THE RIESZ REPRESENTATION THEOREM ON INFINITE DIMENSIONAL SPACES AND ITS APPLICATIONS

Abstract

Let H be a real separable Hilbert space and let E⊂H be a nuclear space with the chain {Em: m=1,2,…} of Hilbert spaces such that E = ∩m∈ℕEm. Let E* and E-m denote the dual spaces of E and Em, respectively. For γ > 0, let [Formula: see text] be the collection of complex-valued continuous functions f defined on E* such that [Formula: see text] is finite for every m. Then [Formula: see text] is a complete countably normed space equipping with the family {‖·‖m,γ : m = 1,2,…} of norms. Using a probabilistic approach, it is shown that every continuous linear functional T on [Formula: see text] can be represented uniquely by a complex Borel measure νT satisfying certain exponential integrability condition. The results generalize an infinite dimensional Riesz representation theorem given previously by the first author for the case γ = 2. As applications, we establish a Weierstrass approximation theorem on E* for γ≥1 and show that the space [Formula: see text] spanned by the class { exp [i(x,ξ)] : ξ ∈ E} is dense in [Formula: see text] for γ>0. In the course of the proof we give sufficient conditions for a function space on which every positive functional can be represented by a Borel measure on E*.