Vector calculus for tamed Dirichlet spaces

In the language of $L^\infty$ modules proposed by Gigli, we introduce a first order calculus on a topological Lusin measure space $(M, m)$ arrying a quasi-regular, strongly local Dirichlet form $E$. Furthermore, we develop a second order calculus if $(M, E, m)$ is tamed by a signed measure in the extended Kato class in the sense of Erbar, Rigoni, Sturm and Tamanini. This allows us to define e.g. Hessians, covariant and exterior derivatives, and Ricci curvature.