  We study sequences of nonlocal quadratic forms and function spaces that are
related to Markov jump processes in bounded domains with a Lipschitz boundary.
Our aim is to show the convergence of these forms to local quadratic forms of
gradient type. Under suitable conditions we establish the convergence in the
sense of Mosco. Our framework allows bounded and unbounded nonlocal operators
to be studied at the same time. Moreover, we prove that smooth functions with
compact support are dense in the nonlocal function spaces under consideration.
