# Fréchet manifold structure of C(M, N)

Let $F = C^{\infty}(M, N)$. I wish to give $F$ the structure of a Fréchet manifold. My plan was to emulate the construction of a smooth manifold. I know that for a finite dimensional smooth manifold M, $T_pM$ will be isomorphic to the model space (i.e if M is m dimensional, then $T_pM \cong \mathbb{R}^m$). Further more, I know that $T_pM$ can be identified with the equivalence classes of curves on M under the relation: For $\gamma, \, \gamma' \in C^\infty((-\epsilon,\epsilon), M)$ with \gamma(o) = p, $\quad$ $\gamma \sim \gamma'$ iff $\dot{\gamma(0)} = \dot{\gamma'(o)}.$ Define a path in $F$ to be a map $C:M\times[0,1] \rightarrow N$ with $C(x,t) = C_t(x)$ such that $C_0(x) = f, \, C_1(x) = g, \, C_{t_0}(x) \text{ is smooth for all } t_0 \in [0,1], \text{ and } C_{t}(x_0) \text{ is smooth for all } x_0 \in M.$ After a little work, ones sees that for some $f \in F, \, T_{f}F = \Gamma_f(M, TN)$ which I have proved is (set)isomorphic to $\Gamma(M, f^{*}TN)$ where $f^{*}TN$ denotes the pullback bundle.

Question(s): (i) What is a good candidate for a semi-norm (or family of semi-norms) on $\Gamma(M, f^{*}TN)\,$? (ii) Is the metric structure on $\Gamma(M, f^{*}TN)\,$ dependent upon the choice of semi-norm (or family of semi-norms)?

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Why do you want to emulate the construction of the tangent space to a manifold when defining a smooth (Fréchet-manifold) structure on $C=C^{\infty}(M,N)$? Am I missing something?
Anyways, here is a construction (from memory): we define charts for $C$ by taking charts $\Phi$ and $\Psi$ of $M$ and $N$, and defining the domain $U_{\Phi,\Psi}$ to be the set of smooth maps $f:M\to N$ such that $f(U_{\Phi})\subset U_{\Psi}$ and then sending $f\in U_{\Phi,\Psi}$ to $\Psi\circ f\circ\Phi^{-1}\in C^{\infty}(U_{\Phi},U_{\Psi})$. The latter space is already a Fréchet space. This gives a well-defined (this needs checking) Fréchet manifold structure on $C$.