Hi there.


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 $L^2(M,E):=\{s\in\Gamma(E)\mid
\int\_M|\langle s,s\rangle|^2dV\_g<\infty\}$, identifying sections being equal almost everywhere. Is this true in general and if yes, do you know a book where this is worked out?


Greetings,
Robert Rauch



Completing The Space Sections in a Vectorbundle
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.


