Let $\mathcal{H}$ be a Hilbert space and $(\Omega, \mathcal{F}, P)$ a probability space with $L^2$-space $L^2(\Omega)$. I am looking at the space of square integrable $\mathcal{H}$-valed random variables defined as
$$L^2(\Omega,\mathcal{H})=\{X:\Omega \to \mathcal{H} \text{ measurable} : \mathbb{E}[\|X\|^2]<\infty\}$$ which is again a Hilbert space. Now, I was wondering if this space can be identified with the Hilbert space tensor product $L^2(\Omega)\otimes \mathcal{H}$ in some way?
I know Hilbert space tensor products are related to the space of Hilbert-Schmidt operators from $L^2(\Omega)\to \mathcal{H}$, which seems useful here, but I'm not familiar enough with these operators to figure it out myself.
Assuming $H$ is separable, the two spaces are naturally isomorphic. The isomorphism $L^2(\Omega) \otimes H \to L^2(\Omega, H)$ sends $f \otimes h$ to the function $\Omega \ni \omega \mapsto f(\omega)h \in H$.