I'm trying to understand the definition of a differential form on $M$ in the context of Fréchet spaces or, more generally, locally convex spaces. The standard procedure defines a k-form as a map $\alpha: M \rightarrow L^k_{\text{alt}}(TM)$ such that the coordinate representation $U \times E^k \rightarrow \mathbb{R}$ is smooth. Here, $L^k_{\text{alt}}(TM)$ denotes the space of k-multilinear, skew-symmetric and continuous maps on $TM$.

Another method ([1], [2]) endows the spaces $L^k_{\text{alt}}(T_mM)$ with the strong topology, constructs a vector bundle structure for $\Lambda^k M := \bigcup_m L^k_{\text{alt}}(T_mM)$ and finally defines differential forms as smooth sections of this bundle.

I have two questions about this later construction:

1) How is the smooth structure of $\Lambda^k M$ defined?

Background: In [3, Remark II.3.5 p. 18], Neeb declares that one can endow the cotangent bundle with the structure of a (topological?) vector bundle, but not with a smooth manifold structure. The smoothness of the transition functions would require that maps of the form $U \times E' \rightarrow E'$, $(x,\alpha) \rightarrow \alpha \circ d(\rho \circ \kappa^{-1})_x$ are smooth for all chart transitions. This seams only to hold for Banach manifolds (Why?). But this no-go-fact would us prevent from talking about

smoothsections of the exterior product.

2) Wurzbacher [2] says that $\Lambda^k M$ is a vector bundle because every continuous linear map $T: E \rightarrow E$ maps bounded to bounded sets. Where is this fact needed?

[1]: Kriegl, Michor: Convenient setting of global anlysis (p. 336 ff.)