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Let $A$ be a commutative unital $\mathbb{R}$-algebra and let $\rho$ be aseminorm on $A$ which satisfies $\rho(ab)\leq\rho(a)\rho(b)$. We apply T.Jacobi's representation theorem to determine the closure of a $\sumA^{2d}$-module $S$ of $A$ in the topology induced by $\rho$, for any integer$d\ge1$. We show that this closure is exactly the set of all elements $a\in A$such that $\alpha(a)\ge0$ for every $\rho$-continuous $\mathbb{R}$-algebrahomomorphism $\alpha : A \rightarrow \mathbb{R}$ with$\alpha(S)\subseteq[0,\infty)$, and that this result continues to hold when$\rho$ is replaced by any locally multiplicatively convex topology $\tau$ on$A$. We obtain a representation of any linear functional $L : A \rightarrow\reals$ which is continuous with respect to any such $\rho$ or $\tau$ andnon-negative on $S$ as integration with respect to a unique Radon measure onthe space of all real valued $\reals$-algebra homomorphisms on $A$, and wecharacterize the support of the measure obtained in this way.

Application of Jacobi's Representation Theorem to locally multiplicatively convex topological real Algebras