A Generalization of the Riesz Representation Theorem to Infinite Dimensions

LetHbe a real separable Hilbert space and letE⊂Hbe a nuclear space with the chain of Hilbert spaces {Ep: p=1, 2, 3, …} such thatE=∩∞p=1 Ep. LetE* andE−pdenote the dual spaces ofEandEp, respectively. Let Cpbe the collection of real-valued functionsfdefined onE−psuch thatfis uniformly continuous on bounded subsets ofE−pand such that ‖f‖∞, p≔supx∈E−p{|f(x)| exp(−12 |x|2−p)} is finite. Set C∞=∩∞p=1 Cp. ThenC∞is a complete countably normed space equipped with the family {‖·‖∞, p: p=1, 2, 3, …} of norms. In this paper it is shown that to every bounded linear functionalFin C*∞, there corresponds a signed measureνFsuch thatF(ϕ)=∫E* ϕ(x) νF(dx) forϕ∈C∞. It is also shown that there exists somepsuch that the measurable support ofνis contained inE−pand ∫E−p exp(12 |x|2−p) |νF| (dx)<∞. This result extends the Riesz representation theorem to infinite dimensions. In the course of the proof, an infinite dimensional analogue of the Weierstrass approximation theorem is also established onE*.