  We investigate a weak space-time formulation of the heat equation and its use
for the construction of a numerical scheme. The formulation is based on a known
weak space-time formulation, with the difference that a pointwise component of
the solution, which in other works is usually neglected, is now kept. We
investigate the role of such a component by first using it to obtain a
pointwise bound on the solution and then deploying it to construct a numerical
scheme. The scheme obtained, besides being quasi-optimal in the $L^2$ sense, is
also pointwise superconvergent in the temporal nodes. We prove a priori error
estimates and we present numerical experiments to empirically support our
findings.
