The Hilbert Space tensor product gives
$L^2(\mathbb R^2,dx\otimes dx;\mathbb R)= L^2(\mathbb R,d x;\mathbb R) \otimes L^2(\mathbb R,dx;\mathbb R)$
My question is: does there exist also a notion of tensor product which gives $C^\infty(\mathbb R^2;\mathbb R)= C^\infty(\mathbb R;\mathbb R) \otimes C^\infty(\mathbb R;\mathbb R)$?