Reflexivity of the Space of Transversal Distributions

Abstract

AbstractFor any smooth, Hausdorff and second-countable manifold N one can define the Fréchet space $$\mathord {\mathcal {C}^{\infty }}(N)$$ C ∞ ( N ) of smooth functions on N and its strong dual $$\mathcal {E}'(N)$$ E ′ ( N ) of compactly supported distributions on N. It is a standard result that the strong dual of $$\mathcal {E}'(N)$$ E ′ ( N ) is naturally isomorphic to $$\mathord {\mathcal {C}^{\infty }}(N)$$ C ∞ ( N ) , which implies that both $$\mathord {\mathcal {C}^{\infty }}(N)$$ C ∞ ( N ) and $$\mathcal {E}'(N)$$ 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 $$\pi :P\rightarrow M$$ π : P → M . We show that the strong $$\mathord {\mathcal {C}^{\infty }_{c}}(M)$$ C c ∞ ( M ) -dual of the space $$\mathcal {E}'_{\pi }(P)$$ E π ′ ( P ) of $$\pi $$ π -transversal distributions is naturally isomorphic to the $$\mathord {\mathcal {C}^{\infty }_{c}}(M)$$ C c ∞ ( M ) -module $$\mathord {\mathcal {C}^{\infty }}(P)$$ C ∞ ( P ) .