Let $\Omega$ be the domain
$\Omega = \lbrace f : [0,1] \rightarrow \mathbb{C} | f \ \text{is abs. cont.}, f' \in L^2 ([0,1]), f(0)=0 \rbrace$.
I am asked whether the operator $T= i \frac{d}{dx}$ is symmetric on this domain? Surely, the answer is no, which can be shown by simple integration by parts, since no condition is given for $f(1)$? I just want to make sure I am not missing something obvious.