Problem: Let $f\in L^2(0,\infty)$ and let $(Tf)(s)=\frac1s\int_0^sf(t)dt$. Find the adjoint, $T^*$.
Attempt: I know that problems like these should be very simple, but oftentimes I find them very difficult, to my shame. I understand that the adjoint in this case is defined as being the $T^*:L^2(0,\infty)\to L^2(0,\infty)$ where, for all $f,g\in L^2(0,\infty)$,
$$\langle Tf,g\rangle=\langle f,T^*g\rangle$$
I understand the method one usually employs; namely to start from the left hand side and via manipulation (in this case, of the integrals) to get the inner product of $f$ with something else.
In this case though, I don't really understand what is going on with regards to the order of integration changing and how, exactly, the limits thereof change as well. I have been told it is to do with Fubini's theorem, but it's not something I have really grappled with, and had exposure to before.
I have had a look at some examples like this one here involving the same inner product, but I am thrown more than usual by the $1/s$ factor.
Could somebody use the above problem to elucidate what is going on, and how the machinations behind these manipulations really work?