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I was reading and trying to understand the proof that the space $\mathcal{L}^2 (\mathcal{H})$ (Hilbert-Schmidt operators) is made by all the $T_K:L^2(X,\mu) \rightarrow L^2(X,\mu)$ with $K \in L^2(X \times X, \mu \times \mu)$.

Basically we assume $T \in \mathcal{L}^2$ and show $K \in L^2(X \times X, \mu \times \mu) $ such that $T = T_K$. I really don't understand pretty much about the proof.. anyone could help?

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