I know that for manifolds we can define smoothness just in terms of local coordinates representation. But for two Sobolev spaces, if we have a map between them, what do we mean by this mapping is smooth?
p.s. I encounter this problem when I see the following theorem.
There is a notion of smoothness for Banach (normed) spaces; there the right concept is the notion of Frechet derivative (which generalizes the usual differentiability on $\mathbb{R}^n$). With these in mind, what that passage means is that the composition map $H:L^2_k\to L^2_k$ is infinitely differentiable (in this Frechet sense).
So for instance, if we want to show first that the map is once differentiable, we need to show that for every $u\in L^2_k$ there exists a bounded linear map $D(u):L^2_k\to L^2_k$ such that $$ \lim_{\| v\|_{L^2_k}\to 0} \dfrac{\| H(u+v)-H(u)-D(u)[v]\|_{L^2_k}}{\| v\|_{L^2_k}}=0. $$