Fréchet manifold

Definition

Let $F$ and $G$ be real Fréchet spaces, and let $U \subseteq F$ be open. A map $f \colon U \to G$ is of class $C^1$ if it is continuous, the Gateaux derivative $Df(x) h = \lim_{t \to 0} (f(x + t h) - f(x)) / t$ exists for all $x \in U$ and $h \in F$, and $Df \colon U \times F \to G$ is continuous. It is of class $C^{k+1}$ if it is $C^1$ and $Df$ is $C^k$, and smooth if it is $C^k$ for all $k$.

Let $M$ be a Hausdorff space. A smooth Fréchet atlas on $M$ is a family of charts $(U_i, \varphi_i)_{i \in I}$ such that:

1. Each $U_i$ is open in $M$ and the sets $U_i$ cover $M$.
2. Each $\varphi_i$ is a homeomorphism from $U_i$ onto an open subset of a Fréchet space.
3. Each transition map $\varphi_j \circ \varphi_i^{-1} \colon \varphi_i(U_i \cap U_j) \to \varphi_j(U_i \cap U_j)$ is smooth.

A Fréchet manifold is a space $M$ with an equivalence class of smooth Fréchet atlases, two atlases being equivalent when their union is an atlas.