Let $M \mapsto\mathcal{F}(M)$ be a map associating topological vector spaces of some type (that I will call "function spaces") to geometric spaces $M$ of some type. 

For $M$, I'm mostly thinking of manifolds with some additional structure, or locally compact topological spaces. $\mathcal F$ may or may not be a functor in some way, though it's better if it's a contravariant functor. I'm mostly interested in the case where $\mathcal{F}(M)$ is a usual function space such as $L^p(M)$, $W^{k,p}(M)$, $\mathrm{Meas}(M)$, like in [this][1] question, and [this][2] one. I want the function spaces of the form $\mathcal{F}(M)$ to have some completed tensor product $\otimes$.

**Question 1:** When does it happen that $\mathcal{F}(M\times N)\simeq\mathcal{F}(M)\otimes\mathcal{F}(N)$ and when does it fail and how badly? 

The above tensor property, when $\mathcal F$ is a functor, would be better intended to hold naturally, i.e. $\mathcal F$ is to be a monoidal functor from spaces with their Cartesian product $\times$ to function spaces with $\otimes$, but the emphasis is not on the categorical aspect.

**Edit:** I'm aware that, as Nik Weaver points out in the comments, I can't expect to get a completely general answer. Rather, the question (which I find very natural) should be intended in "community wiki" style, i.e. partial contributions are ok.  









 


  [1]: https://mathoverflow.net/questions/202323/is-the-space-of-all-borel-measures-on-mathbb-rn-isomorphic-to-the-tensor-pro?noredirect=1&lq=1
  [2]: https://mathoverflow.net/questions/203572/topologies-for-which-mathcalmx-otimes-mathcalmy-is-dense-in-mathca