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There are several different issues happening here. The space of $C^\infty$ is not a Banach space because the topology is generated by a countable family of semi-norms not by one norm. You might try to think about why you can’t generate the topology by some magical norm. OK so why do we care about $C^\infty$ anyway. The point is applying Sard Smale. This theorem (like Sard’s Thm) requires more regularity as the formal dimension ( index) goes up so of you want one perturbation to rule all modulo spaces no matter the index you need it to be smooth. How do you marry these two issues. The idea is to find Banach spaces \emph{contained} in $C^\infty$. The trick (which I recall appears in Floer’s Instanton Homology paper first. Perhaps he mentions that he learned the idea from Taubes. What you do is pick a rapidly decreasing sequence $\epsilon_k$ and take the norm $$\sum_k \epsilon_k \|\cdot\|_{C^k}. $$ Floer The unregularized gradient flow of the symplectic action in “The unregularized gradient flow of the symplectic action” take there are plenty of such but not all of $C^\infty$ for suitable sequences. Thus there a Banach space of perturbations contained the smooth perturbations. We can now apply Sard Smale to a countable family of moduli space problems to get still a dense set of perturbations that are regular for all of these.