Let $\{\varphi_j:j\in\mathbb{N}\}$ be an orthonormal set of functions in $L^2[(a,b), \mathbb{C}]$ and $\lambda_j \in\mathbb{C}$ for $j\in\mathbb{N}$. Let $$K(x,y) = \sum_{j=1}^\infty \lambda_j\varphi_j(x)\overline{\varphi_j(y)}$$ and $$\sum_{j=1}^\infty |\lambda_j|^2 < \infty$$ The question is then to
(a) Show that $\varphi_j$ is an eigenfunction of the operator $$(Af)(x) = \int_a^b K(x,y)f(y) dy$$ with corresponding eigenvalue $\lambda_j$
(b) Show that there are no other non-zero eigenvalues of $A$
(c) Determine when $A$ is self-adjoint.
My progress thus far:
(a) I am unsure how to show that the sum in $K(x,y)$ is finite. I tried using the properties of an orthonormal set and the fact that $\sum_{j=1}^\infty |\lambda_j|^2 < \infty$ but got stuck with the fact that we have two variables $x$ and $y$.
I understand how to show that $\varphi_j$ is an eigenvalue of $A$ with eigenvalue $\lambda_j$, but I have no justification for interchanging the order of summation and the integration. I tried to use the Dominated Convergence Theorem but got very lost.
(b) I have no idea how to show there are no other non-zero eigenvalues of $A$.
(c) Again, I am completely stuck here.
For the first part note that $$ \begin{align*} \iint|K(x,y) f(y)|\,\mathrm d y\,\mathrm d x&\leqslant \iint\sum_{j\geqslant 1}|\lambda _j||\varphi _j(x)||\varphi _j(y)||f(y)|\,\mathrm d y\,\mathrm d x\\ &=\sum_{j\geqslant 1}\iint|\lambda _j||\varphi _j(x)||\varphi _j(y)||f(y)|\,\mathrm d y\,\mathrm d x\\ &\leqslant \sum_{j\geqslant 1}\left(\iint|\lambda _j|^2|\varphi _j(x)|^2|\varphi _j(y)|^2\,\mathrm d y\,\mathrm d x\right)^{1/2}\left(\iint|f(y)|^2\,\mathrm d y\,\mathrm d x\right)^{1/2}\\ &\leqslant \sqrt{b-a}\|f\|_2\sqrt{\sum_{j\geqslant 1}|\lambda _j|^2}\\ &<\infty \end{align*} $$ where we used the monotone convergence theorem first and the Cauchy-Schwartz inequality later. Therefore the set $\{(x,y)\in (a,b)^2:K(x,y)=\infty \}$ have measure zero.