On page 170 of the book "Spin geometry" by Michelson and Lawson, given a compact Riemannian manifold $(M,g)$, a Hermitian vector bundle $E$ over $M$ and a connection $\nabla$ on the bundle, the author defined a norm on the vector space of smooth sections $\Gamma(E)$ given by
$$||f||_k^2=\int_M\sum_{i=0}^k||\nabla...\nabla f||^2$$ here there are $i$ number of $\nabla$'s here.
The authors then claim that is an easy exercise that the norm defined using different Riemannian metric and connections are all equivalent.
However I'm stuck on why the norm is independent of the connection here, because when you change the connection, the function that is being integrated over also changes, and there is no obvious way to compare the two different integrals. I'm aware of the question: Sobolev space on a compact Riemannian manifold does not depend of the metric and Sobolev spaces on manifolds. but I don't really understand the logic provided in the answer of the first link and the second link doesn't have an answer.
EDIT: all I think I see why the norm is independent (I meant equivalence) of the connection now.
