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] M. C. Abbati, R. Cirelli, A. Mania, and P. Michor, *Smoothness of the action of the gauge transformation group on connections*,
J. Math. Phys. **27**, 2469 (1986), DOI:10.1063/1.527404