Functional Properties of Hörmander’s Space of Distributions Having a Specified Wavefront Set
Résumé
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 spacetime. 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 (ultra)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 .