A GENERALIZATION OF THE RIESZ REPRESENTATION THEOREM TO INFINITE DIMENSIONS

摘要

Let H be a real separable Hilbert space and let E subset of H be a nuclear space with the chain of Hilbert spaces {E-p: p = 1, 2, 3,...} such that E=boolean AND(p=1)(infinity) E-p. Let E* and E-p denote the dual spaces of E and E-p, respectively. Let C-p be the collection of real-valued functions f defined on E-p such that f is uniformly continuous on bounded subsets of E-p and such that parallel to f parallel to(infinity,p):= sup(x is an element of E-p){(x) exp(-1/2x(2)(-p))} is finite. Set C-infinity=boolean AND(p=1)(infinity) C-p. Then C-infinity is a complete countably normed space equipped with the family {parallel to.parallel to(infinity,p): p = 1, 2, 3,...} of norms. In this paper it is shown that to every bounded linear Functional F in C-infinity*, there corresponds a signed measure nu F such that F(phi) = integral(E*)psi(x)nu(F)(dx) for phi is an element of C-infinity. It is also shown that there exists some p such that the measurable support of nu is contained in E-p and integral(E-p)exp(1/2x(2)(-p))u(F)(dx)