Let $k \in {L^2}((0,4) \times (0,1))$, $g \in {L^2}(0,1)$.
We consider the following first kind Fredholm equation $$\int\limits_0^4 {k(s,x)f(s)ds=g(x), x\in(0,1).} $$ Where $f$ is the unknown. How can I prove that there exists $g$ in $L^2(0,1)$ such that the above equation doesn't have solution. I thought about the compactness of Hilbert-Schmidt operator, but I think that as the kernel $k$ is not defined on a square, we cannot apply that. Any suggestions? Thanks