Moment problem for algebras generated by a nuclear space

We establish a criterion for the existence of a representing Radon measure for linear functionals defined on a unital commutative real algebra A, which we assume to be generated by a vector space V endowed with a Hilbertian seminorm q. Such a general criterion provides representing measures with support contained in the space of characters of A whose restrictions to V are q−continuous. This allows us in turn to prove existence results for the case when V is endowed with a nuclear topology. In particular, we apply our findings to the symmetric tensor algebra of a nuclear space.