Let $B:L^2(\Omega) \times L^2(\Omega)\to \mathbb{R}$, $\Omega\subset \mathbb{R}^n$, be a bounded bilinear form. Using the tensor product for Hilbert spaces, we can find a bounded linear operator $\tilde B: L^2(\Omega) \otimes L^2(\Omega)\to \mathbb{R}$, such that $B= \tilde B \circ \tau$, where $\tau: L^2(\Omega) \times L^2(\Omega)\to L^2(\Omega) \otimes L^2(\Omega)$ is the universal bilinear mapping for the tensor product.
So far so good. Now, we also know that $L^2(\Omega) \otimes L^2(\Omega) \cong L^2(\Omega^2)$, and $L^2(\Omega^2)$ is a Hilbert space. Therefore, $\tilde B$ can be identified with an element $f\in L^2(\Omega^2)$.
My question is: Can we say anything about $f$? Is there a representation of $f$ in terms of $B$? Maybe for some special cases?
Thank you very much in advance!
Best, Luke