We define the Sobolev space as the space of all functions on $U$ whose distributional derivatives up to the order of $k$ are contained in the space $L^2(U)$.
$H^k(U):=\{u\in L^2(U)| D^\alpha u \in L^2(U) \forall0 \leq |\alpha| \leq k \}$
The space $H_0^k$ contains all functions that can be approximated using bump functions in $D(U)$ in the $H^k(U)$-norm.
$H_0^k:=\{ u \in H^k(U) | \exists \phi_n \in D(U):\lim_{n \rightarrow \infty} ||u-\phi_n||_k = 0\}$
Now the question is which functions are in $H_0^0$
Since $||x||_0=0$ and $0=0$ I'd say we're looking at all functions in $H^0(U)$. This is the same as all functions in $L^2(U)$ until the 0-th derivative hence $H_0^0=L^2(U)$?
Is this true? How could I express this more formally?
Yes it is true: $H_0^0(\Omega)=L^2(\Omega)$. This is just another way of saying that the space of smooth compactly supported functions in $\Omega$ is dense in $L^2(\Omega)$.