  This article concerns the basic understanding of parabolic final value
problems, and a large class of such problems is proved to be well posed. The
clarification is obtained via explicit Hilbert spaces that characterise the
possible data, giving existence, uniqueness and stability of the corresponding
solutions. The data space is given as the graph normed domain of an unbounded
operator occurring naturally in the theory. It induces a new compatibility
condition, which relies on the fact, shown here, that analytic semigroups
always are invertible in the class of closed operators. The general set-up is
evolution equations for Lax--Milgram operators in spaces of vector
distributions. As a main example, the final value problem of the heat equation
on a smooth open set is treated, and non-zero Dirichlet data are shown to
require a non-trivial extension of the compatibility condition by addition of
an improper Bochner integral.
