  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.
