On the Non-nuclearity of a Space of Test Functions on Hilbert Space

Dudin has constructed a space D(H), consisting of all the infinitely differentiable functions with bounded support on a real separable Hilbert space H, all of whose derivatives have bounded range. He establishes that D′(H), the dual of D(H), can be identified with a certain space of generalized measures. In this work we establish the non-nuclearity of Dudin′s space D(H). The result follows by modifying a result of Meise that a certain space of infinitely differentiable functions on a real locally convex space, not necessarily of bounded support, is non-nuclear.