I want to prove the operator $D=\frac{1}{i}\frac{d}{dt}$ is Hermitian/self-adjoint. I saw a claim that said "for any two functions $f,g$", by integration by parts:
$$\int g^*(t)\frac{1}{i}\frac{d}{dt} f(t)dt = \left. \frac{1}{i}f g^* \right|_{-\infty}^{\infty} + \int f(t)\left( \frac{1}{i}\frac{d}{dt} g(t) \right)^*$$
With the first term on the right claimed to be 0.
Now how can I show $\lim_{|t|\to \infty}f(t) g^*(t)= 0$? I would like to make as few additional assumptions as possible.
I guess that we must start by assuming $f,g\in L^2(\mathbb{C})$ since $D$ needs to be defined on some Hilbert space, but even if I assume the left hand side is finite, I can't show $\lim_{|t|\to \infty}f(t) g^*(t)= 0$...
However, this reminds me of some distributional derivatives stuff, or even more basic: for $g\in C^1(\mathbb{R})$, $\int g'(t)f(t)dt=-\int g(t)f'(t)dt, f\in C_c^\infty(\mathbb{R})$. But extending this with density arguments to some Hilbert space seems a bit too much for me...