Let $S\subset\mathbb{R}^n$ be Lebesgue-measurable. For $l\in L^2(S\times S)$ we define the operator $A:L^2(S)\rightarrow L^2(S),u\rightarrow Au$ with: $$(Au)(s):=\int_Sl(s,t)u(t)dt$$ Show that A is well defined, linear and continuous with $||A||_{L(L^2(S),L^2(S))}\leq||l||_{L^2(S\times S)}$. Also show that $A$ is compact.
I already showed everything except for the last property. Can someone help me with the compactness of A?
Approximate $l$ with simple functions, and show that you obtain $A$ as a limite of finite-rank operators.