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Deny introduced in  the notion of functional spaces by generalizing Dirichlet spaces. In this paper, we shall give the following necessary and sufficient conditions for a functional space to be a real Dirichlet space.Letbe a regular functional space with respect to a locally compact Hausdorff spaceXand a positive measure ξ inX. The following four conditions are equivalent.doi:10.1017/s0027763000012319 fatcat:aipgb7znyfcqnh7heyjwbskxo4