We consider a countably generated and uniformly closed algebra of bounded functions. We assume that there is a lower semicontinuous, with respect to the supremum norm, quadratic form and that normal contractions operate in a certain sense. Then we prove that a subspace of the effective domain of the quadratic form is naturally isomorphic to a core of a regular Dirichlet form on a locally compact, separable metric space. We also show that any Dirichlet form on a countably generated measure space can be approximated by essentially discrete Dirichlet forms, i.e., energy forms on finite weighted graphs, in the sense of Mosco convergence, i.e., strong resolvent convergence.
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Published in Zapiski Nauchnykh Seminarov POMI, Vol. 441, 2015, pp. 299–317.
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Hinz, M., Teplyaev, A. Closability, Regularity, and Approximation by Graphs for Separable Bilinear Forms. J Math Sci 219, 807–820 (2016). https://doi.org/10.1007/s10958-016-3149-7
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DOI: https://doi.org/10.1007/s10958-016-3149-7