Let $U,V \subset \mathbb{R}^n$ be open subsets, and suppose there is a diffeomorphism $\phi : U \rightarrow V$.
Then, $\phi$ should induce a linear isomorphism between $C^\infty_c(U)$ and $C^\infty_c(V)$.
My 1st question is, does the bijection above also induce an isomorphism of topological vector spaces (TVSs)? Here we give these "test spaces of functions" the canonical LF topology.
My 2nd question is, for what other common kinds of assignments of test spaces (e.g. Schwartz functions) does $\phi$ induce a TVS isomorphism?