  This paper focuses on parabolic final value problems, and well-posedness is
proved for a large class of these. The clarification is obtained from Hilbert
spaces that characterise data that give existence, uniqueness and stability of
the solutions. The data space is the graph normed domain of an unbounded
operator that maps final states to the corresponding initial states. It induces
a new compatibility condition, depending crucially on the fact that analytic
semigroups always are invertible in the class of closed operators. Lax--Milgram
operators in vector distribution spaces constitute the main framework. The
final value heat conduction problem on a smooth open set is also proved to be
well posed, and non-zero Dirichlet data are shown to require an extended
compatibility condition obtained by adding an improper Bochner integral.
