# Reference for the following problems

I'm a newbie and need reference for the following proofs:

1.Inverse of a Laplacian operator is compact in $L^2(\Omega)$ : Proof

2.If I've a general Hilbert-Schmidt Integral operator defined from one $L^p(\Omega_1)$ to another $L^q(\Omega_2)$ with all the exponents defined appropriately and the Kernel is too defined appropriately.

Detail: Chapter 8 and 8.15 of "H.W. Alt. Lineare Funktionalanalysis", Springer-Verlag, Berlin, $2002$. But this is in German and I'm not able to understand. So here we have to prove that the operator is compact as it's done in $L^2$ case. But here it's for a general $L^p$ space.

Thanks.

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For the first question, math.stackexchange.com/questions/114947/… can help. For the second one, do you want $p$ and $q$ finite. In this case, the spaces are separable, and we can work with sequence, and a diagonal argument. The characterization of the dual of these spaces will help. –  Davide Giraudo Jul 27 '12 at 21:06