Aspects of differential calculus related to infinite-dimensional vector bundles and Poisson vector spaces

We prove various results in infinite-dimensional differential calculus which relate differentiability properties of functions and associated operator-valued functions (e.g., differentials). The results are applied in two areas: 1. in the theory of infinite-dimensional vector bundles, to construct new bundles from given ones, like dual bundles, topological tensor products, infinite direct sums, and completions (under suitable hypotheses). 2. in the theory of locally convex Poisson vector spaces, to prove continuity of the Poisson bracket and continuity of passage from a function to the associated Hamiltonian vector field. Topological properties of topological vector spaces are essential for the studies, which allow hypocontinuity of bilinear mappings to be exploited. Notably, we encounter $k_{{\mathbb R}}$-spaces and locally convex spaces $E$ such that $E\times E$ is a $k_{{\mathbb R}}$-space.