Let $K$ be the integral operator defined by
$$ (Kf)(x)=\int_0^1 u(x)v(y)f(y) dy $$
for some continuous functions $u,v$ in the Hilbert space with inner product $\langle f,g \rangle = \int_0^1 f(x)^* g(x) dx$ on $(0,1)$. I want to find the eigenfunctions and eigenvalues corresponding to $K$. (this is problem 3.4 in http://www.mat.univie.ac.at/~gerald/ftp/book-fa/ )
The exercise is from a chapter about compact symmetric operators (which this operator is), but it only contains existence theorems.
If I could get some helpful hints on how to get started, I'd be thankful. (I have a suspicion this is easier than it looks)
Thanks in advance.