  We propose a least-squares formulation for parabolic equations in the natural
$L^2(0,T;V^*)\times H$ norm which avoids regularity assumptions on the data of
the problem. For the abstract heat equation the resulting bilinear form then is
symmetric, continuous, and coercive. This among other things paves the ground
for classical space-time a priori and a posteriori Galerkin frameworks for the
numerical approximation of the solution of the abstract heat equation.
Moreover, the approach is applicable in e.g. optimal control problems with
(parametrized) parabolic equations, and for certification of reduced basis
methods with parabolic equations.
