MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question
up vote 4 down vote accepted

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

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.

share|cite|improve this answer
Thank you very much – Rami Oct 18 '12 at 22:13

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.