Cartesian product of test function spaces

Question

Mini introduction

Suppose $U \subset \mathbb R^n, V \subset \mathbb R^m$ are two open sets. If we take http://en.wikipedia.org/wiki/Distributions_space#Test_function_space">test functions $f_i \in \mathfrak D (U),~g_i \in \mathfrak D (V)$ for $1 \leq i \leq n$, then $f_1(x)g_1(y) + \dots + f_n(x)g_n(y)$ is an element of $\mathfrak D (U \times V)$, so we have an inclusion: $$\operatorname{span}\left(\mathfrak D (U) \times \mathfrak D (U) \right) \subset \mathfrak D (U \times V)$$ where "span" means linear span.

Question

Is it true that $$\overline{\operatorname{span}\left(\mathfrak D (U) \times \mathfrak D (U) \right)} = \mathfrak D (U \times V)$$ where line means the closure in topology of $\mathfrak D (U \times V)$?

Answer 1

This is true.

By a partition of unity, the proof can be reduced to the case when the test functions have their supports in a unit cube and the result follows from a more or or less straightforward manipultation with the corresponding Fourier series.

See, for instance, Theorem 4.3.1 in "Introduction to the Theory of Distributions" by Friedlander and Joshi (или задачи 423 и 430 в книге Кириллов А.А., Гвишиани А.Д. "Теоремы и задачи функционального анализа").