Reflexivity of the Space of Transversal Distributions

Abstract

For any smooth, Hausdorff and second-countable manifold N one can define the Fréchet space mathord {mathcal {C}^{infty }}(N) of smooth functions on N and its strong dual mathcal {E}'(N) of compactly supported distributions on N. It is a standard result that the strong dual of mathcal {E}'(N) is naturally isomorphic to mathord {mathcal {C}^{infty }}(N), which implies that both mathord {mathcal {C}^{infty }}(N) and mathcal {E}'(N) are reflexive locally convex spaces. In this paper we generalise that result to the setting of transversal distributions on the total space of a surjective submersion pi :Prightarrow M. We show that the strong mathord {mathcal {C}^{infty }_{c}}(M)-dual of the space mathcal {E}'_{pi }(P) of pi -transversal distributions is naturally isomorphic to the mathord {mathcal {C}^{infty }_{c}}(M)-module mathord {mathcal {C}^{infty }}(P).