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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 $C^\infty(\mathbb R^2;\mathbb R)= C^\infty(\mathbb R;\mathbb R) \otimes C^\infty(\mathbb R;\mathbb R)$?

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    For smooth functions with compact support, it is discussed here http://math.stackexchange.com/questions/63416/tensor-products-of-functions-generate-dense-subspace.2012-05-08

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