If $H$ is a Hilbert space with norm $\| . \|$, and $A$ is an operator, we call it a Hilbert Schmidt Operator if $\sum_{n=1}^\infty \|Ax_n\|^2<\infty$ for some orthonormal basis $\{x_i\}.$
Consider $L^2(X,\mu)$. How could one prove that every Hilbert Schmidt Operator on this space is given by $(Af)(x)=\int_X k(x,y)f(y)dy$ for some $k(x,y)\in L^2(X\times X, \mu \times \mu).$
I am not really sure where to start, but I imagine this would be in many textbooks/online notes? Does this fact have a particular name? Ideally I would love it if someone could show why it is true, but a reference is useful as well.
Thanks for any help!