If $M$ is a smooth manifold and assume it is compact if necessary, then can I define a space $W^{k,p}(M,E)$ (for some vector bundle $E$) consisting of $W^{k,p}$-sections (rather than common $C^l$-sections) from $M$ to the total space $E$ ?
How to make this definition clear? and are they Banach spaces or Banach manifolds?
What if we replace the vector bundle $E$ by another smooth manifold?