In the following paper by A.Abbondandolo and M. Schwarz https://arxiv.org/pdf/math/0408280.pdf in section $1.6$ we consider the set of smooth almost complex structures $\mathcal{J}$ on $T^*M$ compatible with $\omega_{can}$ and such that $||J-\hat J||_{\infty}<\infty$, where $\hat J$ is the almost complex structure give by the splitting into the verical and horizontal components.
They claim that one can put a distance on $\mathcal{J}$ defined as $\text{dist}(J_1,J_2)=||J_1-J_2||_{\infty}+dist_{C^{\infty}_{loc}}(J_1,J_2)$, so that it's a complete metric space. I was able to see that this was true. However can't we say something more general ? I.e. that this will be a banach space, if we define a norm as $||J||=||J||_{\infty}+\sum_{k=1}^{\infty}\sum_{l=0}^{\infty}2^{-r-l}\frac{||J||_{C^l(K_k)}}{1+||J||_{C^l(K_k)}}$ where $K_k=\{(t,q,p\in S^1\times T^*M:|p|\leq k)\}$. I also belive this is true but the authors don't say anyhting about this, hence this makes me feel at little suspicious.
The reason I want this to be a banach space is so that when we prove that there exists a dense set of almost complex structures that give us transversality, we need to use the regular value theorem for banach manifolds.
Edit: Nevermind this won't be a banach space since we won't have linearily. However I wonder if we can give this a structure of a banach manifold instead of a Frechet manifold.
Any insight is appreciated, thanks in advance.