I'm studying distributions and just came along the formula $$\langle f\otimes g,\phi\otimes\psi\rangle=\langle f,\phi\rangle\langle g,\psi\rangle.$$
I understand what that means in the context of distributions, but I try to get some more intuition about the tensor product operator. Is there a more general setting (e.g. linear spaces) where we have $\otimes$, $\langle\cdot,\cdot\rangle$ and multiplication and can deduce above equality? Or does this only make sense in a very particular setting?