  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.
