Does direct limit commute with functor of smooth sections? Consider a countable family of finite-rank vector bundles $V_k$ over a finite-dimensional smooth manifold $M$. The direct limit of such a family is still a topological vector bundle even though it may have infinite rank. 
The first question is: Is there a natural structure of smooth manifold on the total space of $\varinjlim V_k$? 
And the second one: Is it true, that the functor of smooth sections commutes with direct limit? I.e, is it true that
$$\mathcal{C}^\infty(M,\varinjlim V_k) = \varinjlim \mathcal{C}^\infty(M,V_k)$$
Or turning things upside down -- is there a choice of smooth structure such that the above equation holds? If so, how does it look like?
 A: The answers are: Yes and No-but-yes-if-M-is-compact.


*

*Kriegl and Michor's A Convenient Setting for Global Analysis describes how to put a smooth structure on an arbitrary locally convex topological vector space, say $V$, by looking first at the smooth curves in $V$ (these can be unambiguously defined).  This works for $V = \lim V_k$ where the $V_k$ are finite dimensional.  As your family is countable, this is just $\sum_{\mathbb{N}} \mathbb{R}$.  Smooth curves are continuous, and therefore an important property of this structure is that if $c \colon \mathbb{R} \to \sum_{\mathbb{N}} \mathbb{R}$ is smooth then $c([a,b])$ is contained in a finite dimensional subspace.

*And that's really the key to the second answer.  If $M$ is compact, then its image lies in a finite dimensional subspace of $\lim V_k$, whence inside one of the $V_k$.  If $M$ is not compact then it will be possible to find a smooth map $M \to \lim_k V_k$ which does not lie in any subspace and so the limit will not commute with the mapping space construction.  Note that it is enough to show that such a map exists for $M = \mathbb{R}$, whereupon you take curves $\alpha_k \colon [0,1] \to V_k$ which map to $0$ on their endpoints, are infinitely slow there, and such that the image of $\alpha_k$ contains a basis for $V_k$.  Then concatenate these paths together to get a smooth path $\alpha \colon \mathbb{R} \to \lim V_k$ with image not contained in any finite dimensional subspace.
