What is the usual topology of $C^\infty_c(M)$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T11:48:55Z http://mathoverflow.net/feeds/question/95372 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/95372/what-is-the-usual-topology-of-c-infty-cm What is the usual topology of $C^\infty_c(M)$ Adterram 2012-04-27T16:18:29Z 2012-04-27T17:28:23Z <p>If $M$ is a smooth paracompact manifold, then what is the usual topology of $C^\infty_c(M)$, i.e., the smooth function with compact support? </p> http://mathoverflow.net/questions/95372/what-is-the-usual-topology-of-c-infty-cm/95376#95376 Answer by Mark Grant for What is the usual topology of $C^\infty_c(M)$ Mark Grant 2012-04-27T16:38:39Z 2012-04-27T16:38:39Z <p>Give $C^\infty(M)$ the <a href="http://en.wikipedia.org/wiki/Whitney_topologies" rel="nofollow">Whitney topology</a>, then topologise $C^\infty_c(M)\subseteq C^\infty(M)$ as a subspace.</p> http://mathoverflow.net/questions/95372/what-is-the-usual-topology-of-c-infty-cm/95381#95381 Answer by unknown (google) for What is the usual topology of $C^\infty_c(M)$ unknown (google) 2012-04-27T17:28:23Z 2012-04-27T17:28:23Z <p>Topologizing $C_c^\infty(M)\subseteq C^\infty(M)$ with the subspace topology (where $C^\infty(M)$ has the Whitney topology, generated by the seminorms $\left|\sup_K\frac\partial{\partial x^\alpha}f\right|$), makes it a dense subspace; in particular it is not itself complete. So I wouldn't really call this the "usual topology" on $C_c^\infty(M)$. (it would be sort of like saying the usual topology on $C(M)$ is given by the $L^2$ norm).</p> <p>To me the usual topology is the inductive limit topology $C_c^\infty(M)=\lim_{K\subseteq M}C_c^\infty(K)$ (which Mariano calls the colimit topology). This topology is not metrizable when $M$ is noncompact (since it's not even first-countable), but is "nicer" in the sense that it gives a well-understood dual space, namely the space of distributions on $M$.</p> <p>In comparison, the dual space of $C^\infty(M)$ with the Whitney topology is the space of compactly supported distributions on $M$.</p>