Here is a way to define Banach manifolds given in Margalef-Differential topology
Definition 1
Let $E$ be a real Banach space and $\Lambda:=\{\lambda_1,...,\lambda_n\}$(possibly an empty set) be linearly independent continuous linear functionals on $E$. Define $E_\Lambda^+:=\{x\in E: \forall 1\leq i \leq n, \lambda_i(x)\geq 0\}$. We call this the $\Lambda$-quadrant of $E$.
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Definition 2
Let $X$ be a set and $E$ be a real Banach space and $\Lambda:=\{\lambda_1,...,\lambda_n\}$ be linearly independent continuous linear functionals on $E$. Let $U\subset X$ and $\phi:U\rightarrow E_\Lambda^+$ be an injection such that $\phi(U)$ is open in $E_\Lambda^+$. We call $(U,\phi,(E,\Lambda))$ a chart on $X$.
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Definition 3
Let $X$ be a set and $(U,\phi,(E,\Lambda)),(V,\psi,(F,\Gamma))$ be charts on $X$. Then they are $C^k$-compactible iff $\phi(U\cap V),\psi(U\cap V)$ are open and $\psi\circ \phi^{-1}:\phi(U\cap V)\rightarrow \psi(U\cap V)$, $\phi\circ \psi^{-1}:\psi(U\cap V)\rightarrow \phi(U\cap V)$ are of $C^k$
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Definition 4
Let $X$ be a set and $\mathscr{A}$ be a collection of charts. Then, $\mathscr{A}$ is a $C^k$-atlas iff the domains cover $X$ and any two of them are $C^k$-compatible
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Definition 5
Let $X$ be a set and $\mathscr{A},\mathscr{A}'$ be $C^k$-atlases. Then, they are equivalent iff there is a $C^k$-atlas containing both of them.
The author defines $C^k$ differentiable structures as the equivalence classes of atlases. However, is it a set and well-defined?
Alternatively, let $X$ be a set and $\mathscr{A}$ be a $C^k$-atlas on $X$. Then, does there exist a maximal atlas with respect to the $C^k$-compatible relation?
Unlike the union of Euclidean spaces form a set (i.e. $\bigcup_{n\in\mathbb{N}} \mathbb{R}^n$), the collection of all Banach spaces form a class. Hence, the usual argument for this problem on euclidean differentiable manifolds cannot be applied here.
How do we define a differentiable structure?
