  This note aims to show a uniqueness property for the solution (whenever
exists) to the moment problem for the symmetric algebra $S(V)$ of a locally
convex space $(V, \tau)$. Let $\mu$ be a measure representing a linear
functional $L: S(V)\to\mathbb{R}$. We deduce a sufficient determinacy condition
on $L$ provided that the support of $\mu$ is contained in the union of the
topological duals of $V$ w.r.t. to countably many of the seminorms in the
family inducing $\tau$. We compare this result with some already known in
literature for such a general form of the moment problem and further discuss
how some prior knowledge on the support of the representing measure influences
its determinacy.
