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We establish a criterion for the existence of a representing Radon measurefor linear functionals defined on a unital commutative real algebra $A$, whichwe assume to be generated by a vector space $V$ endowed with a Hilbertianseminorm $q$. Such a general criterion provides representing measures withsupport 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 thecase when $V$ is endowed with a nuclear topology. In particular, we apply ourfindings to the symmetric tensor algebra of a nuclear space.

Moment problem for algebras generated by a nuclear space