https://en.wikipedia.org/wiki/Nuclear_space#Motivations_from_geometry explains that we need to take the completed tensor product for $\mathcal{C}^{\infty}(\mathbb{R})\otimes\mathcal{C}^{\infty}(\mathbb{R}) = \mathcal{C}^{\infty}(\mathbb{R}^2)$ to hold. I agree with this, but the example they give of a function not in the uncompleted tensor product is $\sin(x+y).$ Isn't $\sin(x+y) = \sin(x)\otimes\cos(y) + \cos(x)\otimes\sin(y),$ though?
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4You're right and the Wikipedia article is incorrect. It says $\sin (x + y)$ isn't a pure tensor but as you say it is a sum of two pure tensors. A correct example of a function not in the tensor product is $e^{xy}$. – Qiaochu Yuan Aug 23 '23 at 02:12
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@QiaochuYuan Thanks! – oggledog Aug 23 '23 at 15:44