The Riesz representation theorem on infinite dimensional spaces and its applications

Let H be a real separable Hilbert space and let E subset of H be a nuclear space with the chain {E-m: m = 1, 2,...} of Hilbert spaces such that E = boolean AND(mepsilonN)E(m). Let E* and E-m denote the dual spaces of E and E-m respectively. For gamma > 0, let C-infinity,C-gamma,C-c be the collection of complex-valued continuous functions f defined on E* such that parallel tofparallel to(m,gamma) := sup(xis an element ofE-m) {\f(x)\ exp(-gamma(-1)\x\(gamma)(-m))} is finite for every m. Then C-proportional to,C-gamma,C-c is a complete countably normed space equipping with the family {parallel to.parallel to(m,y) : m = 1, 2....} of norms. Using a probabilistic approach, it is shown that every continuous linear functional T on C-infinity,C-gamma,C-c can be represented uniquely by a complex Borel measure v(T) satisfying certain exponential integrability condition. The results generalize an infinite dimensional Riesz representation theorem given previously by the first author for the case gamma = 2. As applications, we establish a Weierstrass approximation theorem on E* for gamma greater than or equal to 1 and show that the space D spanned by the class {exp[i(x,xi)]: xi is an element of E} is dense in C-infinity,C-gamma,C-c for gamma > 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*.