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



Cartesian product of test function spaces
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"](http://books.google.co.uk/books?id=9O-pXIVUEFMC&printsec=frontcover&dq=Introduction+to+the+Theory+of+Distributions&hl=en&ei=vmhhTIimAZS7jAe9-OyvCQ&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCwQ6AEwAA#v=onepage&q&f=false) by Friedlander and Joshi (или задачи 423 и 430 в книге Кириллов А.А., Гвишиани А.Д. *"Теоремы и задачи функционального анализа"*).


