Fréchet Topology on $C^\infty(M)$ In the Fréchet space wikipedia article, in the "Examples" section, it is stated that the space of smooth functions $C^\infty(M)$ on a compact smooth manifold $M$ can be made into a Fréchet space "by using as seminorms the suprema of the norms of all partial derivatives". However since of course $M$ need not admit a single coordinate chart, what is meant by "partial derivatives"? I.e. what exactly is this countable family of seminorms on $C^{\infty}(M)$? 
I'm sorry if this is super standard (it seems like it should be!), but my preliminary Google searches were quite unsuccessful. 
 A: You have a couple choices. One is to pick your favorite finite covering by charts $U_i$ and partition of unity $\rho_i$ subordinate to it, and define $\|f\|_{C^k} = \sum \|\rho_i f \|_{C^k}$, where the sum on the right hand side makes sense because we've used the charts to transport ourselves to $\Bbb R^n$. The seminorms you get by using a different covering or partition of unity will differ, but the topology they define will be the same.
More classy, in my mind, is to pick a Riemannian metric on the manifold. This gives a "connection", which is a special kind of map on sections of tensor products of cotangent bundles $\nabla: \Gamma(\otimes^k T^*M) \to \Gamma(\otimes^{k+1} T^*M)$; and these bundles have fiberwise metrics coming from the Riemannian metric. Then I can define $\|f\|_{C^k} = \|f\|_{C^0} + \|\nabla f\|_{C^0} + \dots + \|\nabla^k f\|_{C^0}$. Again, the seminorms depend on the choice of Riemannian metric, but the Frechet topology does not. (I tend to prefer this approach, as it allows for easier coordinate-free comptuations.)
Two quick related notes. 1) Suppose you want a Frechet topology on the space of smooth sections of a bundle $E$. You can play the above game, but now in addition to the covering or metric, you need (respectively) a trivialization over each chart or a connection on $E$. 2) You can use the exact same methods to define Sobolev spaces $W^{p,k}$ for all integral $k$. One advantage of the first method is that it's easier to see how to define non-integral Sobolev spaces here.
