The Moment Problem for Continuous Positive Semidefinite Linear functionals

Let $\tau$ be a locally convex topology on the countable dimensional polynomial $\reals$-algebra \rx:=\reals[X_1,...,X_n]. Let $K$ be a closed subset of $\reals^n$, and let $M:=M_{\{g_1, ... g_s\}}$ be a finitely generated quadratic module in \rx. We investigate the following question: When is the cone \Pos(K) (of polynomials nonnegative on $K$) included in the closure of $M$? We give an interpretation of this inclusion with respect to representing continuous linear functionals by measures. We discuss several examples; we compute the closure of M=\sos with respect to weighted norm-$p$ topologies. We show that this closure coincides with the cone \Pos(K) where $K$ is a certain convex compact polyhedron.Comment: 14 page