I am attempting to understand what types of kernels the standard theory of Fredholm Type-2 integral equations applies to, but I've never taken a course in analysis. Basically, given a kernel, $K(x,t)$, I would like to immediately say whether Fredhlm's Theorems hold and to find the parameter values for which the Liouville–Neumann series converges.
To that end there are some definitions and notation that I do not understand:
Suppose we wish to solve $y(x)- \lambda \int^b_a y(t) K(x,t) dt = f(x)$
1) Let $Q = [a,b]^2$ what does it mean to say $K(x,t) \in L^2_Q $?
2)How do we define $ \| K(x,t) \|_{L^2_Q} $?
I understand the case of a function of one variable: $f(x) \in L^2_{[a,b]}$ if $\int_a^b |f(x)|^2 dx $is finite but the generalization to 2 variables is ambiguous to me.