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If $H$ is a Hilbert space with norm $\| . \|$, and $A$ is an operator, we call it a Hilbert Schmidt Operator if $$\sum_{n=1}^\infty \|Ax_n\|^2<\infty$$ for some orthonormal basis $\{x_i\}.$

Consider $L^2(X,\mu)$. How could one prove that every Hilbert Schmidt Operator on this space is given by $$(Af)(x)=\int_X k(x,y)f(y)dy$$ for some $$k(x,y)\in L^2(X\times X, \mu \times \mu).$$

I am not really sure where to start, but I imagine this would be in many textbooks/online notes? Does this fact have a particular name? Ideally I would love it if someone could show why it is true, but a reference is useful as well.

Thanks for any help!

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How many details do you want?

Let $$A x_n = \sum_k \alpha_{n,k} x_k.$$

Show $\sum |\alpha_{n,k}|^2 < \infty$.

Now define $$k = \sum_{n,k} \alpha_{n,k} (x_n \otimes \overline{x}_k)$$

Then show that this works.

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