Schwartz Kernel theorem for tempred functions - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T20:25:51Z http://mathoverflow.net/feeds/question/86334 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/86334/schwartz-kernel-theorem-for-tempred-functions Schwartz Kernel theorem for tempred functions Rami 2012-01-21T21:44:55Z 2012-10-04T15:00:23Z <p>Let $T(R)$ denote the space of tempered functions on the line,<br> i.e. the smooth functions that give Schwartz function after a<br> multiplication by any Schwartz function, equipped with the natural<br> nuclear topology. e.g. the topology induced from the strong<br> (convergence on bounded sets) topology on the endomorphism space of<br> the space of Schwartz functions.</p> <p>Is it true that tempered functions on the plane is the completed tensor square<br> of tempered functions on the line, i.e. $T(R) \hat{\otimes} T(R)<br> =T(R^2)$?</p> http://mathoverflow.net/questions/86334/schwartz-kernel-theorem-for-tempred-functions/108826#108826 Answer by Peter Michor for Schwartz Kernel theorem for tempred functions Peter Michor 2012-10-04T15:00:23Z 2012-10-04T15:00:23Z <p>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. <a href="http://www.mat.univie.ac.at/~michor/ncdg-smo.pdf" rel="nofollow">pdf</a></p> <p>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. </p>