# Isometric isomorphism between $L^2$ and $\mathcal{L}^2$

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?