I know how to prove that the Fredholm Integral operator $$ T_k : L^2([0,1]) \rightarrow L^2([0,1]) \; \left(T_kf \right)(s)= \int_{[0,1]} k(s,t) \, f(t) \; dt$$ with $$k \in L^2([0,1]^2) \; , f\in L^2([0,1])$$is compact. I use the fact that the operator has finite image.
My question is, if someone has a reference or know a proof in which the Arzela-Ascoli theorem is used ?
Thanks in advance.