Moment problem for symmetric algebras of locally convex spaces

Abstract

It is explained how a locally convex (lc) topology $\tau$ on a real vector space $V$ extends to a locally multiplicatively convex (lmc) topology $\overline{\tau}$ on the symmetric algebra $S(V)$. This allows the application of the results on lmc topological algebras obtained by Ghasemi, Kuhlmann and Marshall to obtain representations of $\overline{\tau}$-continuous linear functionals $L: S(V)\rightarrow \mathbb{R}$ satisfying $L(\sum S(V)^{2d}) \subseteq [0,\infty)$ (more generally, $L(M) \subseteq [0,\infty)$ for some $2d$-power module $M$ of $S(V)$) as integrals with respect to uniquely determined Radon measures $\mu$ supported by special sorts of closed balls in the dual space of $V$. The result is simultaneously more general and less general than the corresponding result of Berezansky, Kondratiev and \v Sifrin. It is more general because $V$ can be any lc topological space (not just a separable nuclear space), the result holds for arbitrary $2d$-powers (not just squares), and no assumptions of quasi-analyticity are required. It is less general because it is necessary to assume that $L : S(V) \rightarrow \mathbb{R}$ is $\overline{\tau}$-continuous (not just continuous on each homogeneous part of $S(V)$).