Let $\pi: E \rightarrow M$ be a fiber bundle over the manifold M and denote by $\Gamma(E)$ the space of smooth sections of $E$.
For compact $M$ it is well known (Hamilton 1982, Part II Corollary 1.3.9), that $\Gamma(E)$ (if not empty) is a (tame) Fréchet manifold with respect to the topology of uniform convergence of all derivates on compacta. E.g. the topology is given by seminorms (shown here for vector bundles):
$$p_{i, K} (\phi) = \sum_{j=1}^i \text{sup}_{x \in K} |\phi^{(j)}(x)|.$$
Where the section $\phi$ is identified with its local representative $\phi: U \subset R^n \rightarrow R^m$ and the compact sets $K$ form a exhaustion of $U$. As for a paracompact manifold there exists a countable atlas, this procedure results in countable many seminorms. Thus $\Gamma(E)$ is a Fréchet manifold. (For the general case of a fiber bundle one has to invoke the tubular neighborhood theorem.)
I`m now interested in the case of non-compact base manifold $M$. To be honest, I do not see why the above construction fails then.
Supportive to this view, in section 2.2. of [1] the authors construct along the above lines a topology for non-compact $M$. But on the other hand in [2] the gauge group $\text{Gau}(P)$ (which is the group of sections of the associated bundle $P \times_G G$ to the principal bundle $P \rightarrow M$) is described only as a strict inductive limit of countable many Fréchet spaces and only for compact $M$ one has the simpler Fréchet structure on $\text{Gau}(P)$.
Where is the error here? Thanks!
[1] Čap, A. & Slovak, J. On multilinear operators commuting with Lie derivatives, eprint arXiv:dg-ga/9409005, 1994
[2] Smoothness of the action of the gauge transformation group on connections
M. C. Abbati, R. Cirelli, A. Mania, and P. Michor, J. Math. Phys. 27, 2469 (1986), DOI:10.1063/1.527404
Best Answer
Your definition depends on the choice of the exhaustion and on the choice of the metric on $E$. To get a meaningful theory you have to add many more assumptions (like: a Riemannian metric of bounded geometry on $M$ where the open sets are geodesic balls ...). For example, if you want to let the diffeomorphism group of $M$ act smoothly on the locally convex space of functions you are defining. Just keep in mind how many different function spaces on $\mathbb R^n$ are useful.
See:
MR2343536 Eichhorn, Jürgen Global analysis on open manifolds. Nova Science Publishers, Inc., New York, 2007. x+644 pp.
for a careful development of Sobolev spaces on non-compact Riemannian manifolds and vector bundles on them.
EDIT: You might check around 10.10 in (this uses a clumsy version of calculus on locally convex spaces):
Peter W. Michor: Manifolds of differentiable mappings. Shiva Mathematics Series 3, Shiva Publ., Orpington, (1980), iv+158 pp., MR 83g:58009 (scanned pdf)
Or you can check chapter IX of:
Andreas Kriegl, Peter W. Michor: The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs, Volume: 53, American Mathematical Society, Providence, 1997. (pdf)
Both references model on spaces of test functions (Choice 1 in the answer of Andrew Stacey below). There are other choices, but they are increasingly complicated. See for example the following paper which discusses the group of diffeomorphisms on $\mathcal R^n$ which fall rapidly towards the identity, or fall like $H^\infty$ (intersection of all Sobolev spaces).
Peter W. Michor and David Mumford: A zoo of diffeomorphism groups on $\mathbb ℝ^n$. arXiv:1211.5704. (pdf)