I have the following two linear spaces of functions: $\mathcal{A}(\mathbb{R}^n)$ contains all $f \in C^{\infty}(\mathbb{R}^{2n})$ such that for each multi-indices $\alpha, \beta$ there is a constant $C_{\alpha, \beta} > 0$ with $$|\partial_x^{\alpha} \partial_{\xi}^{\beta} f(x, \xi)| \leq C_{\alpha, \beta} (1+ |x| + |\xi|)^{-|\alpha| - |\beta|}.$$ And $\mathcal{B}(\mathbb{R}^{n+m})$ in the same way contains all functions $f \in C^{\infty}(\mathbb{R}^{2n + 2m})$ such that $$|\partial_{x}^{\alpha} \partial_{\xi}^{\beta} \partial_{y}^{\gamma} \partial_{\eta}^{\delta} f(x, \xi, y, \eta)| \leq C_{\alpha, \beta, \gamma, \delta} (1 + |x| + |\xi|)^{-|\alpha| - |\beta|} (1 + |y| + |\eta|)^{-|\gamma| - |\delta|}.$$

Here $\mathcal{A}(\mathbb{R}^n)$ is given the Fréchet topology of optimal constants, i.e. the topology generated by the seminorms $$\|f\|_{\alpha, \beta, K} := \sup_{(x, \xi) \in K}|((1 + |x| + |\xi|)^{|\alpha| + |\beta|}) \partial_x^{\alpha} \partial_{\xi}^{\beta} f(x, \xi)|$$ for each $K \subset \mathbb{R}^{2n}$ compact and multi-indices $\alpha, \beta$. Analogously for $\mathcal{B}$.

QUESTION: Is $\mathcal{A}(\mathbb{R}^n) \otimes \mathcal{A}(\mathbb{R}^m)$ dense in $\mathcal{B}(\mathbb{R}^{n+m})$?

IDEAS: I think the answer is yes as would anyone by looking at it. But I don't have a clear argument. I suspect ultimately a reduction to the density $C_c^{\infty}(\mathbb{R}^{n}) \otimes C_c^{\infty}(\mathbb{R}^m) \subset C_c^{\infty}(\mathbb{R}^{m+n})$ by a clever application of bump functions will do the trick. But if anyone has an idea or a good direct proof I'd be delighted.

EDIT: Answered question about topology.


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