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