Topology for test functions - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-25T01:32:31Z http://mathoverflow.net/feeds/question/51477 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/51477/topology-for-test-functions Topology for test functions Bernhard 2011-01-08T12:24:24Z 2011-01-08T13:41:19Z <p>One naive way to define a topology on test functions ${\mathcal D}(\Omega)$ would be to exhaust $\Omega$ by compacts $(K_n)$ and to take the metric induced by the semi-norm system $${\| f \|} _ {n} := \| f \|_{C^n(K _ {n} )},$$ i.e. $$d(f, g) = \sum _ n 2^{-n} \frac{ \|f-g\| _ {n} }{ 1+\|f-g\| _ {n} }$$ I read (without any reference) that this yields a non-complete space. </p> <p>Do you know a reference or a concrete example how to show non-completeness?</p>