  Roughly speaking, regular subspaces are regular Dirichlet forms that inherit
the original forms with smaller domains. In this paper, regular subspaces of
1-dim symmetric $\alpha$-stable processes are considered. The main result is
that it admits proper regular subspaces if and only if $\alpha\in [1,2]$.
Moreover, for $\alpha\in(1,2)$, the characterization of the regular subspaces
is given. General 1-dim symmetric L\'evy processes will also be investigated.
It will be shown that whether it has proper regular subspaces is closely
related to whether its sample paths have finite variation.
