  The space $D'_\Lambda$ of distributions having their $C^\infty$ wavefront set
in a cone $\Lambda$ has become important in physics because of its role in the
formulation of quantum field theory in curved spacetime. It is also a basic
object in microlocal analysis, but not well studied from a functional analytic
viewpoint. In order to compute its completion in the open cone case, we
introduce generalized spaces $D'_{\gamma,\Lambda}$ where we also control the
union of H^s-wave front sets in a second cone $\gamma$ contained in $\Lambda$.
We can compute bornological and topological duals, completions and
bornologifications of natural topologies for spaces in this class. All our
topologies are nuclear, ultrabornological when bornological and we can describe
when they are quasi-LB. We also give concrete microlocal representations of
bounded and equicontinuous sets in those spaces and work with general support
conditions including future compact or space compact support conditions on
globally hyperbolic manifolds, as motivated by physics applications to be
developed in a second paper.
