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. $T(R) \hat{\otimes} T(R)
=T(R^2)$?