On partial integration in infinite-dimensional space and applications to Dirichlet forms

Let £ be a locally convex (Souslinean) topological vector space and fi an arbitrary (not necessarily quasi-invariant) probability measure on E. We prove a characterization of the set of all fce£\{0} for which a partial integration formula holds for the corresponding derivative d/dk. As a consequence we improve a recent result by two of the authors on the maximality problem for classical Dirichlet forms. Consider a system £ of Dirichlet forms on some L2(E,fj) which coincide on some domain D, where D is dense in L2(E; /*). Such a situation is relevant since in practice the action of the Dirichlet forms that one is interested in is usually only known on some nice domain D. Therefore, it is an important question whether one can obtai