Closure of the cone of sums of 2d-powers in real topological algebras

Let $R$ be a unitary commutative real algebra and $K\subseteq Hom(R,$\mathbb{R}$)$,closedwithrespecttotheproducttopology\mathrm{.}Weconsider$R$endowedwiththetopology$$\mathcal{T}$_K$, induced by the family of seminorms ${\rho }_{\alpha }(a):=\mathrm{\mid }\alpha (a)\mathrm{\mid }$, for $\alpha \in K$ and $a\in R$. In case $K$ is compact, we also consider the topology induced by $\mathrm{\parallel }a{\mathrm{\parallel }}_K:={\sup }_{\alpha \in K}\mathrm{\mid }\alpha (a)\mathrm{\mid }$ 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 $Psd(K):={a\in R:\alpha(a)\geq 0,\textrm{for all}\alpha\in K}$\mathrm{.}Inparticular,anycontinuouslinearfunctional$L$ on the polynomial ring $R=\mathbb{R}[X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_n]$ with $L(h^{2d})\ge 0$ for each $h\in R$isintegrationwithrespecttoapositiveBorelmeasuresupportedon$K$. Finally we give necessary and sufficient conditions to ensure the continuity of a linear functional with respect to those two topologies.