Show compactness of an operator with Arzelà–Ascoli

Question

We have $K\colon L^{2}(a,b) \rightarrow L^{2}(a,b)$ such that $ Kf(t)=\sum_{j=1}^{n}\phi_{j}(t) \int_{a}^{b} \psi_{j}(S) f(s)ds$ where $\phi_{j} ,\psi_{j} \in L^{2}(a,b)$. We want to show that K is compact. I tried to use the Arzelà–Ascoli theorem and I started by trying to show that for a sequence $f_{n}$ in the unit ball of $L^{2}(a,b)$ $K(f_{n})$ is uniformly bounded but I failed to prove this point. Can anyone help? Thanks.

Answer 1

Notice that the range of the unit ball is contained in the compact set $$K:=\left\{\sum_{j=1}^na_j\phi_j, |a_j|\leqslant \lVert \psi_j\rVert_2\right\}.$$