I've been searching the Internet for the exact expression of the following Hilbert space isomorphism induced by mapping $\mathbb{R}^3$ homemorphically on itself under the cartesian to spherical coordinates transformation. Can't find it. I hope you can help me. More precisely, I'm looking for the exact unitary operator U which makes possible the following isomorphism:
$$ L^2 \left(\mathbb{R}^3\right) \simeq L^2 \left(\left(0,\infty\right), r^2 {}dr\right) \otimes L^2 \left(S^2\right) $$
Now, once I have this exact operator, I wonder if I can obtain a similar splitting of the Schwartz test functions space by acting with it:
$$ \mathcal{S} \left(\mathbb{R}^3\right) \simeq \mathcal{S} \left(?\right) \otimes \mathcal{S} \left(?\right) $$
Who are the sets marked as "?"
Thank you