On hereditary reflexivity of topological vector spaces. the de rham cochain and chain spaces

For proving reflexivity of the spaces of de Rham coho- mology and homology of C∞-manifolds the author considers the notion of hereditary reflexivity as well as the notion of dual hereditary reflexivity of locally convex topological vector spaces which is interesting in itself. Complete barrelled nuclear spaces with complete nuclear duals turn out to be hereditarily reflexive. The Pontryagin duality in locally convex topological spaces is also considered.