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I am reading Chapter 3 (Moduli Spaces and Transversality) of "J-holomorphic curves and symplectic topology" by McDuff & Salamon. Fixing $k\in\mathbb N,p>1$ such that $kp>2$, they consider $\mathcal B^{k,p} = W^{k,p}(\Sigma,M)$. I understand the Banach manifold structure on $\mathcal B^{k,p}$ (as explained in Palais's "Foundations of Global Non-linear Analysis").

McDuff & Salamon then define a vector bundle $\pi:\mathcal E^{k-1,p}\to\mathcal B^{k,p}$ by its fibres: for $u\in\mathcal B^{k,p}$, they define $\mathcal E_u^{k-1,p} = W^{k-1,p}(\Sigma,\Lambda^{0,1}T^*\Sigma\otimes_Ju^*TM)$. They also describe trivializations of $\mathcal E^{k-1,p}$, but it is not clear to me their transition functions are smooth. Where can I find a proof of why $\pi:\mathcal E^{k-1,p}\to\mathcal B^{k,p}$ is a smooth Banach vector bundle?

Next, they define the section $\bar\partial_J$ of $\mathcal E^{k-1,p}$ by $u\mapsto \bar\partial_J(u) := \frac12(du+J\circ du\circ j)$. They claim that this is a smooth section. Where can I find a proof of this?

I think these are essential to understand the application of the implicit function and Sard-Smale theorem to construct the moduli space of $J$-holomorphic maps $\Sigma\to M$. (I would of course be grateful if somebody could tell me of a way to avoid these considerations and still construct the moduli space.)

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