Let $R$ be a unitary commutative real algebra and $K\subseteq Hom(R,\mathbb{ R})$, closed with respect to the product topology. We consider $R$ endowed with the topology $\mathcal{ T}_K$, induced by the family of seminorms $\rho_{ \alpha}(a):=|\alpha(a)|$, for $\alpha\in K$ and $a\in R$. In case $K$ is compact, we also consider the topology induced by $\|a\|_K:=\sup_{ \alpha\in K}|\alpha(a)|$ for $a\in R$. If $K$ is Zariski dense, then those topologies are Hausdorff. In this paper we prove that the closure of the cone of sums of 2d-powers, $\sum R^{ 2d}$, with respect to those two topologies is equal to $Psd(K):=\{ a\in R:\alpha(a)\geq 0,\textrm{ for all}\alpha\in K\}$. In particular, any continuous linear functional $L$ on the polynomial ring $R=\mathbb{ R}[X_1,...,X_n]$ with $L(h^{ 2d})\ge0$ for each $h\in R$ is integration with respect to a positive Borel measure supported on $K$. Finally we give necessary and sufficient conditions to ensure the continuity of a linear functional with respect to those two topologies.
Categories: math.RA