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