We show that any element from the ($L^2$-)maximal domain of a jump-type symmetric Dirichlet form can be approximated by test functions under some conditions. This gives us a direct proof of the fact that the test functions is dense in Bessel potential spaces.
"On an extension of jump-type symmetric Dirichlet forms." Electron. Commun. Probab. 12 57 - 65, 2007. https://doi.org/10.1214/ECP.v12-1256