Let $T(R)$ denote the space of tempered functions on the line,  

i.e. the smooth functions that give Schwartz function after a  

multiplication by any Schwartz function, equipped with the natural  

nuclear topology. e.g. the topology induced from the strong  

(convergence on bounded sets) topology on the endomorphism space of  

the space of Schwartz functions.


Is it true that tempered functions on the plane is the completed tensor square  

of tempered functions on the line, i.e. $T(R) \hat{\otimes} T(R) 
=T(R^2)$?



Schwartz Kernel theorem for tempred functions
This is proved in 4.1 of:
Michel Dubois-Violette, Andreas Kriegl, Yoshiaki Maeda, Peter W. Michor: Smooth \*-algebras. Progress of Theoretical Physics Supplement 144 (2001), 54-78. arXiv:math.QA/0106150. 
[pdf](http://www.mat.univie.ac.at/~michor/ncdg-smo.pdf)


According to L. Schwartz, there are two kind of tempered spaces, $\mathcal O\_M$, and $\mathcal O\_C$. See the paper, where your question is proved for both of them. 


