Dirichlet spaces on \(H\)-convex sets in Wiener space (Q645943)

Let \(U\) be a subset of the abstract Wiener space \((E,H,\mu)\). In this paper a number of \(H\)-notions are introduced (e.g. \(H\)-convex set, \(H\)-open set, \(H\)-continuous function, \(H\)-Lipschitz function, etc.). The relations between these notions and the corresponding quasi-notions are clarified. From this it is proved that the \((1,2)\)-Sobolev space \(W^{1,2}(U)\) on a subset \(U\) in \(E\) has a dense subset of smooth cylindrical functions if and only if \(U\) is \(H\)-convex and \(H\)-open.