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 Šifrin. 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)).
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Murray Marshall passed away in May 2015. He worked on this manuscript together with us until the very last days of his life. We lost a collaborator of many years and a wonderful friend. We sorely miss him. (M. Ghasemi, M. Infusino, S. Kuhlmann).
The work of M. Infusino was partially supported by a Marie Curie fellowship of the Istituto Nazionale di Alta Matematica (Grant PCOFUND-GA-2009-245492). The work of M. Marshall was supported by the Natural Sciences and Engineering Research Council of Canada (Grant NSERC: 7854-2013).
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Ghasemi, M., Infusino, M., Kuhlmann, S. et al. Moment Problem for Symmetric Algebras of Locally Convex Spaces. Integr. Equ. Oper. Theory 90, 29 (2018). https://doi.org/10.1007/s00020-018-2453-7
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DOI: https://doi.org/10.1007/s00020-018-2453-7