  This article provides a new approach to address Mosco convergence of
gradient-type Dirichlet forms, $\mathcal E^N$ on $L^2(E,\mu_N)$ for
$N\in\mathbb N$, in the framework of converging Hilbert spaces by K.~Kuwae and
T.~Shioya. The basic assumption is weak measure convergence of the family
${(\mu_N)}_{N}$ on the state space $E$ - either a separable Hilbert space or a
locally convex topological vector space. Apart from that, the conditions on
${(\mu_N)}_{N}$ try to impose as little restrictions as possible. The problem
has fully been solved if the family ${(\mu_N)}_{N}$ contain only log-concave
measures, due to L.~Ambrosio, G.~Savar\'e and L.~Zambotti, 2009. However for a
large class of convergence problems the assumption of log-concavity fails. The
article suggests a way to overcome this hindrance, as it presents a new
approach. Combining the theory of Dirichlet forms with methods from numerical
analysis we find abstract criteria for Mosco convergence of standard gradient
forms with varying reference measures. These include cases in which the
measures are not log-concave. To demonstrate the accessibility of our abstract
theory we discuss a first application, generalizing an approximation result by
S.~K.~Bounebache and L.~Zambotti, 2014.
