Currently, I am studying the space of sections $\Gamma(E)$ of a vector bundle $(E,M,\pi)$. If $h$ is a riemannian bundle metric over $E$, and $\mu$ is a volume form in $M$, we can define the following inner-product in $\Gamma(E)$: $$\langle s,s'\rangle=\int_Mh(s,s')\, \mu.$$ This inner product turns $\Gamma(E)$ into a pre-Hilbert space that is not complete: there exists Cauchy sequences that do not converge.
From my (very limited) knowledge of Hilbert spaces, we can consider the complection of $\Gamma(E)$ respect to the distance function determined by $\langle\cdot,\cdot\rangle$. If so, we obtain a Hilbert space that contains $\Gamma(E)$ as a dense subset.
However, in most Differential Geometry books I always find that $\Gamma(E)$ is completed respect to another inner-product, producing a new Hilbert space called Sobolev space.
I don't understand why the autors always consider the Sobolev complection instead of the (more natural) complection that we have presented.
Thanks in advance.