Moment problem for symmetric algebras of locally convex spaces

DOI10.1007/S00020-018-2453-7zbMATH Open1395.44014arXiv1507.06781OpenAlexW3103140205MaRDI QIDQ723535

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Publication date: 24 July 2018

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Abstract: It is explained how a locally convex (lc) topology $au$ on a real vector space $V$ extends to a locally multiplicatively convex (lmc) topology $overlineau$ 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 $overlineau$-continuous linear functionals $L:S(V)ightarrowmathbbR$ satisfying $L(sumS(V{)}^{2d})subseteq[0,infty)$ (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 $L:S(V)ightarrowmathbbR$ is $overlineau$-continuous (not just continuous on each homogeneous part of $S(V)$).

Full work available at URL: https://arxiv.org/abs/1507.06781

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