Functional properties of Hörmander's space of distributions having a specified wavefront set

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The space D'Γ of distributions having their wavefront sets in a closed cone Γ has become important in physics because of its role in the formulation of quantum field theory in curved space time. In this paper, the topological and bornological properties of D'Γ and its dual E'Λ are investigated. It is found that D'Γ is a nuclear, semi-reflexive and semi-Montel complete normal space of distributions. Its strong dual E'Λ is a nuclear, barrelled and bornological normal space of distributions which, however, is not even sequentially complete. Concrete rules are given to determine whether a distribution belongs to D'Γ, whether a sequence converges in D'Γ and whether a set of distributions is bounded in D'Γ.