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 $\gamma(o) = p$, $\gamma \sim \gamma'$ iff $\dot{\gamma(0)} = \dot{\gamma'(0)}.$ 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 $C_0(x) = f$, $C_1(x) = g$, $C_{t_0}(x)$ is smooth for all $t_0 \in [0,1]$, and $C_{t}(x_0)$ is smooth for all $x_0 \in M.$ After a little work, ones sees that for $f \in F$, $T_{f}F = \Gamma_f(M, TN)$ 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)? (iii) Does there exist any literature in which an explicit construction of even a trivial example of a Fréchet manifold can be found? (Hamilton's,"The inverse function theorem of Nash and Moser", I feel, comes closest to an example.)
(I should mention also that this is a copy of a question on the math stackexchange website. http://math.stackexchange.com/questions/202048/frechet-manifold-structure-of-cm-n)
