real analysis - 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 $mathfrak D (U times V)$?

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