(\Omega))$- and $L_{2}(I;H^{1}(\Omega))$-type norms when $I$
is the temporal and $\Omega$ the spatial domain for the PDE. We base our
methodology for the analysis on a novel space-time reconstruction approach. Our
approach is flexible as it works for any type of elliptic error estimator and
leaves their choice of up to the user. It also allows exhibits mesh-change
estimators in a clear an concise way. We also show how our approach allows the
derivation of such bounds in the $H^1(I;H^{-1}(\Omega))$ norm.
