What sorts of books would address $L^p$ spaces of sections of the tangent space of a Riemannian manifold ($C^\infty$, compact, ideally with boundary)?
In particular, I'd like to show that smooth sections are dense in the set of $L^2$-measurable or $L^\infty$ sections, in the $L^2$ norm or $L^\infty$ norm.
I could do this using local orthonormal frames and a partition of unity argument, but I'd prefer a reference which does this directly.