What is an example of a smooth function in $C^\infty(\mathbb{R}^2)$ which is not contained in $C^\infty(\mathbb{R})\otimes C^\infty(\mathbb{R})$

Question

When looking at the tensor product of the ring of smooth functions on $\mathbb{R}^n$, there is only an injection $$ C^\infty(\mathbb{R}^n)\otimes_\mathbb{R}C^\infty(\mathbb{R}^m) \to C^\infty(\mathbb{R}^{n+m}) $$ This motivates the construction of the completed tensor product which gives an isomorphism. What is an example of a smooth function which in $$ C^\infty(\mathbb{R}^2) $$ which does not lie in the standard tensor product?

Answer 1

Assume that $H \colon \mathbb{R}^2 \rightarrow \mathbb{R}$ is a function that has the form $H(x,y) = \sum_{i=1}^k f_i(x)g_i(y)$ for some $k \in \mathbb{N}$ and some functions $f_i,g_i \colon \mathbb{R} \rightarrow \mathbb{R}$. Then for each fixed $x_0 \in \mathbb{R}$ the function $y \mapsto H(x_0,y)$ is a linear combination of the functions $g_1, \dots, g_k$ (with coefficients in $\mathbb{R}$). In particular, for any $n > k$ the functions

$$ y \mapsto H(1,y), y \mapsto H(2,y), \dots, y \mapsto H(n,y) $$

must be linearly dependent (because they all belong to $\operatorname{span} \{ g_1, \dots, g_k \}$).

So consider for example $H(x,y) = e^{xy}$ and assume that $H = \sum_{i=1}^k f_i(x)g_i(y)$ for some $k$ and $f_i,g_i$. It is readily seen that the functions

$$ e^{y}, e^{2y}, \dots, e^{ky}, e^{(k+1)y} $$

are linearly independent over $\mathbb{R}$ and we arrived a contradiction.