I'm studying Sobolev spaces using Evans' PDE book.
I can't figure out this simple fact.
Let $L$ be an operator in this form:
$$Lu= \sum{D_i(a_{ij}D_j(u))+\sum{b_iD_i(u)+cu}}.$$

I can't understand why $L \in H^{-1}$, the dual space of $H_0^1$. I'm struggling because to me it would require that $u$ must have 2 derivatives so $u \in H^2$ to be well defined.
Am I missing something? 
Thanks. 

EDIT:
My problem is about this identity $Lu=f$, $f \in L^2$, in $ \Omega$ plus some boundary conditions for example $u \in H_0^1$.I can define a weak solution as $u \in H_0^1$ such that $B[u,v]=(f,v) $ for all $v \in H_0^1$. Ok that's fine, but when i write $Lu=f $ what does it mean?