On the determinacy of the moment problem for symmetric algebras of a locally convex space

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, τ). Let μ be a measure representing a linear functional L : S(V) ↗ R. We deduce a sufficient determinacy condition on L provided that the support of μ is contained in the union of the topological duals of V with respect to countably many of the seminorms in the family inducing τ. 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.