# Cartesian product of test function spaces

## 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 $\mathfrak D (U \times V)$?

-