For the edited question: 

Let $\{f_k\}$ be a basis of $L^2$ using smooth functions.   
Let $V_n = \mathop{span}(f_1, \ldots, f_n)$.   
Let $T_m$ cyclically permute the $f_1, \ldots, f_m$ with $f_1 \mapsto f_m$. 

Then all the requirements you gave are satisfied. But 
$$ \bigcup_{n = m}^\infty T_n(V_m) = \bigcup_{n = 1}^\infty V_n = \mathop{span}(\{f_k\})$$
is not finite dimensional.