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The space $D'_\Lambda$ of distributions having their $C^\infty$ wavefront setin a cone $\Lambda$ has become important in physics because of its role in theformulation of quantum field theory in curved spacetime. It is also a basicobject in microlocal analysis, but not well studied from a functional analyticviewpoint. In order to compute its completion in the open cone case, weintroduce generalized spaces $D'_{\gamma,\Lambda}$ where we also control theunion of H^s-wave front sets in a second cone $\gamma$ contained in $\Lambda$.We can compute bornological and topological duals, completions andbornologifications of natural topologies for spaces in this class. All ourtopologies are nuclear, ultrabornological when bornological and we can describewhen they are quasi-LB. We also give concrete microlocal representations ofbounded and equicontinuous sets in those spaces and work with general supportconditions including future compact or space compact support conditions onglobally hyperbolic manifolds, as motivated by physics applications to bedeveloped in a second paper.

Functional properties of Generalized H\"ormander spaces of distributions I : Duality theory, completions and bornologifications