Consider $T=-\frac{d^2}{dx^2}$ as an operator on $L^2(\mathbb{R})$ with domain $C_0^\infty(\mathbb{R})$. What's the adjoint of T?
This is an exercise from Reed&Simon's book. I have managed to show that $(D(T),T)$ is densely defined, $ D(T)\subset D(T^*)$ and $T^*=-\frac{d^2}{dx^2}$. I am stuck on determining the domain of $T^*$. I am wondering if anyone could give me a hint?