How space of wavelets is created from {0}?

Question

I have learned about wavelets and there is a space $$... V_{-1} \subset V_0 \subset V_1 $$ and they create as a closure of sum of $V_j$ all space $L^2(\Bbb{R})$. There are some other points, but one is struggling me $\bigcap_{j \in \Bbb{Z}}V_{j}={0}$ which in space of functions is $f(x) = 0$. Then we have two operators $T_k$ (shifting) and $D^j$ (scaling) which are used in construction such that: $$V_{j+1}=D(V_j)$$ and $$f \in V_0 \Rightarrow T_kf \in V_0$$ The questions is how that space is possible if it have to starts from function $f(x) = 0$ which treated by $D$ or $T$ operators stay in zero vector? What I am missing, could you help?