Is $C^{\infty}\left(M,N\right)$ a Frechet space? Let $M$ and $N$ be compact, smooth manifolds. Can we equip a topology
on 
$$C^{\infty}\left(M,N\right):=\left\{ f:M\rightarrow N\;\textrm{smooth}\right\} 
$$
such that $C^{\infty}\left(M,N\right)$ is a Frechet space?
Recall that: A Frechet space is a topological vector space that is locally convex, metrizable, and complete.
 A: If the target $N$ is not $\mathbb{R}^n$, you already fail to give $C^\infty(M,N)$ a vector space structure that is of geometrical significance. 
On the other hand $C^\infty(M,\mathbb{R}^n)$ caries a natural Fréchet space structure. Its topology is for example generated by seminorms of the form $p_k(f) = \sup_{p\in M}\vert \nabla^k f(p)\vert_g$ (for $\nabla$ the Levi-Civita-connection w.r.t. to an arbitrary metric $g$ on $M$). (Provided $M$ is compact, otherwise you can only give it the structure of an LF-space, just as it is the case with Distributions on $\mathbb{R}^m$.)
If $N$ is an arbitrary closed manifold, then it admits a proper embedding into some $\mathbb{R}^n$ and you can identify $C^\infty(M,N)$ with the subset of all functions $f\in C^\infty(M,\mathbb{R}^n)$ whose images lie in the embedding of $N$. Then  $C^\infty(M,N) \subset C^\infty(M,\mathbb{R}^n)$ is closed Fréchet submanifold.
For the construction of charts on $C^\infty(M,N)$ consider https://ncatlab.org/nlab/show/manifold+structure+of+mapping+spaces.
