  We construct equivalent semi-norms of non-local Dirichlet forms on the
Sierpi\'nski gasket and apply these semi-norms to a convergence problem and a
trace problem. We also construct explicitly a sequence of non-local Dirichlet
forms with jumping kernels equivalent to $|x-y|^{-\alpha-\beta}$ that converges
exactly to local Dirichlet form.

Functional Analysis