Completing The Space Sections in a Vectorbundle - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T19:49:22Z http://mathoverflow.net/feeds/question/82213 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/82213/completing-the-space-sections-in-a-vectorbundle Completing The Space Sections in a Vectorbundle Robert Rauch 2011-11-29T20:17:42Z 2011-11-29T21:09:44Z <p>Hi there.</p> <p>Assume $(M,g)$ is a Riemanian manifold and $E\to M$ is a vector bundle with a bundle metric $\langle\cdot,\cdot\rangle$. We then have the pre-Hilbert space $H_0:=\Gamma_c^\infty(E)$ of compactly supported smooth sections with $(s_1,s_2):=\int_M\langle s_1,s_2\rangle dV_g$ . In a paper i'm currently working on it says that the completion $H$ of $H_0$ is (in a rather concrete Situation for $E$) given by the space of square integrable sections $L^2(M,E):=\{s\in\Gamma(E)\mid \int_M|\langle s,s\rangle|^2dV_g&lt;\infty\}$, identifying sections being equal almost everywhere. Is this true in general and if yes, do you know a book where this is worked out?</p> <p>Greetings, Robert Rauch</p> http://mathoverflow.net/questions/82213/completing-the-space-sections-in-a-vectorbundle/82218#82218 Answer by MTS for Completing The Space Sections in a Vectorbundle MTS 2011-11-29T21:09:44Z 2011-11-29T21:09:44Z <p>This is true in general. I don't know a reference for the statement, but it is pretty simple just to work it out. The point is that $L^2(M,E)$ is a Hilbert space which contains $H_0$ as a dense linear subspace, so it must be the completion.</p>